.0 Introduction
In this project, it was decided to use home-made air launcher propelled by air to observe projectile motion. The air launcher is made from plastic water bottle and the air is released in an instant as soon as the air launcher is blasted off. The air launcher is in a projectile motion as soon as it is blasted off as the only force acting on it is gravity. Data such as flight time, range, highest point and others will be collected to determine the constant of gravity acceleration.
Three different materials: rubber, plastic and aluminium, have been used as specified in the requirement of this project. The mass of each material is different but the surface area that is in contact with the air is constant. As the mass of each water bottle is different, the launching pressure for each water bottle is different. Pressure is directly proportional to the launching velocity. To achieve a constant launching velocity, the pressure used has to be varied accordingly.
In the experiment, the behaviour of the projectile motion is different from the theoretical prediction. The water bottle will travel less horizontally or vertically. The effect of air resistance will slow down the bottle while it is on its course up. The direction of air resistance is acting in the same direction of the gravity, thus the water bottle is brought slower at a faster rate. On its way down, the air resistance is acting in the opposite direction of the gravity, thus the water bottle is not slowed down as fast as it should be. This allows the water bottle to gain more travelling time, but the total flight time will still be less than the theoretical value in the absence of air resistance. As the air resistance will always be acting in the direction of the motion of water bottle, the horizontal velocity will be decreased at all times during the course of the flight. Therefore, the range the water bottle covered will be less than the theoretical value. In addition to air resistance, there is a presence of wind in reality. As no sophisticated equipment that can track the direction and velocity of the wind can be obtained, it is difficult to calculate the effects of wind on the course of the projectile motion. Therefore, the result of this experiment will be deviated from its theoretical prediction.
2.0 Objective
* To determine the launching velocity and angle of the projectile.
* To describe the profiles of motions for the projectile.
* To determine the effect of varying launching angle on range.
* To vary the launch height and observe the variation of the maximum range achieved.
* To compare the theoretical and practical value of launching velocity of the projectile.
* To vary the material of the projectile and observe the effects of the air resistance on the projectiles.
* To discuss in detail some possible application of the projectile launcher that has been designed and built.
* To determine constants quantitatively.
3.0 Literature Review
3.1 History of Projectile
Figure 3.1.1: General opinion on projectile before Galileo (http://library.thinkquest.org/2779/History.html)
This illustration reflects the general opinion before Galileo which followed largely Aristotelian lines but incorporating a later theory of "impetus" -- which maintained that an object shot from a cannon, for example, followed a straight line until it "lost its impetus," at which point it fell abruptly to the ground.
Figure 3.1.2: Illustration of a work by Niccolo Tartaglia on projectile (http://library.thinkquest.org/2779/History.html)
Later, simply by more careful observation, as this illustration from a work by Niccolo Tartaglia shows, it was realized that projectiles actually follow a curved path. Yet no one knew what that path was, until Galileo. There was yet another brilliant insight that led Galileo to his most astounding conclusion about projectile motion. First of all, he reasoned that a projectile is not only influenced by one motion, but by two. The motion that acts vertically is the force of gravity, and this pulls an object towards the earth at 9.8 meters per second. But while gravity is pulling the object down, the projectile is also moving forward, horizontally at the same time. And this horizontal motion is uniform and constant according to Galileo's principle of inertia. He was indeed able to show that a projectile is controlled by two independent motions, and these work together to create a precise mathematical curve. He actually found that the curve has an exact mathematical shape. A shape that the Greeks had already studied and called the parabola. The conclusion that Galileo reached was that the path of any projectile is a parabola.
Photo 3.1.1: Replica catapult at Château des Baux, France
(http://en.wikipedia.org/wiki/Catapult)
A catapult is any siege engine which uses an arm to hurl a projectile a great distance, though the term is generally understood to mean medieval siege weapons. The name is derived from the Greek ???? (against) and ß?????? (to hurl (a missile)). (An alternate derivation is from the Greek "katapeltes" meaning "shield piercer," kata (pierce) and pelta (small shield)). Originally, "catapult" referred to a dart-thrower, while "ballista" referred to a stone-thrower, but the two terms swapped meaning sometime in the fourth century AD.
Catapults were usually assembled at the site of a siege. And were made out of wood. An army would carry a few necessary pieces with them because wood was easily available. Although usually incorrectly depicted with a spoon on the end of the arm (as in the picture to the right) catapults were most often equipped with a sling to hold the projectile.
During medieval times, catapults and related siege machines were the first weapons used for biological warfare. The carcasses of diseased animals or even diseased humans, usually those who had perished from the Black Death, were loaded onto the catapult and then thrown over the castle's walls to infect those barricaded inside. There have even been recorded instances of beehives being catapulted over castle walls.
The last large-scale military use of catapults was during the trench warfare of World War I. During the early stages of the war, catapults were used to throw hand grenades across no man's land into enemy trenches.
Until recently, in England, catapults were used by thrill-seekers as human catapults to experience being catapulted through the air. The practice has been discontinued due to fatalities, when the participants failed to land onto the safety net.
Figure 3.1.3: Trebuchet catapult at rest
(http://www.redstoneprojects.com/trebuchetstore/how_a_trebuchet_catapult_works.html)
A trebuchet consists of five basic parts: the frame, counterweight, beam, sling and guide chute. The frame supports the other components and provides a raised platform from which to drop the counterweight. The counterweight, pulled by gravity alone, rotates the beam. The beam pulls the sling. The guide chute guides the sling through the frame and supports the enclosed projectile until acceleration is sufficient to hold it in the sling. The sling accelerates and holds the projectile until release.
How the Beam Accelerates the Projectile
Figure 3.1.4: Trebuchet catapult's sling holding the projectile before release
(http://www.redstoneprojects.com/trebuchetstore/how_a_trebuchet_catapult_works.html)
One end of the sling is fixed to the end of the beam, while the other is tied in a loop and slipped over a release pin extending from the end of the beam. As the beam rotates, it pulls the sling, with its enclosed projectile, down the guide chute. As the sling exits the chute, it accelerates in an arc away from the beam, but because the beam is still pulling the sling behind, the loop is held on the pin.
How the Sling Releases the Projectile
Figure 3.1.5: Trebuchet catapult preparing to launch
(http://www.redstoneprojects.com/trebuchetstore/how_a_trebuchet_catapult_works.html)
The sling continues accelerating through its arc until it eventually swings ahead of the release pin. At this point, known as the release angle, the loop slips off the pin and the sling opens, releasing the projectile.
Figure 3.1.6: Trebuchet catapult at release phase (http://www.redstoneprojects.com/trebuchetstore/how_a_trebuchet_catapult_works.html)
3.2 Introduction to Projectile
A projectile is an object which the only force acting upon it is gravity. There is a variety of examples of projectiles: an object dropped from rest is a projectile (provided that the influence of air resistance is negligible); an object which is thrown vertically upward is also a projectile (provided that the influence of air resistance is negligible); and an object is which thrown upward at an angle is also a projectile (provided that the influence of air resistance is negligible). A projectile is any object which once projected continues in motion by its own inertia and is influenced only by the downward force of gravity.
Figure 3.2.1: An illustration of projectile motion
By definition, a projectile has only one force acting upon it - the force of gravity. If there were any other forces acting upon an object, then that object would not be a projectile. Thus, the free-body diagram of a projectile will show a single force acting downwards and this is labelled "force of gravity" (or simply Fgrav). This is to say that regardless of whether a projectile is moving downwards, upwards, upwards and rightwards, or downwards and leftwards, the free-body diagram of the projectile is still the same.
Figure 3.2.2: Free body diagram of a projectile
In this experiment, air launcher is used to explore the effects of projectile motion. Air launching involves the flight of air launchers using compressed air to provide the needed force. This method for propelling air launcher is chosen as it is interesting to discover the flight of an air launcher while it can still be considered affordable by students as the items used in the construction of these compressed air launchers are inexpensive. This translates into lots of science and math activities for little cost. Moreover, these projects are tons of fun.
Before launching the air launchers, determining the launching pressure is very important as it represents the magnitude of energy the air launcher is going to receive. Launch pressures for low-pressure projectiles should not exceed 80 psi. 40 to 60 psi normally provides enough pressure to display the distinct features of a projectile flight. If excessive pressure is used during launch, it is likely that the air launchers will be unable to withstand the launch force and might be blasted off into the air although the locking mechanism is still engaged or just tore into pieces. For this reason, it is important to keep everyone behind the launcher during flight operations. Besides that, when firing air launchers in a confined area, the pressure needs to be limited to keep the air launchers within the assigned space.
In addition, because tremendous forces and velocities are generated during the flights of these air launchers, caution should be exercised to prevent injury and damage to property. It is important to take note that one should never aim these air launchers in a direction that may possibly cause injuries or damages as once the pressure is released there is no possible way to stop the flights of these projectiles. These compressed air launchers fly as a result of the force released during lift off. As they are merely projectiles, they do not continue to produce thrust during the course of the flight unlike real air launchers where by fuel will be burned to provide thrust. In a real air launcher, should any accident happen, the flight direction of the air launcher can be changed by firing retro thrusters to change its direction to manoeuvre it efficiently through tight spots. In the case of an air launcher, the air launcher does not have this function and this in turn makes it extremely dangerous if the air launcher is driven off course as the landing speed can be very high (Rodriguez C 2000). Other examples of projectile motion are BB gun ammunition, pellets, bullets, slingshot projectiles, artillery shells, etc and they all fly in a similar fashion.
An inclinometer is used to measure the angle formed between the launcher tube and the launcher pad in order to vary the angle (Rodriguez C 2000). To construct an inexpensive inclinometer, a segment of string is inserted through the hole in the centre of the ruler portion of a common, six-inch protractor. A knot is tied in one end of the string and a suitable weight is attached to the other end of the string. The ends of a rubber band are threaded through both holes of the protractor so that a small loop protrudes from the holes. Once the rubber band is in place, a milkshake size straw is slipped through each loop of the rubber band. The inclinometer can provide an accurate measurement of angle compared to using the protractor alone as the thread can be fixed more easily when compared to the big protractor.
Figure 3.2.3: A Simple Inclinometer (http://www.henrytudor.co.uk/page7.htm)
Photo 3.2.1: A Modern Inclinometer (http://www.spadout.com/wiki/index.php/Clinometer)
Figure 3.2.4: How to use an Inclinometer (http://arb.nzcer.org.nz/nzcer3/MATHS/GEOMETRY/4100-199/Gm4164.htm)
Photo 3.2.2: A photo showing a simple inclinometer in use. (http://www.rondexter.com/professional/sun/home-made_clinometer.htm)
3.3 Newton's Law Associated with the Air Launchers
In terms of the laws of motion, Newton's First Law states that a body at rest remains at rest, or a body in motion continues its motion in a straight line until acted on by an external force. When the air launcher is poised for lift off on the launcher, it is in a state of rest. It remains so until the pneumatic charge acts upon it or some other force disturbs it. When the air launcher is in flight, it basically travels in a straight line. However, external forces such as gravity, drag, wind, and other aerodynamic and gyroscopic forces act on the air launcher during the entire duration of its flight. This causes the trajectory of the air launcher to change into a parabolic-like curve.
Newton's Second Law is regarding the acceleration. The acceleration of a given body is proportional to the force acting on it. When the pneumatic charge is released, the force of the blast acts on the projectile, causing it to accelerate. For all other conditions, the greater the force, the greater the acceleration. On the other hand, other conditions do not remain the same. The quantity of drag changes with the size of the air launcher and its velocity. In addition, the relationship between velocity and drag is not linear. As velocity increases, drag increases at an exponential rate. This can be shown through the experiment where the data obtained from the experiment is different from the theoretical calculation (Rodriguez C 2000).
Newton's Third Law, for every action there is an equal magnitude of reaction in the opposite direction, is demonstrated during lift off. As the air pressure is released, the mass of pressurized air contained in the launcher rapidly escapes the chamber where it was housed. As the air launcher is placed in the path of the rushing mass of air, it is given a propelling moment that results in flight. The reaction to this rapid discharge of air is absorbed by the launcher. However, the reactive force applied during lift off is insufficient to overcome the weight and fiction presented by the launcher because the launcher is relatively heavy and resting on a solid surface (e.g., the ground). If it were mounted so that it was weightless and frictionless, the launcher would have a visible reaction when releasing the air charge. It would travel in a direction opposite that of the air launcher. A similar reaction is obtained when firing a rifle, it "kicks" in a direction opposite that of the bullet.
Photo 3.3.1: A rocket is being launch by utilising Newton's Third Law of Motion
(http://www.qrg.northwestern.edu/projects/vss/docs/Navigation/1-how-put-into-orbit.html)
In theory, without an appropriate measuring device, to measure the height of an air launcher launched in a vertical trajectory, another mathematical approach can be used. A right triangle is formed between the tracking station or viewing area, the launch pad, and the apogee of the air launcher. The apogee of the air launcher occurs when it reaches its maximum altitude. In reality, there may be some error in this method as the air launcher may not travel in a perfect vertical path above the launching pad. Any error, however, does not detract from the mathematical and science benefits provided by this experiment. When only one tracker is used to measure the height of the air launcher, select a location slightly downwind of the launch area. For best results, situate the spotter so that the wind is blowing directly on the person's left or right shoulder, as appropriate. To minimize error for the purpose of this experiment, the measurements taken should be conducted in still air (Rodriguez C 2000).
3.4 Theory
Part 1
Firstly, to determine the constant of gravity acceleration, an experiment using pendulum is conducted. A simple pendulum is consisting of a mass, m, suspended by a light string or rod of length L, as shown in figure below. The pendulum has a stable equilibrium when the mass is directly below the suspension point. It will oscillate about this position if the pendulum is displaced to any other point.
Figure 3.1: The Simple Pendulum
To understand the behaviour of the pendulum, consider the forces acting on the mass m. In Figure 3.1, the forces which act on the mass are the force of gravity, , and the tension force in the supporting string, T. The net tangential force acting on m, which is the tangential component of its weight:
F = mg sin? ................ (1)
For small angles, ? (measured in radians) the sine of ? is approximately equal to the angle itself. That is, sin ? .
The arc length displacement of the mass is s = L?. Therefore, ? = s / L. Equation (1) can be written as:
............. (2)
Compare the pendulum to a mass on a spring. The restoring force of a spring is
F = kx. The restoring force acting on the pendulum has precisely the same form, if let
x = s and .
Therefore, the period of a pendulum is simply the period of a mass on a spring,
, with k replaced by mg/L:
Cancelling the mass, m, finally period of a pendulum (small amplitude) is found.
Period of a pendulum (small amplitude)
From here, the gravitational constant can be determined by rearranging the equation,
Part 2
After the gravitational constant is determined, the project moved on to the second stage which is to observe the effect of various factor such as launching angle, height and velocity to the trajectory of the projectile. A projectile is the motion of an object projected near to the Earth's surface at a certain angle. Projectile motion is applied in this project where the rocket undergoes a pure projectile motion. From the projectile, the time of flight, the maximum range, the maximum height and angle of projection can be measured by different means. The initial velocity of the motion of the rocket at every launched is fixed to aid in determining the value of the experimental value for initial velocity which will be determined from the calculation of range later on. The following shows the derivation of the formula used to calculate the practical value of initial velocity:
The graph t versus sin? is used to obtain the value of t / sin? by using the gradient of the graph plotted. The value is then replaced into the formula to obtain the u. Then, the theoretical and practical value of u is compared.
The angles used in the project are complementary angles. The angles tested in the projects are 0?, 15?, 30?, 45?, 60?, 75? and 90?. The following proves that complementary angles are identical:
The Complementary Angle
Hence, all angles are symmetrical about 45? such that 15? and 75? are complementary angles. 30? and 60? obey the formula as well. The ranges of the complementary angle launches are compared in the discussion.
4.0 Methodology
Determining the gravitational acceleration:
Apparatus:
Metal sphere (as pendulum bob), vernier callipers, string of length approximately 1.5m, retort stand with clamp, photo-gate circuit and timer.
Procedures:
. The measurements of the diameter of the metal sphere was taken for several times using vernier calliper. The average value for this quantity was calculated and recorded.
2. The string was clamped so that the length of the string () and the radius () of the pendulum bob which is attached to the string is 0.5 m below the point of suspension. The total length of the pendulum system, will be the length of the string plus the radius of metal sphere ().
3. The position of the metal bob was arranged to be at the same level as the photo-gate timer. The metal bob was made sure to move through the sensor of the photo-gate timer, not the string of the pendulum.
4. The sphere was displaced to one side through an angle not more than 5 and the pendulum was let to be oscillated. The time given by the photo-gate timer was recorded. This was the time taken to make half an oscillation by the pendulum. The step was repeated several times to obtain an average value for .
5. Step 2, 3 and 4 was repeated. This time, the length of the pendulum, was changed into 0.6 m, 0.7 m, 0.8 m, 0.9 m, 1.0 m, 1.1 m and 1.2 m. Data was recorded in Table 3.1.
6. Graph against was plotted from the data obtained from Table 3.1. The value of the gradient of the straight line plotted was found, which is given by and hence the value of was compare.
Determining the initial velocity, u, of the launch:
The reason of choosing an air launcher:
) The air launcher was chosen in the project as it meets the requirements of the project.
2) Firstly, the air from the air launcher is assumed to be let out instantaneously upon launching. Therefore, there is no fuel which will provide an external force and is a pure projectile motion.
3) Besides, the pressure of the air launcher can be easily standardized and adjusted according to the needs of the project, thus giving a more reliable data.
Things needed to build launcher:
Use of all equipments:
* The air launcher
- Provides a base to the air launcher for it to be projected. It is ...
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2) Firstly, the air from the air launcher is assumed to be let out instantaneously upon launching. Therefore, there is no fuel which will provide an external force and is a pure projectile motion.
3) Besides, the pressure of the air launcher can be easily standardized and adjusted according to the needs of the project, thus giving a more reliable data.
Things needed to build launcher:
Use of all equipments:
* The air launcher
- Provides a base to the air launcher for it to be projected. It is also the place where the air launcher is pressurized.
* Inclinometre
- To calculate the angle of the trajectory of the air launcher.
* Air launcher
- A means to obtain data by varying the angle of trajectory, height of launch, and material
* Foot Air Pump
- To obtain the intended pressure for each launch
* Locking Mechanism
- To lock down the air launcher prior to each launch to prevent premature launch
* Cell Phone
- To time each flight using the stopwatch function
* Measuring tape
- To measure the range of the projectile.
* The usages of the remaining equipments are explained below.
The Construction of the Air launcher Pad
) The air launcher pad is built using PVC pipe as it is widely available and is can withstand very high pressure (Had been tested to last up to 70 psi).
2) The main body of this launcher is then locked down to a wooden plank to provide stability. However, the launcher was not made too rigid to allow changes in angle of launch by moving the tip of the pipe not connected to the wooden plank.
3) The components of the launcher are sealed tightly with PVC glue. Therefore, there is no significant air leak in the launcher. There is also a simple locking mechanism applied to prevent it from launching prematurely. With this mechanism, the air launcher can be pressurized to the intended pressure before releasing the air launcher.
4) The release lever is located under the mouth of the bottle and can be released instantaneously by snapping it.
The Construction of the Air launcher Bottle
) Plastic bottles are used to make the main body of the air launcher because it is light, durable and cheap. It is overturned with the mouth facing downward as the mouth will be acting as the nozzle of the air launcher.
2) The golf ball is placed on the bottom of the base of the bottle. This allows the golf ball's position to be slightly higher than the centre of gravity of the intended air launcher because the centre of mass has to be slightly higher than the centre of gravity of the air launcher to ensure a smooth parabolic flight is obtained by preventing the bottle from wobbling in the projectile motion.
3) The hemispheric part was added to the tip of the air launcher to increase its aerodynamics properties, hence ensuring a smooth flight by minimizing the effect of wobbling of the air launcher in the air.
4) The wings are placed below the golf balls as it also has to be above the centre of gravity. The shape of the wings, which is rectangular, is made similar for all three air launchers. The surface area in contact with the air is fixed to obtain a constant drag coefficient so that the effect of varying material can be observed clearer.
5) After the air launchers are built, they are weighed separately using a balance scale.
Varying the Materials Used
) The materials used for the air launcher wings were varied to meet the project's requirement. The materials chosen were rubber, aluminum foil and plastic. These materials were used as they can be easily shaped and cheap.
2) The rubber wing air launcher has the highest mass followed by the aluminum wing air launcher and then the plastic wing air launcher.
3) Although they have different masses, the pressure of the air launchers is varied to ensure they have the same constant initial velocity of 60. Without fixing a constant initial velocity, it will be hard to compare the track of trajectory of the air launchers.
4) By comparing the trajectory of the air launchers, the effects of air drag can be seen and a relationship between air drag and the mass of the air launcher can be made.
Varying the Angle of Launch
) In this project, the angle of projectile is also varied. Each of the air launcher is launched at 0, 15, 30,45,60, 75 and 90 degrees respectively.
2) The trajectory is observed and the total flight time and maximum range is calculated. These angles of trajectory are determined using an inclinometer.
Varying the Height of Launch
) The project has two phases, which are the uni-level projection and the bi-level projection.
2) For uni-level projection, there is a complication due to the initial height of the air launcher from ground level due to the elevation caused by the launching angle.
3) This problem is solved by altering the landing site itself; 'raising' the ground level by means of raffia string. This is explained in a more detailed manner in the latter part of the methodology.
4) As for bilevel projection, the problem with varying the height of launch is solved by launching on different places. This is explained in a more detailed manner in the latter part of the methodology.
Unilevel Projection
) The rubber-winged air launcher is fixated onto the pipe using the locking mechanism.
2) The launching angle, ?°, is fixed at 0° using a inclinometre.
3) By means of a dual-cylinder air pump, the air pressure in the bottle is placed 55 Psi.
4) A raffia string is tied around 4 poles, which acts as the edges, to form a rectangular of 4 x 5 metres.
5) The raffia string loop in rectangular shape is placed at the estimated landing site. The height of the string from the ground is made sure to be exactly the same with the air launcher's height above the ground.
6) The air launcher is then launched by pulling the trigger (locking mechanism). The cell phone's stopwatch function is activated at the same time.
7) When the air launcher reaches the maximum height, the time is taken.
8) As the air launcher proceeds in its motion in air and finally crosses the raffia string, the time is taken again. The data is recorded.
9) Steps 1-9 are repeated by changing the angle into 15°, 30°, 45°, 60°, 75°, and 90°.
0) Then, the graph total time of flight versus sin ? is plotted.
1) Finally, Steps 1-10 are repeated by changing the material of the wings into aluminium and plastic. The air pressure used is also changed accordingly to obtain the same initial velocity for all launches: 35 Psi (aluminium) and 30 Psi (plastic). By predetermined the gravitational constant, the initial velocity can be calculated and this practical value can be compared to the theoretical initial velocity. By using , the objective can be achieved. Also, the effects of varying the materials can be compared relatively.
Bi-level Projection
) For 1 metre elevation from ground level, the launcher is placed on books since there is already an initial height of 0.8 metres due to the elevation of the pipe at 45°.
2) The rubber-winged air launcher is fixated onto the pipe using the locking mechanism.
3) The launching angle, ?°, is fixed at 45° using a clinometre.
4) By means of a dual-cylinder air pump, the air pressure in the bottle is placed 55 Psi.
5) The air launcher is then launched by pulling the trigger (locking mechanism). The cell phone's stopwatch function is activated at the same time.
6) When the air launcher reaches the maximum height, the time is taken.
7) As the air launcher proceeds in its motion in air and finally touches the ground, the time is taken again. The data is recorded.
8) Then, the graph total time of flight versus sin ? is plotted.
9) Steps 1-8 are repeated by varying the height of launch: 2 metres, (on top of the staircase) and 3 metres (on top of a skateboard ramp).
Obtaining the Data
) A few measurements were taken during the course of the project. The following measurements were taken: time of flight, maximum range, maximum height, time to reach maximum height, angle of launch, and air pressure in terms of psi.
2) The time of flight and the time to reach maximum height were measured by means of the timer function of a cell phone.
3) The maximum range of the launch was measured using a measuring tape of 5-metre length. The tape was used multiple times in a single measurement to compensate for its short length.
4) The air pressure was read out from the pressure gauge installed on the air pump.
5) The maximum height achieved by the air launcher can be measured using the height of the building as an estimate. This project was conducted within the Cendana residential area and the students' dormitory (Block 1), which was located just beside the launching site, was chosen as the height estimate. Therefore the maximum height of launch can be estimated by comparing with certain marks. It is estimated that the height of the bottom of the window to the next window is approximately 4 metres while the height of the base to the bottom of the window of the first floor is about 2.5 metres. Two viewers are placed at the first floor and the second floor to hold the extended measuring tape (by 60cm). This is to increase accuracy in estimating the heights, as the air launchers are not launched very nearby to the block. The launches are recorded using a digital camera from the angle of the launching site to estimate the height of each launch based on these marks. The maximum heights of the launches are then recorded.
6) A simulator is used to obtain the theoretical values of the measurements and these values are compared with the measurements obtained during the project.
5.0 Data & Calculation
5.1 Data
Part 1
Diameter of sphere, (average) : 2.82 cm
Radius of sphere, : 1.41 cm
Length of string ()
Length of pendulum ()
Time for half oscillation ()
Period of one complete oscillation ()
()
0.500
0.5141
0.6023
.2046
.451
0.600
0.6141
0.6863
.3726
.884
0.700
0.7141
0.7520
.5040
2.262
0.800
0.8141
0.8129
.6258
2.643
0.900
0.9141
0.8722
.7444
3.043
.000
.0141
0.9348
.8696
3.495
.100
.1141
0.9847
.9694
3.879
.200
.2141
.0436
2.0872
4.356
Part 2
Rubber
Angle
(°)
Time to
reach
maximum
height (s)
Total
flight
time
(s)
Total
Range
(m)
Range to
maximum
height
(m)
Maximum
height
(m)
0
-
2.00
3.18
-
0
0
-
2.05
4.58
-
0
5
0.29
0.46
6.43
4.83
0.5
5
0.25
0.45
6.57
4.81
0.5
30
0.40
0.91
3.31
6.09
3.50
30
0.44
0.92
3.50
6.14
3.5
45
0.58
.58
4.38
7.93
6
45
0.65
.59
4.76
7.89
6
60
.51
2.28
9.65
5.09
7
60
.51
2.26
9.87
4.96
7
75
.81
2.82
3.06
.49
8.5
75
.86
2.79
2.98
.54
8.5
90
.86
3.08
0.27
-
9
90
.23
3.07
0.34
-
9
Aluminium
Angle
(°)
Time to
Reach
maximum
height (s)
Total
Flight
Time
(s)
Total
range
(m)
Range to
maximum
height
(m)
Maximum
height
(m)
0
-
.48
2.52
-
0
0
-
.73
2.52
-
0
5
0.27
0.52
7.01
6.28
0.5
5
0.29
0.56
7.18
6.31
0.5
30
0.65
.21
6.13
9.27
.5
30
0.68
.25
5.96
9.31
.50
45
0.81
.91
7.66
0.12
3
45
0.85
.89
7.97
0.21
3
60
.24
2.27
1.52
8.47
5.5
60
.26
2.25
0.58
8.14
5.5
75
.34
2.51
3.96
4.68
6.5
75
.39
2.54
4.13
4.51
6.5
90
.36
2.43
0.17
-
7.5
90
.55
2.53
0.12
-
7.5
Plastic
Angle
(°)
Time to
reach
maximum
height (s)
Total
flight
time
(s)
Total
Range
(m)
Range to
maximum
height
(m)
Maximum
height
(m)
0
-
.48
2.62
-
0
0
-
.54
2.62
-
0
5
0.39
0.81
7.74
2.98
0.50
5
0.42
0.84
7.81
3.08
0.50
30
0.59
.23
3.12
8.32
.50
30
0.61
.19
3.84
8.43
.50
45
0.82
.59
4.19
9.89
3.50
45
0.84
.54
4.92
0.13
3.50
60
0.99
.90
3.50
8.23
5.00
60
0.81
.79
4.02
8.41
5.00
75
.21
2.12
8.74
2.05
6.50
75
.35
2.16
8.12
2.14
6.50
90
.43
2.80
0.16
-
7.00
90
.05
2.90
0.15
-
7.00
Bi-level Rubber
Angle
(°)
Height
(m)
Total
Flight
Time
(s)
Time to
Maximum
Height
(s)
Range
(m)
Range to
Maximum
Height
(m)
Maximum
Height
(m)
0
2.1
0.81
-
1.14
-
2.1
0
2.1
0.75
-
0.24
-
2.1
45
2
.98
0.99
4.91
9.95
7
45
2
2.07
.05
4.52
0.12
7.5
45
2
2.02
.09
4.74
0.06
8
45
3
2.34
.24
6.42
2.24
8.5
45
3
2.51
.16
6.14
1.59
9
45
3
2.48
.34
5.24
2.01
8.5
45
4
3.12
.57
8.11
3.59
9
45
4
3.16
.61
8.24
3.89
9.5
45
4
3.24
.58
7.98
4.32
9.5
5.2 Calculation
Calculation for Pendulum Experiment:
Gradient of the straight line plotted =
=
=
Given that gradient = ,
Acceleration due to gravity, =
=
Percentage error =
=
Calculation for Air Launcher:
Calculation for Determining Initial Velocity of Air Launcher:
Slope of the graph is , the g value has been determined I the pendulum experiment to be 9.587 ms-2. Thus the initial velocity can now be calculated.
For rubber,
For Aluminium,
For Plastic,
6.0 Discussion
6.1 Comparison between theoretical and practical value
The practical values that are obtained from the experiment are differed from theoretical value. In this case, initial velocity, u is the point of reference.
Rubber
Aluminium
Plastic
Theoretical Value (ms-1)
5
5
5
Practical Value (ms-1)
5.866
2.367
8.676
* Percentage of Difference for Rubber
* Percentage of Difference for Aluminium
* Percentage of Difference for Plastic
6.2 Assumptions made
a) The velocity of the air released from the nozzle
* It is assumed that the air is released from the rocket instantaneously, which is 0.05 second. The air has to be released instantaneously so that the motion is pure projectile. If the air is still continuously released from the rocket when in motion, the air is treated as a fuel. Therefore, the rocket is no longer in a projectile motion.
b) Efficiency of energy transfer from the air pressure to the rocket's initial velocity
* The theoretical value of 55.8 is not used throughout the calculation, as the values of range and maximum height are not the same as shown in a simulator. From the simulator, it is determined that the theoretical value for the initial velocity should be with the presence of air drag. The air drag coefficient value is predetermined to be 1.3. From this, the efficiency of the transformation of energy from air pressure to kinetic energy of the rocket is not at 100%. The efficiency is just 27% as calculated.
c) Loss of energy
* According to the law of conservation of energy, energy cannot be created or destroyed but can be converted from one form into another form. In this project, the potential energy in the rocket is mostly converted into sound, heat and is used to overcome friction. Some energy is also used to overcome air drag while the rocket is in flight.
* As the pressure increase, the rate of evaporation will decrease. Thus, the rate of condensation will increase. At high pressure, the water vapour will be forced to condense into water again. The conversion of water vapour to water molecule will cause the number of air molecules inside the rocket to reduce. This will cause the air pressure to drop and resulted in an inaccurate experiment practice. To solve this problem, the pressure is pumped slightly higher than the intended pressure so that the timing of release of locking mechanism can be done precisely once the pressure drop to the intended pressure.
6.3 Errors
a) The effects of wind
* The effect of wind is significant in this project as it asserts a considerable change to the horizontal component and the vertical component of the velocity of the rocket in motion when the forces are resolved. This subsequently changes the time of flight and range as well. It is also substantially hard to measure the changes as the direction of the wind towards the rocket in flights changes at all times due to the change in direction of rocket at all times in flight. The direction of the wind too may shift unexpectedly during the project. The wind effects on the projectile can be observed in three different directions: the opposite direction of the motion of the rocket, the same direction as the motion of the rocket, and from the sides of the motion of the rocket.
A. The Opposite Direction
(Along the horizontal (? = 0?))
* No change in flight time
* Range is decreased
(Below the horizontal)
Generally, the maximum height at angles 0?? ? ? 90? of the wind in the
opposite direction will be reduced
i) 0?? ? ? 45?
* Range is decreased more
* Time of flight is decreased less
ii) ? = 45?
* Range and time of flight are equally decreased in magnitude
iii) 45?? ? ? 90?
* Range is decreased less
* Time of flight is decreased more
(Above the horizontal)
Generally, the maximum height at angles 0?? ? ? 90? of the wind in the
opposite direction will be increased
i) 0?? ? ? 45?
* Range is decreased more
* Time of flight is increased less
ii) ? = 45?
* Range and time of flight are equally decreased in magnitude
iii) 45?? ? ? 90?
* Range is decreased less
* Time of flight is increased more
B. The Same Direction
(Along the horizontal (? = 0?))
* No change in flight time
* Range is increased
(Below the horizontal)
Generally, the maximum height at angles 0?? ? ? 90? of the wind in the same
direction will be reduced
i) 0?? ? ? 45?
* Range is increased more
* Time of flight is decreased less
ii) ? = 45?
a. Range is increased and time of flight is decreased equally in magnitude
iii) 45?? ? ? 90?
b. Range is increased less
c. Time of flight is decreased more
(Above the horizontal)
Generally, the maximum height at angles 0?? ? ? 90? of the wind in the
opposite direction will be increased
i. 0?? ? ? 45?
d. Range is increased more
e. Time of flight is increased less
ii) ? = 45?
a. Range and time of flight are equally increased in magnitude
iii) 45?? ? ? 90?
b. Range is increased less
c. Time of flight is increased more
C. Sideways
(Along the horizontal (? = 0?))
* No change in flight time
* No change in range
(Below the horizontal)
Generally, the maximum height at angles 0?? ? ? 90? of the wind in the
opposite direction will be reduced
i. 0?? ? ? 90?
* Time of flight is decreased.
* Therefore, the range is decreased too.
(Above the horizontal)
Generally, the maximum height at angles 0?? ? ? 90? of the wind in the opposite direction will be increased
ii. 0?? ? ? 45?
* Time of flight is increased
* Therefore, the range is increased too
b) The wobbling of the rocket
* The data is affected when the rocket is wobbling during the flight. The total flight time is increased because the flight time for the downward motion is increased. The time of flight to reach the maximum height on a contrary decreases slightly.
c) Unstable air pressure before launching
* As the pressure increase, the rate of evaporation will decrease. Thus, the rate of condensation will increase. At high pressure, the water vapour will be forced to condense into water again. The conversion of water vapour to water molecule will cause the number of air molecules inside the rocket to reduce. This will cause the air pressure to drop and resulted in an inaccurate experiment practice. To solve this problem, the pressure is pumped slightly higher than the intended pressure so that the timing of release of locking mechanism can be done precisely once the pressure drop to the intended pressure.
6.4 Observations done
A) Significance of certain launching angles
a) 0 degree bi-level projection
* The 0 degree bi-level projection has no initial vertical component and is a total initial horizontal velocity. As time passes, due to the constant action of the gravitational acceleration, the vertical component gradually increases in velocity until it reaches the ground. The time of flight is the shortest compared to other angles of projection since time of flight is dependent to the vertical component of velocity.
b) 0 and 90 degree uni-level projection
* Both the motions are not considered as a projectile is defined as a projection of an object near by the surface of the earth at a certain angle, 0?<?<90?. The 0 degree launch cannot be accepted as there is friction between the rocket and the concrete surface. As for the 90-degree launch, the total range should be 0 as there is no vertical velocity acting on the rocket however the data reviews that there is a range of 0.03 metres due to the influence of wind when conducting the project.
B) Effects of air resistance
* The descending part of the path shows a different pathway. The range travelled by the object when it is falling is less than the previous part.
* When they are rising, the air drag is in downward direction. This coincides with the direction of the gravitational force thus compounding a greater downward acceleration to the projectile. Therefore, the maximum height that it reaches is lower compared to the object without air drag. When the object is falling, the air drag is in opposite direction to the object. Thus the air drag is also in opposite direction to the gravitational force. The air drag would partially cancel the gravitational force creating a lesser downward acceleration towards the projectile. Therefore the object would take more time to travel the same distance.
* As the projectile does not reach as high as the projectile in a situation without air resistance, the total flight time will be overall less than the projectile motion without the existence of air resistance. As the travelling time is less, the range covered will be less too. The horizontal component of the rocket experiences air drag and its velocity is ever decreasing. Therefore, the range covered will be less.
* Ranges from complementary angles are supposed to be identical, but it is not reflected in the experiment. For the higher angles, the projectile will encounter longer flight time which means longer exposure to air resistance. The air drag and wind experienced by the rocket causes it to have different ranges and it does not follow the theoretical prediction where complementary angles should have the same range.
C) Effects of varying launching angle
* When the angles are varied, so as the horizontal and vertical components as well. Since the initial velocity is fixed at a constant value, the horizontal and vertical components of the initial velocity changes due to the change in angles of launch only. This change subsequently affects the maximum range of the projectile, time of flight, and the maximum height achieved by the rockets.
D) Effects of varying height of launch in bi-level projection
* Height is manipulated by launching at uni-level (0 metre at 45°) and bi-level (2, 3 and 4 metres at 45°).
* When the height is added, the time of flights increases proportionally. Thus, the maximum height and range increases. The maximum height increases by the exact value of the height of the launching base from ground level. The maximum range, on the other hand, increases based on the formula s = vt. However, the range is not affected by the height of launch as greatly as by the angle of projection.
* During the project, it is noticed that it occurred twice when the uni-level projection (0 metre at 45°) launch's range is further than the bi-level projection (2 metre at 45°) launch's range. This error might be due to the effect of wind.
E) Effects of varying materials for the wings of the rockets
* Different materials show the different trajectory as they have the different inertia in their motion. Thus some trajectory seems to be longer for a heavier object as they have the higher inertia to impede the object from changing the angle throughout the projectile motion.
6.5 Structure of the rocket
a) Aerodynamic design
* The rocket is built to an aerodynamic design to minimize the effect of wobbling of the rocket in the air. Therefore the curve of projection is smooth and can be seen clearly.
Photo 6.5.1 : Aerodynamic design of the air launcher
b) Reliance on golf ball
* The golf ball is placed at the front part of the rocket to shift the centre of mass slightly higher than the centre of gravity of the bottle in order to prevent the bottle from wobbling in the projectile motion.
Photo 6.5.2: Golf Ball Used to Stabilise the Rocket during Flight
c) Surface area of wings
* The surface area in contact with the air is fixed to obtain a constant drag coefficient so that the effect of varying material can be observed clearer.
Photo 6.5.3: Large surface areas of the material chosen to create the air drag effect.
7.0 Application
7.1 Ballistic Test
* In many crime cases that involve arms, the police usually run a ballistic test on the weapon used to commit the crime.
* In criminology, ballistic test is applied to the identification of the weapon from which a bullet was fired.
* The ballistic test is carried out to determine the type of artilleries used by the criminal and the position of the criminal at that specific time of firing.
* For a short range of firing, the pattern of the wound caused by the bullet is carefully studied as it will reveal the types of gun used by the criminal. The pattern also reveals the distance of firing between the criminal and the victim.
* For a long range of firing, the projectile motion of the bullet plays a vital role in ballistic test. After examining the pattern of the wound, the projectile path of the bullet can be determined; this will in turn reveal the exact location of the criminal when the bullet was fired.
* With this information building up, the police will soon obtain enough information to track the crime suspect.
Photo 7.1.1: A Forensic Ballistics Experiment
Photo 7.1.2: Controversial Bullet from the John F. Kennedy Assassination
Photo 7.1.3: Post-test Impact on NIJ 3 Transparent Armour Window - 24 x 24 x 2.75 `Inches
7.2 Ballistic
Ballistics is the study of the firing, flight, and effect of ammunition. A fundamental understanding of ballistics is necessary to comprehend the factors that influence precision and accuracy and how to account for them in the determination of firing data. Gunnery is the practical application of ballistics so that the desired ejects are obtained by fire. To ensure accurate predicted fire, we must strive to account for and minimize those factors that cause round-to-round variations, particularly muzzle velocity. Ballistics can be broken down into four areas: interior, transitional, exterior, and terminal. Interior, transitional, and exterior ballistics directly affect the accuracy of artillery fire and are discussed as follows:
* Interior Ballistics
Interior ballistics is the science that deals with the factors that affect the motion of the projectile within the tube. The total effect of all interior ballistic factors determines the velocity at which the projectile leaves the muzzle of the tube, which directly influences the range achieved by the projectile. This velocity, called muzzle velocity (MV), is expressed in meters per second (m/s).
The following general rules show how various factors affect the velocity performance of a weapon projectile family-propellant type-charge combination:
a) An increase in the rate of propellant burning increases the resulting gas pressure developed within the chamber. An example of this is the performance of the multi perforated propellant grains used in white bag (WB) propellants. The result is that more gases are produced, gas pressure is increased, and the projectile develops a greater muzzle velocity. Damage to propellant grains, such as cracking and splitting from improper handling, also affect the rate of burn and thus the muzzle velocity.
b) An increase in the size of the chamber without a corresponding increase in the amount of propellant decreases gas pressure; as a result, muzzle velocity will be less (Boyles Law).
c) Gas escaping around the projectile decreases chamber pressure.
d) An increase in bore resistance to projectile movement before peak pressure increases the pressure developed within the tube. Generally, this results in a dragging effect on the projectile, with a corresponding decrease in the developed muzzle velocity. Temporary variations in bore resistance can be caused by excessive deposits of residue within the cannon tube and on projectiles and by temperature differences between the inner and outer surfaces of the cannon tube.
* Transitional Ballistics
Sometimes referred to as intermediate ballistics, this is the study of the transition from interior to exterior ballistics. Transitional ballistics is a complex science that involves a number of variables that are not fully understood; therefore, it is not an exact science. What is understood is that when the projectile leaves the muzzle, it receives a slight increase in MV from the escaping gases. Immediately after that, its MV begins to decrease because of drag.
* Exterior Ballistics
Exterior ballistics is the science that deals with the factors affecting the motion of a projectile after it leaves the muzzle of a piece. At that instant, the total effects of interior ballistics in terms of developed muzzle velocity and spin have been imparted to the projectile. Were it not for gravity and the effects of the atmosphere, the projectile would continue indefinitely at a constant velocity along the infinite extension of the cannon tube. The discussion of exterior ballistics in the following paragraphs addresses elements of the trajectory, the trajectory in a vacuum, the trajectory within a standard atmosphere, and the factors that affect the flight of the projectile.
a. Trajectory in a Standard Atmosphere.
(1) The resistance of the air to projectile movement depends on the air movement, density, and temperature. As a point of departure for computing firing tables, assumed conditions of air density and air temperature with no wind are used. The air structure is called the standard atmosphere.
(2) The most apparent difference between the trajectory in a vacuum and the trajectory in the standard atmosphere is a net reduction in the range achieved by the projectile. A comparison of the flight of the projectile in a vacuum and in the standard atmosphere is shown in Figure 3-8.
(3) The difference in range is due to the horizontal velocity component in the standard atmosphere no longer being a constant value. The horizontal velocity component is continually decreased by the retarding effect of the air. The vertical velocity component is also affected by air resistance. The trajectory in the standard atmosphere has the following characteristic differences from the trajectory in a vacuum:
(a) The velocity at the level point is less than the velocity at the origin.
(b) The mean horizontal velocity of the projectile beyond the summit is less than the mean velocity before the projectile reaches the summit; therefore, the projectile travels a shorter horizontal distance. Hence, the descending branch is shorter than the ascending branch. The angle of fall is greater than the angle of elevation.
(c) The spin (rotational motion) initially imparted to the projectile causes it to respond differently in the standard atmosphere because of air resistance. A trajectory in the standard atmosphere, compared to a trajectory in a vacuum, will be shorter and lower at any specific point along the trajectory for the following reasons:
* Horizontal velocity is not a constant value; it decreases with each succeeding time interval.
* Vertical velocity is affected by both gravity and the effects of the atmosphere on the projectile.
* The summit in a vacuum is midway between the origin and the level point; in the standard atmosphere, it is actually nearer the level point.
* The angle of fall in a vacuum is equal to the angle of elevation; in the standard atmosphere, it is greater.
b. Deviations from Standard Conditions. Firing tables are based on actual firings of a piece and its ammunition correlated to a set of standard conditions. Actual firing conditions, however, will never equate to standard conditions. These deviations from standard conditions, if not corrected for when computing firing data will cause the projectile to impact at a point other than the desired location. Corrections for nonstandard conditions are made to improve accuracy.
(1) Range effects. Some of the deviations from standard conditions affecting range are:
* Muzzle velocity.
* Projectile weight.
* Range wind.
* Air temperature.
* Air density.
* Rotation of the earth.
(2) Deflection effects. Some of the deviations from the standard conditions affecting deflection are:
* Drift.
* Crosswind.
* Rotation of the earth.
7.3 Military Defence
* In a war, soldiers need to determine the correct angle and height of the cannon to make sure the target is hit. The cannonball will travel in a projectile motion before it hits the target as the cannonball does not have any form of fuel to provide energy for its motion.
Photo 7.3.1: Modern Tanks fire two main types of shells known as HEAT (high-explosive anti-tank) & Sabot rounds. HEAT rounds use explosive firepower rather than momentum to penetrate armour. Sabot rounds (above) don't have any explosive power; they penetrate armour using shear momentum - working like an arrow. Sabot rounds provide ballistic advantages over simply using a lightweight projectile, since the smaller diameter projectile will have a better ballistic coefficient for a given weight. On firing the expanding gas pushes the sabot and attached penetrator down the barrel. The sabot is attached to the penetrator with relatively flimsy plastic, so it falls away as soon as the round leaves the cannon.
(http://www.fas.org/man/dod-101/sys/land/docs/fig3-2.gif)
Photo 7.3.2: A ship is test firing its ballistic missile. (http://www.usswisconsin.org/Pictures/bigguns.htm)
Photo 7.3.3: A tank is firing a missile which its range has been predetermined using the concept of projectile motion. (http://www.strangemilitary.com/content/item/104049.html)
Photo 7.3.4: The photo is showing how the bombs are projected from the plane and detonated at different times, animating the layering of timings can help to create a more natural looking explosion.
(http://features.cgsociety.org/stories/2005_09/explosions/images/munitions_08.jpg)
7.4 Sports
* Projectile motion is applied in badminton and tennis as force is provided to the badminton shuttlecock and tennis ball to create a projectile motion so that the balls can be passed from one player to the other player.
* In baseball, projectile motion also plays an important role as the player could pass the ball in shortest time if the player can throw the ball horizontally.
* In basketball, a player has to throw the ball in the correct angle in order to score.
7.4.1 Basketball
The long throw-in is an important "set play" in basketball, particularly when used as an attacking manoeuvre near to the goal mouth. The farther a player can throw the ball the larger the area in which his/her team mates may receive the ball and the greater the scoring opportunities. To produce a long throw the player must project the ball at high speed and at an appropriate angle with respect to the horizontal. In laboratory studies of the throw-in, male players have recorded release speeds of 12-19 m/s and release angles of 22-40º. Release speed is the primary determinant of the range attained by a projectile, and to achieve the greatest possible distance a basketball player should release the ball with the greatest possible speed. Identifying the optimum release angle for a long throw-in is less straightforward. The optimum release angle of a sports projectile is influenced by:
(1) The physical properties of the projectile
(2) The release conditions
(3) The anatomical and musculoskeletal constraints of the player's body
In a basketball throw-in, the release conditions that affect the optimum release angle are the release speed, release height, and rate of spin. The effects of release speed and release height on the distance and optimum release angle for a moderately aerodynamic projectile such as a basketball ball are well established. For a ball that is released from a typical release height of 2.3m, the throw distance increases rapidly with increasing release speed and the throw distance is a maximum at a release angle of about 40º. Changes in release height do not greatly affect the throw distance or the optimum release angle. The spin imparted to the ball during the release can have a strong influence on the throw distance and the optimum release angle.
It was found that the player produced greater release speeds at lower release angles, and that this relation reduced the optimum release angle to about 30º. Although the player's release height increased with increasing release angle because of changes in his body position at the instant of release, this relation had only a small influence on the optimum release angle. The calculated optimum release angle was in good agreement with the player's preferred release angle
Methods
In a long throw-in, the throw distance (or horizontal range) R is the horizontal distance the ball's centre of mass travels from the instant of release to the instant of landing (Figure 1). The release parameters that determine the throw distance are the release speed v, the release angle ?, the relative release height h (the height difference between the release and the landing), and the spin rate of the ball (?). To calculate a player's optimum release angle we require the mathematical expressions for the relation between release speed and release angle, v(?), and between relative release height and relative release angle, h(?). Intervention is therefore required to obtain measurements of a player's release speed and relative release height over a wide range of release angles, rather than just at the player's preferred release angle. The expressions for v(?) and h(?) may be obtained by fitting mathematical relations to plots of release speed and relative release height as a function of release angle.
Figure 7.4.1: Diagram of a long basketball throw-in showing the release conditions that determine the horizontal range of the ball.
Figure 7.4.2: Anthropometric model of the player at the instant of release. This model was used to explain the observed relation between the release height and release angle.
Figure 7.4.3: Release height as a function of release angle for a male player.
Figure 7.4.4: Release speed as a function of release angle.
Conclusions and Recommendations
This study showed that the optimum release angle in the long basketball throw-in is considerably less than 45°. The release speed a player can generate increases at lower release angles, and this bias reduces the optimum release angle to about 30°. The height difference between release and landing has only a small effect on the optimum release angle. The optimum release angle is probably slightly different for each player, and may depend on the player's body size, muscular strength, and throwing technique. To produce a longer throw we recommend that the player work on increasing the release speed by developing explosive strength in the muscles used in the throwing movement, and by improving his/her throwing technique. It is not essential to launch the ball at precisely the optimum release angle as deviations of several degrees do not substantially reduce the distance of the throw. The distance achieved in a long throw is greatest when the player receives the ball at ground level.
When a throw is received at chest-height or head-height the throw distance is a little less and the ball should be launched a few degrees higher. In an attacking long throw, launching the ball for maximum distance may not be the most appropriate strategy. We recommend that the player should deliberately launch the ball a few degrees below the optimum release angle. This will reduce the flight time of the throw without substantially reducing the throw distance, and hence provide the attacking team with a greater element of surprise. We also recommend that the player should launch the ball with a high backspin. This will increase the distance of the throw, as long as the player's technique for producing backspin does not reduce the release speed. The player should reduce the release angle by a few degrees if s/he launches the ball with substantial backspin.
Photo 7.4.1: A child is applying the projectile motion concept in throwing the basketball.
Photo 7.4.2: A professional player need to judge the release angle and release velocity of the ball to make sure the basketball hit the target.
7.5 Daily Life
* When watering the plants, the angle of the hose is varied to reach the different part of the garden without having to move around. The diameter of the end of the hose is reduced too to obtain a greater initial velocity for the water so that it can travel further in a projectile motion to reach the further part of the garden.
* Besides that, projectile motion is also applied into daily life when it comes to bizarre pondering down to the simplest matters in life.
* This problem was defined in "The Last Word" of New Scientist, "My five-year old daughter wants to know if you spit a cherry stone while swinging on a swing, would it go farthest if you spit it while you are at the lowest point of the swing (because you are moving so fast), or would it be better to spit it at the highest point?"
* It is actually, in a physics approach, requiring calculating the maximum distance attained by a ballistic projectile launched from a free swinging rigid pendulum.
7.5.1 Assumptions
. The swing is rigid in tension (i.e. it will trace a circular arc)
2. Frictional effects are neglected
3. The swing is not being pushed at the time of release
4. The projectile will be released tangentially to the motion of the swing
5. Air resistance and wind are neglected
6. The ground is flat and there are no obstructions in the flight path
7. The projectile will not bounce on impact
7.5.2 Flight profile
There are three distinct phases in the flight of our projectile.
. For the first part of the flight, the projectile is constrained to follow a circular trajectory (inside the head of the child) and can be considered in a similar manner to that of a pendulum of length L where L is the distance between the pivot of the swing and the child's head. The motion of the projectile may be described by the equations of "Simple Harmonic Motion".
2. The second phase of the flight is the release. It can be assumed that the projectile is ejected tangentially to the motion of the swing and that a constant additional velocity VR will be imparted to it, regardless of the angle of the swing at the time of release.
3. The final phase of the flight is ballistic, similar to a thrown ball. Neglecting air resistance and wind, the only force acting on the projectile in this phase is the gravitational attraction of the Earth.
7.5.3 Results:
Figure 7.5.1: Maximum distance as a function of release angle
(http://homepage.ntlworld.com/geraint.bevan/swing/swing.pdf)
Photo 7.5.1: A fireman is putting out the fire by applying the concept of projectile motion on the flight path of the water jet.
Photo 7.5.2: A fireman is taking a rest while putting out fire
7.6 Rescue Mission
* In search and rescue mission, sometimes the aid cannot be delivered by land or water. Therefore, the only possibility is by air. The plane carrying the aid will have to calculate the trajectory of the aid after it is released from the plane precisely to make sure that the aid arrives in the hand of the victims.
Photo 7.6.1: An airplane used for rescue mission about to take off to deliver aid from air. (www.spectre-association.org/library/documents/2004Spotlight.pdf)
8.0 Photos
Photo 8.0.1: The Air Launcher Pad and The Air Launcher
Photo 8.0.2: The Air Inlet
Photo 8.0.3: The Plastic Sticks to Hold The Bottle
Photo 8.0.4: Plastic Material Air Launcher in Position
Photo 8.0.5: Rubber Material Air Launcher in Position
Photo 8.0.6 : Aluminium Material Air Launcher in Position
Photo 8.0.7: Locking Mechanism in Position
Photo 8.0.8: Another View of Locking Mechanism in Position
Photo 8.0.9: Plastic Sticks in Holding Position
Photo 8.0.10: Foot Air Pumps
Photo 8.0.11: PVC Glue Used to Stick PVC Pipes
Photo 8.0.12: Aluminium Foil Used for Material
Photo 8.0.13: Golf Ball Used to Stabilise the Rocket during Flight
Photo 8.0.14: Up close view of aluminium wing
Photo 8.0.15: Aluminium Winged Air Launcher from Afar
Photo 8.0.16: End of pipe used to connect to foot air pump
Photo 8.0.17: Golf ball in position
Photo 8.0.18: Nylon sticks up close
Photo 8.0.19: Nylon sticks used to bind pipe to base
Photo 8.0.20: PVC pipe connecter
Photo 8.0.21: Rubber wing up close
Photo 8.0.22: Tightener fastened around nylon sticks
Photo 8.0.23: Rubber winged air launcher from afar
Photo 8.0.24: View of air launcher and its base from above
8.1 Practical Scenes
Photo 8.1.1: The Rocket Just left the Launcher
Photo 8.1.2: The Rocket Just Passed Its Maximum Height
Photo 8.1.3: The Rocket Reached Its Maximum Height
Photo 8.1.4: Air launcher is launched when condensed air propelled outward at the initial phase of the launch
Photo 8.1.5: Preparing for first launch attempt
9.0 Conclusion
As the air drag is always in the opposite direction to the motion of the rocket, when they are rising, the air drag is in downward direction, yielding a greater downward acceleration. Therefore, the maximum height is reduced and subsequently the range and the time of flight are also reduced. When the object is falling, the air drag is in upward direction thus partially cancelling gravitational force creating a lesser downward acceleration towards the projectile. Therefore the object would take more time to travel the same distance but the distance is shorter.
From this investigation, the gravitational acceleration value for the rubber-winged rocket is slightly lower than the actual value as the experiment data is affected by wind. For the aluminium-winged rocket, the value is slightly higher than the actual value as the wind inflicted a more noticeable effect onto the rocket since it is considerably lighter than the rubber-winged rocket. The plastic-winged rocket, however, has a very high gravitational acceleration value which deviates from the actual gravitational acceleration value by far. This is due to the wobbling of the rocket.
0.0 Reference Lists
. http://en.wikipedia.org/wiki/Catapult
2. http://www.redstoneprojects.com/trebuchetstore/how_a_trebuchet_catapult_works.html
3. http://www.killology.com/art_weap_sum_age.htm
4. http://arxiv.org/ftp/physics/papers/0601/0601149.pdf
5. www.iop.org/EJ/article/0957-0233/7/12/010/e61208.pdf dap
6. http://www.patentstorm.us/patents/5029866-description.html
7. http://www.mechsol.com/html/ch/chmilitary.html
8. http://www.grc.nasa.gov/WWW/RT1997/5000/5910pereira.htm
9. joe.buckley.net/papers/Sonatest_PASS2000.pdf dap
0. http://homepage.ntlworld.com/geraint.bevan/swing/swing.pdf
1. www.ngsir.netfirms.com/englishhtm/ThrowABall.htm
2. http://physics.rutgers.edu/ugrad/381/aim.pdf
3. http://www.edu.gov.mb.ca/k12/cur/science/found/physics40s/topic1_4.pdf
4. http://www.answers.com/topic/projectile-motion
5. http://arxiv.org/ftp/physics/papers/0506/0506032.pdf
6. http://arxiv.org/ftp/physics/papers/0601/0601148.pdf
7. http://www.rose-hulman.edu/Class/CalculusProbs/Problems/CATAPULT/CATAPULT.html
8. http://mcc.org/clusterbombs/resources/research/death/chapter1.html
9. http://ahs.dsd.k12.ut.us/~callred/Air%20Rockets.html
20. http://www.spaceed.org/seiclassroom/cappdf/air_rocketry.pdf
21. http://www.astronautix.com/data/saenger.pdf
22. http://exploration.grc.nasa.gov/education/rocket/flteqs.html
23. http://exploration.grc.nasa.gov/education/rocket/rktsim.html
24. http://www.hydroflite.net/Altitude_Measurement.html
25. http://www.hydroflite.net/Computer_Simulation.html
26. http://www.physics.upenn.edu/courses/gladney/mathphys/subsubsection3_1_3_3.html
27. http://www.hydroflite.net/Rocket_Science.html
28. http://exploration.grc.nasa.gov/education/rocket/Downloads/index.htm
Appendix:
Appendix (a)
LAB PRACTICAL: RELATIONSHIP BETWEEN PERIOD AND LENGTH OF A SIMPLE PENDULUM
Introduction:
As recorded in the 5th century Chinese Book of Later Han, one of the earliest uses of the pendulum was in the seismometer device of the Han Dynasty (202 BC - 220 AD) scientist and inventor Zhang Heng (78-139). Its function was to sway and activate a series of levers after being disturbed by the tremor of an earthquake far away. After this was triggered, a small ball would fall out of the urn-shaped device into a metal toad's mouth below, signifying the cardinal direction of where the earthquake was located (and where government aid and assistance should be swiftly sent).
An Egyptian scholar, Ibn Yunus, is believed to have described an early type of pendulum in the 10th century. Some claimed that he used it for making measurements of time, but this is now believed to be a misinterpretation on the part of Edward Bernard, an English historian.
Among his scientific studies, Galileo Galilei performed a number of observations of the properties of pendulums. His interest in pendulums may have been sparked by looking at the swinging motion of a chandellier in the Pisa cathedral. He began serious studies of the pendulum around 1602. Galileo noticed that period of the pendulum is independent of the bob mass or the amplitude of the swing. He also found a direct relationship between the square of the period and the length of the arm. The isochronism of the pendulum suggested a practical application for use as a metronome to aid musical students, and possibly for use in a clock.
Perhaps based upon the ideas of Galileo, in 1656 the Dutch scientist Christiaan Huygens patented a mechanical clock that employed a pendulum to regulate the movement. This approached proved much more accurate than previous time pieces, such as the sundial and hourglass.
Following an illness, in 1665 Huygens made a curious observation about pendulum clocks. Two such clocks had been placed on his mantlepiece, and he noted that they had acquired an opposing motion. That is, they were beating in unison but in the opposite direction-an anti-phase motion. Regardless of how the two clocks were adjusted, he found that they would eventually return to this state, thus making the first recorded observation of a coupled oscillator.
During his Académie des Sciences expedition to Cayenne, French Guiana in 1671, Jean Richer demonstrated that the periodicity of a pendulum was slower at Cayenne than at Paris. From this he deduced that the force of gravity was lower at Cayenne. Huygens reasoned that the centripetal force of the Earth's rotation modified the weight of the pendulum bob based on the latitude of the observer.
In his 1673 opus Horologium Oscillatorium sive de motu pendulorum, Christian Huygens published his theory of the pendulum. He demonstrated that for an object to descend down a curve under gravity in the same time interval, regardless of the starting point, it must follow a cycloid (rather than the circular arc of a pendulum). This confirmed the earlier observation by Marin Mersenne that the period of a pendulum does vary with amplitude, and that Galileo's observation was accurate only for small swings in the neighborhood of the center line.
The English scientist Robert Hooke devised the conical pendulum, consisting of a pendulum that is free to swing in both directions. By analyzing the circular movements of the pendulum bob, he used it to analyze the orbital motions of the planets. Hooke would suggest to Isaac Newton in 1679 that the components of orbital motion consisted of inertial motion along a tangent direction plus an attractive motion in the radial direction. Isaac Newton was able to translate this idea into a mathematical form that described the movements of the planets with a central force that obeyed an inverse square law-Newton's law of universal gravitation. Robert Hooke was also responsible for suggesting (as early as 1666) that the pendulum could be used to measure the force of gravity.
In 1851, Jean-Bernard-Leon Foucault suspended a pendulum (later named the Foucault pendulum) from the dome of the Panthéon in Paris. The mass of the pendulum was 28 kg and the length of the arm was 67 m. Once the pendulum was set in motion, the plane of motion was observed to precess 360° clockwise once per day. To Foucault, the precession was most easily explained by the Earth's rotationsss.
The National Institute of Standards and Technology based the U.S. national time standard on the Riefler Clock from 1904 until 1929. This pendulum clock maintained an accuracy of a few hundredths of a second per day. It was briefly replaced by the double-pendulum W. H. Shortt Clock before the NIST switched to an electronic time-keeping system.
(http://en.wikipedia.org/wiki/Pendulum)
One Sunday in 1583, as Galileo Galilei attended a service in a cathedral in Pisa, Italy, he suddenly realized something interesting about the chandeliers hanging from the ceiling. Air currents which was circulating through the cathedral had set them in motion with small oscillation, and Galileo noticed that chandeliers of equal length oscillated with equal periods, even if their amplitudes were different. Indeed, as any given chandelier oscillated with decreasing amplitude its period remained constant. He verified this observation by timing the oscillations with his pulse!
Galileo was struck by this observation and after rushing home he experimented with pendula constructed from different lengths of string and different weights. Continuing to use his pulse as a stopwatch, he observed that the period of a pendulum varies with its length, but is independent of the weight attached to the string. In modern terms, we would say that the chandeliers observed by Galileo were undergoing simple harmonic motion, as expected for small oscillations. As we know, the period of simple harmonic motion is independent of amplitude. The fact that the period is also independent of the mass is a special property of the pendulum.
Discussion:
Pre-lab Question:
. Name the type of vibrational motion that is both periodic (to and fro) and planar which is executed by the simple pendulum.
* This type of motion is simple harmonic motion.
2. What is the other name for the unit of 'cycles per second'?
* The other name for the unit of 'cycles per second' is Frequency, Hertz.
3. Which two factors does the period of oscillation of a simple pendulum depend on?
* The period of oscillation of a simple pendulum depends on the length of the pendulum and the acceleration of the gravity.
4. Does the value of the acceleration due to gravity, g varies with altitude and latitude?
* Yes. The value will vary because at different altitude and latitude, the acceleration of gravity will be different because acceleration of gravity is getting smaller as it is farther from the centre of the Earth. As for the latitude, the acceleration of gravity is greater when approaching the North and South Pole than the Ecuador as North and South Pole are closer to the centre of the Earth.
5.
T2 is directly proportional to L and the gradient of the graph of T2 versus L can be used to determine the gravitational constant, g.
(Shown)
Appendix (b)
(http://homepage.ntlworld.com/geraint.bevan/swing/swing.pdf)
Equations of motion
Simple Harmonic Motion
Figure 1: Pendulum (Simple Harmonic Motion)
The position of the projectile while constrained to follow the circular arc of a pendulum is:
x(µ) = L sin(µ)
y(µ) = L (1 ¡ cos(µ)) + h0 (1)
where x is the horizontal position, y is the vertical position, L is the distance from the pivot, µ is the angle of the pendulum measured from the vertical axis and h0 is the height above the ground of the projectile when the pendulum is vertical. The velocity at any point in the trajectory can be determined by the equation:
V (µ) = p2 g L (cos(µ) ¡ cos(µmax)) (2)
where g is the acceleration due to gravity and µmax is the maximum angle that the pendulum makes with the vertical axis, i.e. the angle at which it reverses direction.
Derivation of particle velocity
The velocity of the particle may be determined from consideration of its
energy. The particle's potential energy EP is
EP (µ) = m g y(µ) (3)
and its kinetic energy EK is
EK(µ) =m V 2(µ) (4)
where m is the mass of the particle, g is the gravitational attraction due to the mass of the Earth (g 1/4 9:81m=s), V is the speed of the particle (velocity tangential to its motion) and y its height above the ground. The total energy of the particle at any point in its trajectory is equal to sum of its potential energy and its kinetic energy
E(µ) = EK(µ) + EP (µ) = m V 2(µ) + m g y(µ) (5)
Substituting for y(µ) from equation (1), the energy can be expressed as
E(µ) =m V 2(µ) + m g (L (1 ¡ cos(µ)) + h0) (6)
Now, we know that at its maximum displacement, the swing's velocity is zero, i.e.
E(µmax) = m g (h + L (1 ¡ cos(µmax))) + 0 (7)
Conservation of energy means that in the absence of friction,
E(µ) = E(µmax) (8)
We can therefore equate the right hand sides of equations (6) and (7)
m g (h+L (1¡cos(µ)))+ m V 2(µ) = m g (h+L (1¡cos(µmax))) + 0 (9)
Dividing both sides by m and re-arranging gives the speed V as a function of angle µ
V (µ) = p2 g L (cos(µ) ¡ cos(µmax))
Release
Ballistic Motion
Appendix (c)
(http://www.fas.org/man/dod-101/sys/land/docs/fm6-40-ch3.htm)
a. Nature of Propellant and Projectile Movement.
(1) A propellant is a low-order explosive that burns rather than detonates. In artillery weapons using separate-loading ammunition, the propellant burns within a chamber formed by the obturator spindle assembly, powder chamber, rotating band, and base of the projectile. For cannons using semifixed ammunition, the chamber is formed by the shell casing and the base of the projectile. When the propellant is ignited by the primer, the burning propellant generates gases. When these gases develop enough pressure to overcome initial bore resistance, the projectile begins its forward motion.
(2) Several parts of the cannon tube affect interior ballistics. (See Figure 3-1.)
(a) The caliber of a tube is the inside diameter of the tube as measured between opposite lands.
(b) The breech recess receives the breechblock. The breech permits loading the howitzer from the rear.
(c) The powder chamber receives the complete round of ammunition. It is the portion of the tube between the gas check seat and the centering slope.
* The gas check seat is the tapered surface in the rear interior of the tube on weapons firing separate-loading ammunition. It seats the split rings of the obturating mechanism when they expand under pressure in firing. This expansion creates a metal-to-metal seal and prevents the escape of gases through the rear or the breech. Weapons firing semifixed ammunition do not have gas check seats since the expansion of the ease against the walls of the chamber provides a gas seal for the breech.
* The centering slope is the tapered portion at or near the forward end of the chamber that causes the projectile to center itself in the bore during loading.
(d) The forcing cone is the tapered portion near the rear of the bore that allows the rotating band to be gradually engaged by the rifling, thereby centering the projectile in the bore.
(e) The bore is the rifled portion of the tube (lands and grooves). It extends from the forcing cone to the muzzle. The rifled portion of the tube imparts spin to the projectile increasing stability in flight. The grooves are the depressions in the rifling. The lands are the raised portions. These parts engrave the rotating band. All United States (US) howitzers have a right-hand twist in rifling.
(f) The bore evacuator is located on enclosed, self-propelled howitzers with semiautomatic breech mechanisms. It prevents contamination of the crew compartment by removing propellant gases from the bore after firing. The bore evacuator forces the gases to flow outward through the bore from a series of valves enclosed on the tube.
(g) The counterbore is the portion at the front of the bore from which the lands have been removed to relieve stress and prevents the tube from cracking.
(h) The muzzle brake is located at the end of the tube on some howitzers. As the projectile leaves the muzzle, the high-velocity gases strike the baffles of the muzzle brake and are deflected rearward and sideways. When striking the baffles, the gases exert a forward force on the baffles that partially counteracts and reduces the force of recoil.
(3) The projectile body has several components that affect ballistics. (See Figure 3-2.) Three of these affect interior ballistics--the bourrelet the rotating band and the obturating band.
(a) The bourrelet is the widest part of the projectile and is located immediately to the rear of the ogive. The bourrelet centers the forward part of the projectile in the tube and bears on the lands of the tube. When the projectile is fired, only the bourrelet and rotating band bear on the lands of the tube.
(b) The rotating band is a band of soft metal (copper alloy) that is securely seated around the body of the projectile. It provides forward obturation (the forward gas-tight seal required to develop pressure inside the tube). The rotating band prevents the escape of gas pressure from around the projectile. When the weapon is fired, the rotating band contacts the lands and grooves and is pressed between them. As the projectile travels the length of the cannon tube, over the lands and grooves, spin is imparted. The rifling for the entire length of the tube must be smooth and free of burrs and scars. This permits uniform seating of the projectile and gives a more uniform muzzle velocity.
(c) The obturating band is a plastic band on certain projectiles. It provides forward obturation by preventing the escape of gas pressure from around the projectile.
(4) The sequence that occurs within the cannon tube is described below.
(a) The projectile is rammed into the cannon tube and rests on the bourrelet. The rotating band contacts the lands and grooves at the forcing cone.
(b) The propellant is inserted into the chamber.
(c) The propellant explosive train is initiated by the ignition of the primer. This causes the primer, consisting of hot gases and incandescent particles, to be injected into the igniter. The igniter burns and creates hot gases that flow between the propellant granules and ignite the granule surfaces; the igniter and propellant combustion products then act together, perpetuating the flame spread until all the propellant granules are ignited.
(d) The chamber is sealed, in the rear by the breech and obturator spindle group and forward by the projectile, so the gases and energy created by the primer, igniter, and propellant cannot escape. This results in a dramatic increase in the pressure and temperature within the chamber. The burning rate of the propellant is roughly proportional to the pressure, so the increase in pressure is accompanied by an increase in the rate at which further gas is produced.
(e) The rising pressure is moderated by the motion of the projectile along the barrel. The pressure at which this motion begins is the shot-start pressure. The projectile will then almost immediately encounter the rifling, and the projectile will slow or stop again until the pressure has increased enough to overcome the resistance in the bore. The rotating band and obturating band (if present) or the surface of the projectile itself, depending on design, will be engraved to the shape of the rifling. The resistance decreases, thereby allowing the rapidly increasing pressure to accelerate the projectile.
(f) As the projectile moves forward, it leaves behind an increasing volume to be filled by the high-pressure propellant gases. the propellant is still burning, producing highpressure gases so rapidly that the motion of the projectile cannot fully compensate. As a result, the pressure continues to rise until the peak pressure is reached. The peak pressure is attained when the projectile has traveled about one-tenth of the total length of a full length howitzer tube.
(g) The rate at which extra space is being created behind the rapidly accelerating projectile then exceeds the rate at which high-pressure gas is being produced; thus the pressure begins to fall. The next stage is the all-burnt position at which the burning of the propellant is completed. However, there is still considerable pressure in the tube; therefore, for the remaining motion along the bore, the projectile continues to accelerate. As it approaches the muzzle, the propellant gases expand, the pressure falls, and so the acceleration lessens. At the moment the projectile leaves the howitzer, the pressure will have been reduced to about one sixth of the peak pressure. Only about one-third of the energy developed pushes the projectile. The other two-thirds is absorbed by the recoiling parts or it is lost because of heat and metal expansion.
(h) The flow of gases following the projectile out of the muzzle provides additional acceleration for a short distance (transitional ballistics), so that the full muzzle velocity is not reached until the projectile is some distance beyond the muzzle. The noise and shock of firing are caused by the jet action of the projectile as it escapes the flow of gases and encounters the atmosphere. After this, the projectile breaks away from the influence of the gun and begins independent flight.
(i) This entire sequence, from primer firing to muzzle exit, typically occurs within 15 milliseconds but perhaps as much as 25 milliseconds for a large artillery howitzer.
(5) Pressure travel curves are discussed below.
(a) Once the propellant ignites, gases are generated that develop enough pressure to overcome initial bore resistance, thereby moving the projectile. Two opposing forces act on a projectile within the howitzer. The first is a propelling force caused by the high-pressure propellant gases pushing on the base of the projectile. The second is a frictional force between the projectile and bore, which includes the high resistance during the engraving process, that opposes the motion of the projectile. The peak pressure, together with the travel of the projectile in the bore (pressure travel curve), determines the velocity at which the projectile leaves the tube.
(b) To analyze the desired development of pressure within the tube, we identify three types of pressure travel curves:
* An elastic strength pressure travel curve represents the greatest interior pressure that the construction of the tube (thickness of the wall of the powder chamber, thickness of the tube, composition of the tube or chamber, and so on) will allow. It decreases as the projectile travels toward the muzzle because the thickness of the tube decreases.
* A permissible pressure travel curve mirrors the elastic strength pressure travel curve and accounts for a certain factor of safety. It also decreases as the projectile travels through the tube because tube thickness decreases.
* An actual pressure travel curve represents the actual pressure developed during firing within the tube. Initially, pressure increases dramatically as the repelling charge explosive train initiated and the initial resistance of the rammed projectile is overcome. After that resistance is overcome, the actual pressure gradually decreases because of the concepts explained by Boyle's Law. (Generally, as volume increases, pressure decreases.) The actual pressure should never exceed the permissible pressure.
Figure 3-3 depicts different actual pressure travel curves that are discussed below.
* Initial Excessive Pressure. This is undesirable pressure travel curve. It exceeds the elastic strength pressure and permissible pressure. Causes of this travel curve would be an obstruction in the tube, a dirty tube, an "extra" propellant placed in the chamber, an unfuzed projectile, or a cracked projectile.
* Delayed Excessive Pressure. This is an undesirable pressure travel curve. It exceeds the elastic strength pressure and remissible pressure. Causes that would result in this travel curve would be using wet powder or powder reversed.
* Desirable Pressure Travel Curve. This curve does not exceed permissible pressure. It develops peak pressure at about one-tenth the length of the tube.
(6) The following general rules show how various factors affect the velocity performance of a weapon projectile family-propellant type-charge combination:
(a) An increase in the rate of propellant burning increases the resulting gas pressure developed within the chamber. An example of this is the performance of the multiperforated propellant grains used in white bag (WB) propellants. The result is that more gases are produced, gas pressure is increased, and the projectile develops a greater muzzle velocity. Damage to propellant grains, such as cracking and splitting from improper handling, also affect the rate of burn and thus the muzzle velocity.
(b) An increase in the size of the chamber without a corresponding increase in the amount of propellant decreases gas pressure; as a result, muzzle velocity will be less (Boyles Law).
(c) Gas escaping around the projectile decreases chamber pressure.
(d) An increase in bore resistance to projectile movement before peak pressure increases the pressure developed within the tube. Generally, this results in a dragging effect on the projectile, with a corresponding decrease in the developed muzzle velocity. Temporary variations in bore resistance can be caused by excessive deposits of residue within the cannon tube and on projectiles and by temperature differences between the inner and outer surfaces of the cannon tube.
b. Standard Muzzle Velocity.
(1) Applicable firing tables list the standard value of muzzle velocity for each charge. These standard values are based on an assumed set of standard conditions. These values are points of departure and not absolute standards. Essentially, we cannot assume that a given weapon projectile family-propellant type-charge combination when fired will produce the standard muzzle velocity.
(2) Velocities for each charge are indirectly established by the characteristics of the weapons. Cannons capable of high-angle fire (howitzers) require a greater choice in the number of charges than cannons capable of only low-angle fire (guns). This choice is necessary to achieve range overlap between charges in high-angle fire and the desired range-trajectory combination in low-angle fire. Other factors considered are the maximum range specified for the weapon, the maximum elevation and charge, and the maximum permissible pressure that the weapon can accommodate.
(3) Manufacturing specifications for ammunition include a requirement for velocity performance to meet certain tolerances. Ammunition lots are subjected to test firings, which include measuring the performance of a tested lot and comparing it to the performance of a control (reference) lot that is tested concurrently with the same weapon. An assumption built into the testing procedure is that both lots of ammunition will be influenced in the same manner by the performance of the tube. This assumption, although accurate in most instances, allows some error to be introduced in the assessment of the performance of the tested lot of propellant. In field conditions, variations in the performance of different projectile or propellant lots can be expected even though quality control has been exercised during manufacturing and testing of lots. In other words, although a howitzer develops a muzzle velocity that is 3 meters per second greater (or less) than standard with propellant lot G, it will not necessarily be the same with any other propellant lot. The optimum method for determining ammunition performance is to measure the performance of a particular projectile family-propellant lot-charge combination (calibration). However, predictions of the performance of a projectile family-propellant lot-charge group combination may be inferred with the understanding that they will not be as accurate as actual performance measurements.
c. Factors Causing Nonstandard Velocities. Nonstandard muzzle velocity is expressed as a variation (plus or minus so many meters per second) from the accepted standard. Round-to-round corrections for dispersion cannot be made. Each of the following factors that cause nonstandard conditions is treated as a single entity assuming no influence from related factors.
(1) Velocity trends. Not all rounds of a series fired from the same weapon and using the same ammunition lot will develop the same muzzle velocity. Under most conditions, the first few rounds follow a somewhat regular pattern rather than the random pattern associated with normal dispersion. This phenomenon is called velocity trends (or velocity dispersion), and the magnitude varies with the cannon, charge, and tube condition at the time each round is fired. Velocity trends cannot be accurately predicted; thus, any attempt to correct for the effects of velocity trends is impractical. Generally, the magnitude and duration of velocity trends can be minimized when firing is started with a tube that is clean and completely free of oil. (See Figure 3-4.)
(2) Ammunition lots. Each ammunition, projectile, and propellant lot has its own mean performance level in relation to a common weapon. Although the round-to-round variations within a given lot of the same ammunition (ammo) types are similar, the mean velocity developed by one lot may differ significantly in comparison to that of another lot. With separate-loading ammunition, both the projectile and propellant lots must be identified. Projectile lots allow for rapid identification of weight differences. Although other projectile factors affect achieved muzzle velocity (such as, diameter and hardness of rotating band), the cumulative effect of these elements generally does not exceed 1.5 m/s. As a matter of convenience and speed, they are ignored in the computation of firing data.
(3) Tolerances in new weapons. All new cannons of a given caliber and model will not necessarily develop the same muzzle velocity. In a new tube, the mean factors affecting muzzle velocity are variations in the size of the powder chamber and the interior dimensions of the bore. If a battalion equipped with new cannons fired all of them with a common lot of ammunition a variation of 4 meters per second between the cannon developing the greatest muzzle velocity and the cannon developing the lowest muzzle velocity would not be unusual. Calibration of all cannons allows the firing unit to compensate for small variations in the manufacture of cannon tubes and the resulting variation in developed muzzle velocity. The MVV caused by inconsistencies in tube manufacture remains constant and is valid for the life of the tube.
(4) Tube wear. Continued firing of cannon wears away portions of the bore by the actions of hot gases and chemicals and movement of the projectile within the tube. These erosive actions are more pronounced when higher charges are fired. The greater the tube wear, the more the muzzle velocity decreases. Normal wear can be minimized by careful selection of the charge and by proper cleaning of both the tube and the ammunition.
(5) Nonuniform ramming. Weak ramming decreases the volume of the chamber and thereby theoretically increases the pressure imparted to the projectile. This occurs because the pressure of a gas varies inversely with volume. Therefore, only a partial gain in muzzle velocity might be achieved. Of greater note is the improper seating of the projectile within the tube. Improper seating can allow some of the expanding gases to escape around the rotating band of the projectile and thus result in decreased muzzle velocity. The combined effects of a smaller chamber and escaping gases are difficult to predict. Weak, nonuniform ramming results in an unnecessary and preventable increase in the size of the dispersion pattern. Hard, uniform ramming is desired for all rounds. When semifixed ammunition is fired, the principles of varying the size of the chamber and escape of gases still apply, particularly when ammunition is fired through worn tubes. When firing semifixed ammunition, rearward obturation is obtained by the expansion of the cartridge case against the walls of the powder chamber. Proper seating of the cartridge case is important in reducing the escape of gases.
(6) Rotating bands. The ideal rotating band permits proper seating of the projectile within the cannon tube. Proper seating of the projectile allows forward obturation, uniform pressure buildup, and initial resistance to projectile movement within the tube. The rotating band is also designed to provide a minimum drag effect on the projectile once the projectile overcomes the resistance to movement and starts to move. Dirt or burrs on the rotating band may cause improper seating. This increases tube wear and contributes to velocity dispersion. If excessively worn, the lands may not engage the rotating band well enough to impart the proper spin to the projectile. Insufficient spin reduces projectile stability in flight and can result in dangerously erratic round performance. When erratic rounds occur or excessive tube wear is noted, ordnance teams should be requested to determine the serviceability of the tube.
(7) Propellant and projectile temperatures. Any combustible material burns more rapidly when heated before ignition. When a propellant burns more rapidly than would be expected under standard conditions, gases are produced more rapidly and the pressure imparted to the projectile is greater. As a result, the muzzle velocity will be greater than standard and the projectile will travel farther. Table E in the tabular firing tables lists the magnitude of change in muzzle velocity resulting from a propellant temperature that is greater or less than standard. Appropriate corrections can be extracted from that table; however, such corrections are valid only if they are determined relative to the true propellant temperature. The temperature of propellant in sealed containers remains fairly uniform though not necessarily at the standard propellant temperature (70 degrees Fahrenheit [F]). Once propellant has been unpacked, its temperature more rapidly approaches the air temperature. The time and type of exposure to the weather result in temperature variations from round to round and within the firing unit. It is currently impractical to measure propellant temperature and apply corrections for each round fired by each cannon. Positive action must be taken to maintain uniform projectile and propellant temperatures. Failure to do this results in erratic firing. The effect of an extreme change in projectile or propellant temperature can invalidate even the most recent corrections determined from a registration.
(a) Ready ammunition should be kept off the ground and protected from dirt, moisture, and direct rays of the sun. At least 6 inches of airspace should be between the ammunition and protective covering on the sides, 6 inches of dunnage should be on the bottom, and the roof should be 18 inches from the top of the stack. These precautions will allow propellant and projectile temperatures to approach the air temperature at a uniform rate throughout the firing unit.
(b) Propellant should be prepared in advance so that it is never necessary to fire freshly unpacked ammunition with ammunition that has been exposed to weather during a fire mission.
(c) Ammunition should be fired in the order in which it was unpacked.
(d) Propellant temperature should be determined from ready ammunition on a periodic basis, particularly if there has been a change in the air temperature.
(8) Moisture content of propellant. Changes in the moisture content of propellant are caused by improper protection from the elements or improper handling of the propellant. These changes can affect muzzle velocity. Since the moisture content cannot be measured or corrected for, the propellant must be provided maximum protection from the elements and improper handling.
(9) Position of propellant in the chamber. In fixed and semifixed ammunition the propellant has a relatively fixed position with respect to the chamber, which is formed by the cartridge case. In separate-loading ammunition, however, the rate at which the propellant burns and the developed muzzle velocity depends on how the cannoneer inserts the charge. To ensure proper ignition of the propellant he must insert the charge so that the base of the propellant bag is flush against the obturator spindle when the breech is closed. The cannoneer ensures this by placing the propellant flush against the Swiss groove (the cutaway portion in the powder chamber). The farther forward the charge is inserted, the slower the burning rate and the lower the subsequent muzzle velocity. An increase in the diameter of the propellant charge can also cause an increase in muzzle velocity. Loose tie straps or wrappings have the effect of increasing the diameter of the propellant charge. Propellant charge wrappings should always be checked for tightness, even when the full propellant charge is used.
(10) Weight of projectile. The weights of like projectiles vary within certain zones (normally termed square weight). The appropriate weight zone is stenciled on the projectile (in terms of so many squares). Some projectiles are marked with the weight in pounds. In general terms, a heavier-than-standard projectile normally experiences a decrease in muzzle velocity. This is because more of the force generated by the gases is used to overcome the initial resistance to movement. A lighter-than-standard projectile generally experiences an increase in velocity.
NOTE: Copperhead projectiles are not marked with weight in pounds. The precision manufacturing process used guarantees a weight of 137.6 pounds.
(11) Coppering. When the projectile velocity within the bore is great, sufficient friction and heat are developed to remove the outer surface of the rotating band. Material left is a thin film of copper within the bore and is known as coppering. This phenomenon occurs in weapons that develop a high muzzle velocity and when high charges are fired. The amount of copper deposited varies with velocity. Firing higher charges increases the amount of copper deposited on the bore surfaces, whereas firing lower charges reduces the effects of coppering. Slight coppering resulting from firing a small sample of rounds at higher charges tends to increase muzzle velocity. Erratic velocity performance is a result of excessive coppering whereby the resistance of the bore to projectile movement is affected. Excessive coppering must be removed by ordnance personnel.
(12) Propellant residue. Residue from burned propellant and certain chemical agents mixed with the expanding gases are deposited on the bore surface in a manner similar to coppering. Unless the tube is properly cleaned and cared for, this residue will accelerate tube wear by causing pitting and augmenting the abrasive action of the projectile.
(13) Tube conditioning. The temperature of the tube has a direct bearing on the developed muzzle velocity. A cold tube offers a different resistance to projectile movement and is less susceptible to coppering, even at high velocities. In general, a cold tube yields more range dispersion; a hot tube, less range dispersion.
(14) Additional effects in interior ballistics. The additional effects include tube memory and tube jump.
(a) Tube memory is a physical phenomenon of the cannon tube tending to react to the firing stress in the same manner for each round, even after changing charges. It seems to "remember" the muzzle velocity of the last charge fired. For example, if a fire mission with charge 6 M4A2 is followed by a fire mission with charge 4 M4A2, the muzzle velocity of the first round of charge 4 may be unpredictably higher. The inverse is also true.
(b) Tube jump occurs as the projectile tries to maintain a straight line when exiting the muzzle. This phenomenon causes the tube to jump up when fired and may cause tube displacement.
Appendix (d)
(http://arxiv.org/ftp/physics/papers/0601/0601149.pdf)
INTRODUCTION
he long throw-in is an important "set play" in soccer, particularly when used as an attacking manoeuvre near to the goal mouth. The farther a player can throw the ball the larger the area in which his/her team mates may receive the ball and the greater the scoring opportunities. To produce a long throw the player must project the ball at high speed and at an appropriate angle with respect to the horizontal. In laboratory studies of the throw-in, male players have recorded release speeds of 12-19 m/s and release angles of 22-40º. Release speed is the primary determinant of the range attained by a projectile, and to achieve the greatest possible distance a soccer player should release the ball with the greatest possible speed. Identifying the optimum release angle for a long throw-in is less straightforward.
Physical properties of the ball
Although the effects of the physical properties of the ball on the throw distance and the optimum release angle have not been investigated, the range of values allowed by the rules is relatively narrow. We therefore expect to see only small changes in the throw distance and optimum release angle arising from changes in the physical properties of the ball. In any case, the physical properties are to a large extent fixed once the ball to be used in the match is selected, and they cannot easily be manipulated by the player in an effort to increase the distance of a throw.
Participant and throwing protocol
One collegiate male soccer player (age 21 years; height 1.77 m) was recruited to participate in the study. The study was approved by the Human Ethics Committee of Brunel University, the participant was informed of the protocol and procedures prior to his involvement, and written consent to participate was obtained. The throws were conducted in still air conditions in an outdoor stadium using a FIFA approved match ball. All throws were performed from a flat
synthetic surface and the landing area was level with the release surface. The participant wore athletic training clothes and sports shoes. According to FIFA competition rules the player must be facing the field of play when
coefficient is expected to increase at a rate of about 0.01 per 1 rev/s increase in spin rate
The calculated effects of backspin on the player's maximum throw distance and optimum release angle are shown in Figure 8. Most of the effects are due to the increase in the lift coefficient. The increase in lift coefficient tends to produce a longer throw and a lower optimum release angle, whereas the increase in drag coefficient tends to produce a shorter throw and a lower optimum release angle. A spin of 3 rev/s is probably close to the maximum that a player can achieve in a throw-in. If a player were to achieve a spin rate of 3 rev/s th
e throw distance would be increased by about 1.2 m (8%) and the optimum release angle would be lowered by about 3°. In our calculations the throw distance reaches a maximum at a spin rate of about 5 rev/s. At higher spin rates the gain in distance due to the greater lift is outweighed by the loss in distance due to greater drag.
Figure 8 Calculated effect of backspin on the throw distance and optimum release angle.
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