The coefficient of restitution is a particular characteristic of any ball. When a ball bounces off a hard surface, the surface cannot move therefore the only object that can move is the ball. The result of this is a partially elastic collision between the ball and the table where some of its kinetic energy is turned into sound and heat energy as well as still propelling itself back upwards (because it is a partially elastic). The coefficient of restitution is a measure of the balls elasticity and is the ratio between the ball's collision speed and its rebound speed. Its scale ranges from 0 to 1, 0 being completely inelastic and 1 being elastic.
Source: http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Momentum/CoeffOfResititution.html
The coefficient of restitution can be deduced from the equation for gravitational potential energy GPE = MgΔh
h = height the ball bounces back up to
H = height the ball was dropped from
COR = √2gh
√2gH
COR = √2 x 9.81 x h
√2 x 9.81 x H
COR = √h
√H
There only two measurements or values that are needed to calculate the coefficient of restitution and these are the height at which the ball has been dropped from and the height of the bounce.
Relating this to velocity (on impact with the surface):
E.g. Acceleration = 9.81ms-2
Initial Velocity (u) = 0.00 ms-1
Distance = 1.00m
Applying v2 = u2 + 2as
v2 = 02 + 2as
v2 = 2as
v2 = 2 x 9.81 x 100
v2 = 1962
v = √1962
v = 44.29 ms-1
Preliminary
- Clamp Stand
- Clamp
- Metre rule
- Table tennis ball
- Hard surface
I performed a preliminary investigation in which I dropped a table tennis ball from different heights in 20cm intervals recording the height I dropped the ball from and the height the ball bounced back up to. I repeated each height three times for accuracy and so I could take an average from all 3 results. Before I dropped the ball for the preliminary, I dropped it once to find roughly where I should be looking to see the bounce height.
From the preliminary experiment I found that the lowest accurate height I could read the bounce height from was 20cm otherwise, any lower than that, and the results would not be very accurate. Because of the physical constraints of the physics lab I was working in it was not feasible to carry out any drops higher than 100cm because it was unsafe.
To ensure accuracy will be maintained throughout the experiment I will use the same table tennis ball for the actual experiment and I will make sure that the ball is not damaged by checking it regularly.
All the equipment I used was fairly accurate but could be made better by a number of instruments or methods. The possibility of human error is quite large because the ball is traveling at quite a high speed. This error could virtually eliminated by using a high speed camera or possibly just a high quality home video camera. A preliminary drop could be done and the camera could be positioned so that it read vertically off the metre rule. The video could then be played back and the frame could be found where the ball ceases to be in motion. In the actual experiment I shall test closer increments of dropping heights and increase the number of drops to 5 to increase accuracy and get a better average. I will also include a larger number of dropping heights so that I can try and locate the height at which the ball will bounce same height each time the dropping height increases as this will confirm my prediction. But because I am working in a lab, there is a restriction on the height that I will be able to drop the ball from.
The experiment is not a very dangerous experiment so safety precautions are needed although if the ball does fall of the table, care must be taken not to stumble or dive under the ball to catch it. Also, all coats, bags or any other equipment that is not necessary for the experiment should be put in a safe place away from anyone carrying out the experiment.
Equipment for actual experiment
- Clamp Stand
- Clamp
- Metre rule
- Table tennis ball
- Hard surface
The reason I will be using the same equipment is because it worked in the preliminary and it was accurate. If the equipment was available to me, I would use a high speed camera or a good quality video camera to record the height the ball bounces back up to.
Method
- Setup the equipment so that the clamp stand has one clamp clamped onto the bar and a ‘claw’ clamp on the end of the bar holding onto the metre rule, just so it doesn’t fall over.
- The next step is to start by dropping the table tennis ball (or getting someone else to drop it for you) from a measured height. The ball should be dropped so that the bottom of the ball is at the measured height.
- The first drop should be a ‘test’ drop in which the ball should be dropped and the person watching to see how high the ball bounces should look in what area the ball is bouncing back up to.
- Then, a number of drops (minimum 3 drops) should be performed. All the results should be recorded and should then be averaged.
- The average results can then (from the tabulated data) be plotted on a graph to assess the outcome of the results.
These are the results I gained from the experiment and I have tabulated them to make reading them much easier. When I tabulated the data I have used an average of the 3 drops. I worked out the average by dividing the sum of the 3 bounce heights by 3. I have also plotted a scatter graph to show the trend in bounce height against drop height. The graph is on a separate sheet of graph paper for accuracy and is at the back of the document but a computer generated graph is also included below with the extrapolated result of the terminal velocity. I can plot this point because the graph produces a slight curve due to air resistance which stops the ball continually accelerating at 9.81ms-2 until it hits the ground.
The scatter graph illustrates the results from 20-100cm comparing the drop height with the bounce height.
The bounce height was never greater than the drop height at any point.
As well as proving this prediction I also said that after a certain drop height the bounce height would be the same because the ball would reach its terminal velocity. In the experiment I did not reach this height but it could be predicted by extrapolated from the data using a trend line (or a curve in this case). The drop height that the terminal velocity would be reached at would be around 270cm. I found this by extending the curve of the graph until it leveled out. Any height greater than 270cm should have the same (or nearly the same due to air resistance) bounce height because the terminal velocity would be reached each time.
In the experiment there were any sources of error. The main source of error would be the human eye and hand. This is because the ball is traveling at quite a high speed and it is sometimes hard to determine exact distances, hence the accuracy to +/- 1cm.
I started off with the smallest measurement (of 20cm) and worked my way up in 10cm increments until I reached 100cm.
