Making Assumptions and relating them to both the model and the experiment To be able to use the mechanics theory that I know, the following assumptions must be made, listed in order of importance. All motion is vertical – Although the horizontal motion could be taken into account, it cannot really be measured with the equipment we have, and since it isn’t wanted anyway it is best just to redo trials that result in some horizontal motion.
This requires two further assumptions:o There is no spin – spin causes the movement of the ball in other directions. o The floor is perfectly horizontal – As it not being so would cause the ball to bounce in other directions (although in practise this can’t be helped anyway) There is no air resistance – As I cannot calculate air resistance using the theory I currently know, and the acceleration would no longer be constant. Since the ball is small, spherical and heavy, it should encounter little air resistance and be only lightly affected by it.
The ball is uniform and perfectly spherical – The ball could appear to have different values of e because of having different densities or small deformations at different points on the surface, although the chance of these remaining unnoticed is minimal. Acceleration due to gravity is constantly 9. 8ms-2 – Gravity must be constant so that the constant acceleration formulae can be used. Fortunately the gravity will deviate very little from this value so this assumption makes little difference. Conducting the experiment.
The experiment is set up by attaching two metre rules to a wall using a plumb line to ensure that they are vertical. For the first part of the experiment a ball was dropped from a number of heights, and the height it bounced back up to was recorded. For the second part the ball was dropped from different heights and the time taken for it to bounce three times was measured. Any measurements in which the ball bounced at an angle of more than 10 degrees were ignored, as were any in which the ball hit the wall or when the person dropping the ball felt that they had accidentally given it some spin.
In both parts results were taken for every 10cm between 1 and 2 metres, and 5 results were taken for each height. The heights that the ball bounced to were recorded by the height that the top of the ball reached, as this was easier to measure than the height that the bottom of the ball reached, so the results have been adjusted by 4. 38cm, the diameter of the ball. Initial Height (cm) Height after first bounce (cm) Average (cm) 1.
Height of the First Bounce v=speed when the ball hits the ground (ms-1) u=speed when ball is released (ms-1) a=acceleration due to gravity (ms-2) s=initial distance from the ground (m) v2=u2+2as v2=2as v=V(2as) V=speed when ball has bounced to it’s highest point (ms-1) U=speed when ball leaves the ground (ms-1) A=acceleration due to gravity(ms-2)(negative as gravity is now slowing the ball down).
S=highest point the ball bounces to (m) e=the coefficient of restitution V2=U2+2AS U=eV(2as) 0=2ase2+2AS A=-a 0=2ase2-2aS 2aS=2ase2 S=se2 ? ) Since the graph of T against s0 will be of the form y=kVx it will have the following shape: In other words it will be a slightly curve towards the x-axis. Task 1:Find e Using the dashed lines on the graph, I wish to find the gradient of the line. The vertical line is 10. 09cm long and the horizontal line is 8. 12cm long. Since the two axes have the same units and scale the figures don’t need to be adjusted.
Task 2:Find the time taken up to the third bounce Predicted Results Experimental Results Discussion of variation in the experimental results The fact that the second part of the experiment was so much less precise than the first leads me to believe that the majority of the variation was due to the human error in the use of the stopwatch. I believe it was so large because : 1.
Stopwatches in experiments are usually used to measure times of around 20s so that the effect of their inaccuracy is less important. The experiment is usually adjusted so that such measurements can be possible, but that was not really possible with our experiment as the ball would probably have come to rest within 10s. 2. The stopwatches that we used had a tendency to take 1 button press as three quick simultaneous ones, which could have resulted in about a tenth of a seconds difference either way (as the two unwanted presses add a bit of time on at the end or take a bit off at the beginning.
Most of the rest of the variation was assumably due to accidental differences in the way the ball was dropped, as giving the ball some amount of energy could not be avoided while the ball was being dropped by hand. This energy will partly have been in the form of spin, causing the ball to bounce at a bit of an angle so that the height reached was lower, and partly translational kinetic energy which will have affected how fast and hence how far it went. The graphs for both tasks have error bars to show the spread of the results.
Comparison between the experimental results and the predictions of the model For task 2 I have drawn in the predicted results using the value of e found in task 1. They are pleasingly similar to the experimental results and it would at first seem quite possible that that difference was merely due to some constant error on the part of the person timing, for example their having a tendency not to click the stopwatch until a second after the ball had been released.
However, the fact that the difference between the two lines is small at first and gradually becomes greater suggests that some other factor is involved. As the ball is dropped from greater heights, the greater speeds it reached will have caused air resistance to have a greater effect, although I would not have thought the difference was large on a small object at such small speeds. In fact, it is possible that the unknown factor was at work in task 1, giving us a faulty value of e. Another possible (though not likely) cause is that gravity may not be 9.
8ms-2 at the altitude at which we did the experiment, and this will not have had allowance made for it by our value of e, as the way it was worked out made the result independent of gravity. A higher value of g would explain why it took less time, as the ball would have bounced as high but travelled faster. However, this slight difference does not seem to have any importance to the overall results. The two graphs have the general shapes that the model predicted and so are explained by the model.
Revision of the process I believe that the greatest contributor to the imprecision of the experiment was the error in timing, so if it were feasible(i. e. cheap) I would use some kind of electronic equipment that detected when the ball bounced or was released. I would also have thought it better so use some simple mechanical device to drop the ball to avoid giving the ball any energy, and it may have been better to test the value of acceleration due to gravity rather than just giving it an assumed value.
After all, the position of the moon has enough of a gravitational effect to cause the changing of the tides, so it may have had some noticeable effect on task 2 of the experiment, when gravity was significant. I think testing the value of g beforehand might have helped to reduce the difference between the experimental and predicted results for task 2. I am certain that better timing equipment would reduce the size of the error bars in task 2, and I think that using some mechanical device to release the ball would slightly reduce the error bars in both tasks.