After thorough research on the internet and books, I found that a moving ball has both linear and rotational kinetic energy. This new kind of energy is called Rotational Kinetic Energy or RKE. To make a body which is initially at rest start rotating about a fixed axis, it is necessary to apply a torque to the body. The torque does work on the body and in the absence of friction; the work done increases the KE of the body. Therefore the body rotates faster and faster. The kinetic energy of a rotating body is given by:
RKE = ½ Iw2
“I” is its Moment of Inertia about the given axis and “w” is its angular speed, hence the new advanced formula is:
mgh = ½mv2 + ½ Iw2
For any solid spherical object, “I” is given by:
I = 2/5 mr2
In addition we also know that:
v = wr
w = v / r
As a result, the new calculation is:
mgh = [½mv2 ]+ [½ (2/5)(m)( r2)] * [v2 / r2]
2gh = v2 + 2/5v2
2gh = v2 (1 + 2/5)
v2 = 2gh * 5 / 7
v = √ (10gh / 7)
The time of flight remains the same and ultimately the more precise formula is:
Range = √ (10gh / 7) * √ (2s / g)
Range = √ (20sh / 7)
Plan:
Before starting my investigation, I will set up a plan in which all my work will be based on. This plan is summarized in the following points:-
- I will first set up my experiment, and take all measurements required.
- I will take these measurements, and place them in tables.
- Then I will take these results and draw a graph of them and analyse them, and find from them the equation of range, and compare the results with the theoretical equations.
Apparatus:
- Plastic Ramp.
- Ball.
- Sand Tray.
- Set of Rulers.
- Stool.
- Support Pillar.
I used the plastic ramp as the course in which the ball will take while sliding down. The ball is the ski jumper, and I used a sand tray to measure the landing spot of the ball.
Rulers where used to measure the drop height on the ramp, and the height of tables. The pillars where used to keep the ramp in its place so that it wouldn’t affect the results later on.
Diagram:
Method:
- First of all I collected all the equipments need for my experiment, and started setting up the apparatus as shown in the picture above.
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The second step is to take measurements. We will need to measure the height of the table from the ground and the height of the sand from the ground as well. By subtracting the height of the ramp and the sand from the ground, we could find out the drop height mentioned earlier on and is indicated to “S” in the above diagram.
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The value I obtained for “S” was 38.5 cm (or 0.385 m).
- Then we placed the stool away from the edge of the table and it’s indicated as “A” in the above diagram.
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The value “A” depends upon dropping the ball at the lowest and highest height on the slope. If the ball lands - in the two cases- in the sand tray then the value “A” is measured.
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Then we will use the value of “A”, to add it to the value shown on the sand. For example: If A = 10cm, and I dropped a ball and the value that I read on the sand tray was 5cm, therefore the total range would be 10cm + 5cm = 15cm (or 0.15m).
- I also used a protractor to measure the angle of the slope.
- After getting the apparatus ready, we begin our experiment. We grade the slope using a ruler to identify where we will drop the ball. The ball is dropped from heights that are measured up the ramp by aligning an 80 cm-long ruler with 10 cm intervals to the slope. To do this, we hold the ball at the height on the ramp and release it. It will then travel down and into the sand tray leaving a mark similar to half a sphere.
- We then measure from the centre of the mark to the end of the sand tray closer to the edge of the table. By adding this value we obtain with 15cm, we will acquire the final value of the range.
- I have also then applied the same techniques to different angles, so that I can have a good variety of data.
Safety:
There was no need to take safety precautions while conducting this experiment as it does not entail any risk.
Preliminary Tests and Experiment Errors:
After setting up my experiment, I did some preliminary tests to insure that my experiment is set up perfect and using the best equipments, this will help increase the accuracy of my results.
The first problem I encountered was setting the ramp in the right position. At first, it was difficult for me to hold the ramp to the table. I used scotch tape to hold it to the table but I found that it could obliterate the results because while the ball is moving down the ramp it could alter the movement of the ball, thus affecting the results. I, therefore, used an adhesive putty which works by sticking this material to the ramp and then to table. This method worked extremely well.
Once the ramp was set, I began my trial run. I tried using balls of different sizes and found that the smaller one seems to hit the target better than larger ones. In other words, when you dropped a ball with large mass, the result is more gravity pushing it down while travelling in the air, so the difference between dropping is at 10 cm up the ramp and 20 cm will not make much difference in the range. So I decided to use a smaller ball with smaller mass so that the results can be spaced apart.
The height of the stool seemed to me a little high. When I dropped the ball from a higher level, it jumped over the other side without landing in the sand tray, so I decided to use a smaller stool, with shorter height. This perfectly matched what I needed to do in my experiment.
The final check up was to add more sand to the sand tray; because I saw that the ball was rolling without even leaving a good sign to read. For that reason I add more sand to the sand tray which resulted that the sand would be able to absorb more of the ball’s kinetic energy.
There were some errors in my experiment which had to be solved; one of the main errors is the ruler. I used a 30 cm ruler to measure the distance travelled by the ball on sand but I found that the edge of the ruler didn’t start from 0 cm, but 0.5 cm. This kind of error was solved by subtracting 0.5 cm from all the results.
The other kind of error, which I encountered in my experiment, is dropping the ball. When I use to hold the ball in a certain position up the ramp, sometimes it used to give me some strange measurements on the sand tray; I discovered that these uncertainties were caused by a push which is applied by my hand. This problem was solved by using a ruler to hold the ball in position, then by removing the ruler from its place the ball will roll down without any other force has been applied to it.
Results:
Angle 34º
Angle 20º
Angle 15º
Comment on results:
As you can see, the results of my experiment were entered in the above tables. These tables consist of:
- Height up the ramp.
- Range.
- Velocity.
- Time of flight.
- Predicted range (with and without RKE).
Height of ramp and range were calculated from the experiment and entered in the tables, while the velocity was calculated using the theoretical equation.
The time of flight was also calculated and found to be constant, because the time is fixed to the drop height “S”.
The Predicted ranges with RKE and without RKE were also calculated using the theoretical equations. And the one “with RKE” should be compared to the equation which I will find from my graphs, this way I can ensure the success of my experiment.
Analysing the graphs:
From my results, I have done a couple of graphs that can, help and provide equations that may help prove the success of my work.
Using the theoretical equations, it became clear to me that the relationship between the dropping height on slope and the range is a square root relationship. And in order to show and prove that this relationship also applies to my results, I have plotted first of all the dropping height on slope against the range. Then if we have a square root relationship and squared root the values on the x-axis, we should get a straight line.
Analysing graph -1- for angle 34º (page ), we can clearly see that it’s a square root relationship, this is because the theoretical equation told me that r = √ (20sh / 7), and by substituting in s, which is equal to 0.385 m, therefore r = 1.0488√ h. From this equation we can conclude, that r is proportional to the square root of h.
This relationship means that as the greater the height dropped on the slope, the greater the range. This is because greater the drop is the more gravitational potential energy is converted into kinetic energy. Also the greater the height of drop, the ball gains more horizontal velocity before leaving the ramp.
If you compare my results with the results I got from the theoretical equation, you will find a big difference this is because the way our theoretical results are calculated uses formula which assumes no energy is wasted. And this difference can be seen from the results in the tables, were if you compare the Actual range with the Predicted range, you will see that our actual result is lower than the Predicted range.
In graph -2- for angle 34º, I have plotted the root of the dropping height on the slope, against the range. What we were expecting was a straight line which goes through the origin. This is because if we have a square root relationship and we squared root the values on the x-axis we should get a straight line.
What we expected was exact, and the points were joined together to give us a straight line, but didn’t cross at the origin.
At this stage I started thinking why didn’t it cross at the origin?
This question was clarified, and found that some factors might have affected the range while doing the experiment.
Why crossing through the origin:
Y=mx + c
r = gradient * h + (range when root of dropping height on ramp is equal to zero)
When we say that the straight line does not cross at origin, we mean that there will be a range when the root of dropping height on slope is equal. And by thinking of this idea, it sounds bizarre, because we can’t have a range if we don’t have a height.
Analysing the graphs (continue):
Two more different angles were investigated in addition to angle 34º, these two angles where done in order to prove that the graph will cross through the origin.
The result of graphs 4 and 6 were positive, which means that they did cross through the origin.
Comparing the Equations:
After plotting the results for the square root of dropping height on slope against the range on a graph paper, I did some calculations to find the equation of the straight line.
Angle 34º Angle 20º Angle 15º
r = 0.4√h + 0.0954 r = 1.0125√h – 0.0614 r = 0.7√h -0.00018
The theoretical equation is:
r = 1.0488√ h
If we compare the above equations with the theoretical one, we will see that there isn’t much big different between them. The closest one to the theoretical equation is the equation of the straight line of angel 20º. Therefore the graph of angle 20º can be considered the most appropriate.
Factors that could have affected the results:
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The slope: Because I used a plastic ramp, I discovered that as I place the ball on the ramp, the ramp vibrates because it’s a plastic ramp like shown in the picture and no holder to hold it from all points. This is one of the first factors I thought that they might affect the range.
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Angle of ramp: The launch angle is another factor affecting the range. The greater the angle of ramp, the ball will accelerate more; in other words, it gives the particle a vertical velocity component rather than just a horizontal component. Therefore a greater angle causes an increase in the range and vice versa.
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Vertical dropping height on slope “h”: As I said before if we assume that no energy is wasted, therefore we are saying that GPE lost causes a gain in KE, eventually, the greater the height dropped the greater the velocity it has when leaving the ramp and consequently greater the range.
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Drop height “S”: Theoretically the time of the jump is equal to the time it takes the ball to reach the sand tray, if dropped vertically. So this information tells us that for a shorter stool the time is more. In other words, more time more distance to travel, less time less distance to travel, thus affecting the range.
Conclusion
From my predicted range I have shown the relationship between the dropping height on slope and the range. Although I new from the predicted range that it’s going to be a square root relationship, as shown below.
I still had to prove that my results are correct. So I obtained three different results, for three different angles -34º, 20º and 15º- which I have plotted the graph of dropping height on slope against range for each angle, and the root of dropping height on slope against the range.
For angle 34º, I had a problem with the second graph –which is the root of dropping height on slope against range-; the straight line didn’t cross at the origin. I concluded that because I was using my hand to drop the ball, I may have applied a force to the ball, this force have caused a change in range.
For angles 20º and 15º, I used a ruler so that it can hold the ball in place, then by removing the ruler the ball will slide down without any force applying to the ball (hand force). Therefore, my graphs for angle 20º and 15º turned to be positive, and the second graph did cross at the origin.
Also I can say that the theoretical equation for the range with Rotational Kinetic Energy is the most suitable for my experiment.
Range = √ (20sh / 7)
Where”s” in the equation is the drop height and “h” is the vertical dropping height on the slope, as a result we can see that as “h” increases, whereas other values remain constant, the magnitude of the range would increase.
Evaluation:
The experiment generally went according to plans with no significant problems to speak of. The results I obtained were very satisfying except for angle 34º which appeared to produce some bizarre results because I dropped the ball with my hand, which must have given it a push. But that did not affect the general outcome as the results obtained from dropping the ball when the ramp was at angles 15° and 20° which gave perfect results.
Because it is a practical investigation, then equations have to be derived from the experiment results. And in order to make sure that these equations make sense, they must be compared to other theoretical equations which are used for comparison reasons only.
The equation obtained from the experiment matched the theoretical equation when the slope was at 20° by up to 96.5%. If we take into consideration human error, friction and air resistance, this result can be considered a perfect one.
My aim in this investigation was to determine the relationship between the range of the jump achieved by the ski jumper and the dropping height on the ramp; this was done by establishing the equations of the straight line which was calculated from the graphs of root of dropping height on slope against range. These equations were:
Angle 34º Angle 20º Angle 15º
r = 0.4√h + 0.0954 r = 1.0125√h – 0.0614 r = 0.7√h -0.00018
Now if you look at these equations, you will see that the range is directly proportional to the root of h (which is the dropping height on slope). Therefore we can say that the relationship between the range of the jump achieved by the ski jumper and the dropping height on the ramp is a square root relationship.
Bibliography:
Modern physics – by Frederick E. Trinklein (information about GPE and KE)
http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html (RKE equation)
http://physics.about.com/cs/puzzles/a/070603_2.htm (information about RKE)