Kinetic energy = ½ mv2
In the equation ‘m’ represents mass and ‘v’ represents velocity. The mass of the object will be easy to obtain however, the velocity in which the object travels will be difficult without the aid of expensive equipment. I will have to take into consideration the idea of projectile motion.
In projectile motion, the displacement, velocity and acceleration of the projectile, in my case the ball bearing, are all vectors. The horizontal components can be treated as separate to the vertical components. If the ball laves the catapult horizontally, the initial velocity vector has no vertical component, in the vertical direction the force of gravity provides a uniform acceleration, which I will consider as 9.81 m/s2. In the horizontal direction there is no component of the gravitational force, so there is no acceleration and the velocity therefore remains equal to its initial value. The horizontal and vertical components combine to give the projectile a trajectory that has the shape of a parabola. The only manual way, therefore in which the value for the velocity of the object can be obtained is to use the following equations,
Speed = Distance x Time
s = ut + ½ at2
(v = final velocity, u = initial velocity, a = acceleration, t = time, s = displacement)
I am trying to find the speed of the ball bearing as it leaves the catapult so I could use the well known equation ‘speed = distance x time’. However I would first need to obtain a value for time by using the equation of uniform acceleration ‘s = ut + ½ at2’ as I have values for all the factors of this equation except of course ‘t’. The value for ‘u’ we know is 0 m/s as we are measuring the vertical components, ‘a’ therefore will equal the gravitational force of 9.81 m/s2 and ‘s’ will equal the height at which the ball bearing is being projected. I could then substitute this value of ‘t’ into my equation ‘speed = distance x time’, giving me a value for speed which can be used in the kinetic energy equation to give a value to be substituted in as the ‘useful energy transferred’ in the efficiency equation.
Accuracy & Sensitivity Considerations:
Originally, I was planning on firing my catapult vertically upwards however, there were certain factors of the experiment which I considered and made me alter my initial idea. I was planning to conduct a similar method of finding efficiency as my current plan, I was going to find the kinetic and elastic potential and substitute them into the efficiency equation. The difference was that I was going to use the equation of uniform acceleration to obtain a value for initial velocity. The equation I would have used would have been ‘v = u + at’ which could be rearranged to ‘u = v – at’. I knew that if I was firing vertically upwards the acceleration would have been –9.81m/s2 because the ball would be travelling against the pull of gravity. I also knew that final velocity (v) would have been 0m/s as the ball reached it’s peak. I would therefore only have to obtain a value for time (t) by using a timer to record the time the ball was in the air. I would then only have to substitute my values into the above equation to find initial velocity (u), and add this into the equation to find kinetic energy which I could then use to find the efficiency. However, I thought that this would be too inaccurate in comparison to firing horizontally for many reasons. My calculation would only be accurate if the ball was fired exactly vertically upwards, this would have been very difficult to do manually and would have led to inaccuracies. I would have had to control the timer, starting from the release and stopping as soon as I heard the sound of the ball hitting the floor, and the human error involved in that would have meant very inaccurate results. If the ball had been fired vertically it would have also been affected by wind that would have delayed or increased the time period the ball was in the air. I also considered the safety hazard that if the metal ball bearing went out of sight it could cause serious injury if it hit someone below. I therefore decided to alter my methods and shoot the catapult horizontally due to accuracy and safety reasons.
In the planning of this experiment there are many considerations I have taken into account concerning accuracy and sensitivity. For example when I take the values for force and extension of the elastic I will be taking 13 recordings as I believe this is a suitable amount which will allow me to plot an accurate force-extension graph. I also took into consideration the weights I would be using in order to check the extension of the elastic. Each weight is said to be 100g however I decided to check the weight of them and record the weight to 2 decimal places so as I could plot a more accurate graph. This is why I have recorded both the ‘force’ and the ‘true force’ in my table of force and extension measurements below. I chose scales which I though would be sensitive enough to give accurate measurements of both the ball bearing and the weights which I measured. I selected scales which gave a reading in grams to 2 decimal places. This I believe will be an appropriate degree of accuracy for the sensitivity of this experiment. In recording the readings I have taken of the distance which the catapult projected the ball bearing, I will make sure that I have made 15 repeats of the firing and therefore obtained 15 different distances which I will record in metres to 3 decimal place. This will provide me with an appropriate amount of results in order to make a accurate average of the effective range of the catapult and therefore give me a more accurate value for the elastic potential energy.
Method:
- Collect all apparatus from the list of equipment required below.
- Set up stand and attach boss and clamp, select the elastic of the catapult and attach it to the clamp so as it dangles vertically without any obstruction.
- Add weight in lots of 100g to the end of the elastic, and for each weight put added measure and record the extension of the elastic.
- Continue adding weights until the elastic is holding 1300g in order to obtain enough points to plot an accurate force against extension graph.
- Plot recorded co-ordinates onto a graph with the x-axis representing extension and the y-axis representing force.
- Measure the area underneath the graph to obtain a value for the elastic potential energy, which will then represent ‘total energy transferred’ in the equation for efficiency.
- Attach clamp stands to bench using ‘G’ clamps, setting the stands 30cm apart, attach the elastic to the top of the stands.
- Measure the distance of the catapult from the floor and record value.
- Hold the ball bearing on the elastic and pull back so as the elastic 20cm from its unstretched state. Fire the ball bearing and measure from where the ball was released to the point at which it touched the floor. Repeat this process 15 times in order to get a more accurate average.
- Take an average of the distance it reached by the ball bearing.
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Substitute all known values into the equation of uniform acceleration ‘s = ut + ½ at2 in order to find ‘t’.
- Substitute value for ‘t’ into equation ‘speed = distance x time’ using the value for time obtained and the average distance the catapult fired the ball bearing.
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Substitute this value for speed into the equation for kinetic energy, ½mv2 as well as the value for the mass of the ball bearing which I previously measured as 28.25g.
- Take the values for elastic energy and kinetic energy and put them into the equation: efficiency = total energy transferred (kinetic energy) / useful energy transferred (elastic potential). This will give a value for the efficiency.
- Put value to an appropriate degree of accuracy, I will put it as three significant figures, and then convert value into percentage.
Apparatus:
- Approximately 70cm length of elastic.
- 2 clamp stands
- Boss & clamp
- Ball bearing
- Bench
- Metre stick & chalk
- Weighing scales
- 1300g of 100g weights
Safety:
As I will be conducting experiments catapults and elastic there are safety considerations which I must take into account:
- Goggles must be worn when I am stretching the elastic to find out the extension against the force applied.
- When I set up the catapult I will use a clear, location which will not have any other people present.
- I will make sure the location is spacious so as I do not hit anything when firing the catapult.
- When I launch the ball bearing I will check that no other people are in front of the catapult.
- I will stand behind the catapult when it is being fired.
- I will practice firing the catapult with short extension so as I am aware of the range.
- I will wear goggles whilst conducting my experiment.
- I will make sure no other students are firing their catapults when retrieving my ball bearing.
Results:
Below is the table of recordings showing the values of force against extension of the elastic which I used in my catapult. The table shows the ‘force’ and the ‘true force’ because I considered that the weights would not equal exactly 100g, so I weighed each weight and recorded the true weights which I will plot on my graph giving more accurate points. I then converted the weight to force using the equation ‘Force = Mass x Gravity’, where I considered gravity to be 9.81m/s2. I made my answers to 3 decimal places.
On the graph paper I have plotted a force-extension graph from which the area underneath is the value of elastic potential energy. I found the area of the graph by first calculating the area of one square and then estimating the amount of squares under the graph.
Area of one square = 0.005 x 1
Approximate amount of squares under graph = 446
Area = 446 x 0.005 = 2.23 square units
As I mentioned in my introduction, the area underneath the graph should equal the elastic potential energy of the catapult. Therefore,
Elastic potential energy = 2.23 Joules
The following table shows the values I got from the 15 shots I took with the catapult. The range is measured in metres to 2 decimal places. I have recorded my average to 4 significant figures in order to get as accurate a final answer as possible, when I come to substitute this value into my equation for kinetic energy.
The average distance reached therefore was:
73.45 / 15 = 4.896667
= 4.897 m
In order to get a value for time, so as I can substitute the values of time and distance into my equation to find speed, I must solve the equation of uniform acceleration below.
s = ut + ½ at2
s = 0.91m (distance from the point at which ball would be fired to the floor)
u = 0m/s
a = 9.81m/s2 (equal to gravitational force)
s = ut + ½at2
0.91 = (0 x t) + (½ x 9.81 x t2)
0.91 = ½ x 9.81 x t2
2 x 0.91 = t2
9.81
t = √0.185524
t = 0.43072497
t = 0.431 seconds
After obtaining this value for ‘t’ I can now substitute the values for time and distance (distance being the average range reached by the projectile), into the following equation,
Speed = Distance/ Time
Speed = 4.897/0.431
Speed = 11.361948…
Speed = 11.36 m/s (to 2 d.p)
The value I have found for velocity can now be substituted into the following equation for kinetic energy,
Mass of ball bearing = 0.02825kg
Velocity of ball bearing = 11.36 m/s
Kinetic Energy = ½mv2
Kinetic Energy = ½ x 0.02825 x 11.362
Kinetic Energy = 1.823 J
Kinetic Energy = 1.823 Joules (to 3 d.p)
I now have the two values for kinetic and elastic energy which I can substitute into the efficiency formula in order to obtain the efficiency of my catapult.
Efficiency = Energy usefully transferred
Total energy transferred
Efficiency = Kinetic energy transferred
Elastic potential energy transferred
Efficiency = 1.823 x 100
2.23
Efficiency = 81.7%
Evaluation:
Overall I think that my experiment was a success. The final efficiency of the catapult appears to be rational suggesting that all my calculations were all accurate. However there were some factors of my experiment which led to certain inaccuracies. The main problem came when I thought that an extension of 0.1m would be sufficient but it was not and I had to increase the extension in order to get a feasible set of results for the range which I required in order to obtain the elastic potential energy. This explains why on my graph I have been forced to extrapolate the line by attempting to follow the trend of the points plotted as accurately as possible. This meant that the area underneath the extrapolated section of the graph would have been inaccurate due to the fact that it was a pure estimation. The value of the area was equal to the elastic potential energy which was a factor of the efficiency equation, therefore the inaccuracy of the area would have directly led to inaccuracy of the final efficiency calculation.
There were many factors of my experiment that I considered involving accuracy & sensitivity. For example when I took the values for force and extension of the elastic I took 13 recordings in order to get a more accurate force-extension graph. I also took into consideration the weights I would be using in order to check the extension of the elastic. Each weight is said to be 100g however I decided to check the weight of them and record the weight to 2 decimal places so as I could plot a more accurate graph. I chose scales which I though would be sensitive enough to give accurate measurements of both the ball bearing and the weights which I measured. I selected scales which gave a reading in grams to 2 decimal places. This I thought was an appropriate degree of accuracy for the sensitivity of this experiment. I also made sure that I took 15 recordings of the distance which the catapult projected the ball bearing, which I recorded in metres to 3 decimal place. This was done to provide me with a sufficient amount of results in order to make a accurate average of the effective range of the catapult and therefore give me a more accurate value for the elastic potential energy. All these factors I have considered improved the accuracy of my experiment however, if it would be possible to increase the sensitivity of these factors to obtain a more accurate final result. For example I could round my calculations to 4 or 5 decimal places as oppose to 2 and 3 decimal places, I could use more sensitive scales and take more recordings of distance and extension against force in order to obtain a more precise average. I could have also used mathematical techniques in order to find the exact area under the graph of force – extension instead of estimation method.
Despite some inaccuracies and alterations I have had to make to the experiment which have had a direct effect on the final efficiency answer, I believe that I have deduced from the experiment a rational value for the efficiency of the catapult which was 81.7%.
