Examine the relationship between the height a ball is dropped from and the vertical distance from the point of impact that its first bounce travels.

Investigation to find out what affects the height that a golf ball bounces to

Aim: I wish to examine the relationship between the height a ball is dropped from and the vertical distance from the point of impact that its first bounce travels.

Apparatus: Tape Measure, A Golf Ball, Ping Pong Ball, Floor

Plan:

Using a tape measure suspended and weighted at the bottom (to ensure verticality), measure the appropriate starting height. The centre of the ball should be aligned with the relevant markings on the tape and dropped so that the tape does not foul the path of the ball. A spotter should be ready to sight up the height of the bounce and as level with the apex of the balls path as possible. A tolerance of 10mm is realistic. This should be repeated five times each at heights of 0.5m, 1m, 1.5m, 2m and 2.5 meters.

In this experiment I will measure the height reached after the bounce, because that is what I wish to investigate. I will also need to measure the independent variable, the height at which the ball is dropped from. To measure both of these I will use metre sticks and I think that they will be accurate enough considering that we will only have someone’s judgement to go by with regards to how high it bounces to. Each of the five heights should be repeated five times so that the results are more reliable and any anomalous results won’t be so noticeable when results are worked out. I chose the five heights because I think that they are appropriate and it will be easier to see the height reached when it is on a larger scale.

Control of Variables:

The following points should be adhered to in order to reduce the influence of preventable variables upon the results:

The same golf ball should be used throughout the experiment so that any differences in material, construction or finish from one ball to the next are not reflected in the results.

The same contact surface should be used in each drop. A softer surface will tend to absorb more energy than a hard one, resulting in a much shorter bounce, as energy will have been dispersed in the form of heat, sound, and significantly, deformation of the contact surface.

The person observing the height reached should always measure to the bottom of the ball because if they sometimes measure to the middle or the bottom of the ball the heights recorded might not be correct.

Diagram

Preliminary Study.

For our preliminary work we experimented with different types of balls in order to find out which would be most appropriate for the experiment. These are the results from our preliminary work.

From these results it is clear that the golf ball bounces highest and for this reason the golf ball will be used in our experiment as the higher the ball bounces, the greater the difference in the results, therefore the easier it is to see differences.

Forces: AS the ball falls, gravity acts on it causing it to accelerate downwards. The earth will also move towards the ball, (because every action has an equal and opposite reaction) but because F=ma the earth will move only a tiny bit because of it’s immense mass. When the ball hits the floor the floor then pushes up with the same force that the ball hit it with. As the ball falls the air resistance acting against it increases causing the rate of acceleration of the ball to decrease slightly until it reaches a point where the gravity and air resistance ...

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Forces: AS the ball falls, gravity acts on it causing it to accelerate downwards. The earth will also move towards the ball, (because every action has an equal and opposite reaction) but because F=ma the earth will move only a tiny bit because of it’s immense mass. When the ball hits the floor the floor then pushes up with the same force that the ball hit it with. As the ball falls the air resistance acting against it increases causing the rate of acceleration of the ball to decrease slightly until it reaches a point where the gravity and air resistance are balanced and the ball travels at a constant speed (terminal velocity). If the ball was bouncing without air resistance, whether the ball is travelling up or down the downward velocity will increase by 10m/s (because of gravity) every second unless it reaches terminal velocity. Therefore, a ball dropped from a higher height having taken longer to fall will have a higher velocity on hitting the floor.

Energy: Before the ball is dropped it has gravitational potential energy, for which the formula is mass × gravity × height. As the ball falls the height decreases and therefore the gravitational potential does also, however, since energy cannot be created or destroyed, only transferred from one type to another, this energy must still exist. Since the ball is now moving it must have kinetic energy (1/2 mass × velocity 2). Velocity will increase as the ball falls until the air resistance balances the weight (terminal velocity). If the velocity is increased then the kinetic energy of the ball is also increasing, the gravitational potential energy is being transferred to kinetic energy. In the moment before the ball hits the floor all the gravitational potential energy has been transferred to kinetic energy since the height is now 0 and anything multiplied by 0 is 0. A very small amount of energy would be transferred to heat energy from friction between the particles as the ball falls but this would not be a significant amount. As the ball hits the floor the kinetic energy is transferred to heat energy due to the friction on the floor, sound energy and elastic energy within the ball. The heat and sound energy is no longer useful to the ball so after it has bounced it will have less kinetic energy, for this reason I think that it will be difficult to return to the height from which it was dropped. As the ball bounces back up again the elastic energy within the ball is transferred to kinetic energy.

Momentum: The formula for momentum is Momentum = Mass × Velocity. It is possible to work out the average velocity of the falling ball using the formula v=√ (2gh)(a rearranged version of the formulas for gravitational potential energy and kinetic energy). Since the only thing within this formula that will change is the height it is possible to see that a ball dropped from a higher height will have a higher average velocity, unless it reaches terminal velocity. Therefore a ball dropped from a higher height, having a bigger velocity will subsequently have more momentum. Before colliding with the floor the momentum of the ball is its mass multiplied by its velocity and the momentum of the floor is 0. When the ball hits the floor it stops momentarily but because of the conservation of momentum the momentum must still exist after the collision in the particles of the floor and the ball, which are now moving. When the ball bounces up again the particles are still moving so they have some momentum which means that the ball cannot have as much momentum as it did before the collision and therefore, since its mass has not changed it must have a smaller velocity.

Elasticity: Elasticity is the ability of a solid to recover its shape once deforming forces are removed. A golf ball has this ability due to the elastic bands inside it. Most elastic material obeys Hooke’s Law, however rubber is an exception, as the deformation of rubber is not proportional to the applied stress. It is the elastic property of a golf ball that enables it to bounce so well. As the ball collides with the floor it will change shape, because of the force with which it hits it. However, because of elasticity it will recover its shape after the collision and bounce back up again, providing it hasn’t reached its elastic limit (the point at which stress deforms the object permanently).

Prediction: Considering all the theory, I predict that the ball will never bounce to the height at which it started, because energy is transferred to heat and sound on its collision with the floor and it won’t then have as much kinetic energy after the bounce. Also the force of gravity acts against the falling ball.

I think that the higher a ball is dropped from the higher it will bounce back up to because it has more gravitational potential energy at the start and will therefore have more kinetic energy as it falls and probably more kinetic energy after the bounce (although some will have been transferred to sound and heat energy).

I should imagine that the height from which the ball falls and the height that it bounces to are in some way proportional, however, with the limited theory that is known I cannot see exactly how they are related. It may well be that they are directly proportional with a set percentage of energy being lost on the collision with the floor, for example, a 30% energy loss on each bounce, in which case I would expect the graph of he heights before and after the bounce to show a straight line. Another possibility is that on collision with the floor a set amount of energy transferred to sound and heat energy. However, I do not see that this is likely considering that an object dropped from a higher height tends to make a louder sound, which probably means that more energy is being transferred from kinetic to sound energy.

Unless there is some kind of quadratic relationship between the heights before and after, I would expect the graph of results to look something like this, providing they are proportional as I predict.

Analysis: The positive correlation of the graph for heights reached by a golf ball shows that as the height the ball is dropped from increases, the height of the bounce does also, this is as I predicted. This can be explained because a golf ball having been dropped from a higher height has more gravitational potential energy at the beginning (because mass x

Gravity x height and the only thing changing is height) and therefore the higher ball will have more energy on collision with the floor. When the ball collides with the floor the kinetic energy is transferred to sound and heat energy as well as kinetic energy in the floor particles and elastic energy within the ball. It seems that a set percentage of the energy is being transferred to each type of energy and therefore a ball dropped from a higher height, because it begins with more energy will have more energy after the bounce. This is one of the possibilities that I suggested in my prediction.

The ball never reached the height it started from, as I predicted, and managed to reach approximately 72.3% of its starting height each time. This makes sense because energy is transferred to sound and heat energy when it hits the floor and this then becomes useless to the ball. Also the force of gravity acts with the ball, causing it to accelerate as it falls (gaining 10m/s every second until it reaches terminal velocity) but after the ball has bounced, gravity is acting against it and causes it to lose 10m/s of it’s velocity every second.

We also repeated the experiment with ping-pong balls as we had some spare time. The results for this were surprising because they didn’t show the straight line on the graph, as I would have expected. Instead the line was straight for the first part and began to curve over as it went up. This may be because as it falls from a higher height it has a higher velocity, because as I mentioned in my theory section, the formula V= √(2gh) can be used to calculate velocity and since the only thing in this that is changing is the height, a ball falling from a higher height must have a bigger average velocity. Its high velocity may mean that it cannot cope so well with the impact and is unable to bounce as high. If this is the reason I wonder if the same would be true of the golf ball if it was dropped from a much higher height. I expect the reason that it happened to the ping-pong ball and not the golf ball at the heights that we tried was because a golf ball is a stronger structure and it also has a greater degree of elasticity, which enables it to change shape on collision and return to it’s former shape as it bounces back up.

My results support my prediction in as much as the ball didn’t reach it’s starting height ever, bounced higher when dropped from a higher height and because there was some kind of proportionality between the starting height and the bounce. However my prediction wasn’t very specific and I was unsure in what way they would be proportional. Apart from the proportionality the other points I made in my prediction were proven by this experiment.

Evaluation:

There are several aspects of this experiment that make it unfair and inaccurate. Here are a few:

Landing on a different bit of the floor, which may affect the height reached.
The ball being thrown not dropped, giving it a higher initial velocity.
The difficulty of accurately reading the measurement each time because it is hard for the eye to see the height reached.

Despite all the inaccuracies in the method the results seem surprisingly accurate. The averages fall in a near perfect straight line, which would suggest that they are accurate. Assuming the results were read fairly accurately it would lead me to believe that the other issues of inaccuracy are fairly negligible. To achieve such a straight line probably means that our results are accurate enough to base a conclusion on.

The experiment could be made more accurate by filming each bounce and playing it back in slow motion to obtain more accurate measurement, for example to the nearest 5mm. I could also go to extra lengths to make sure that the ball landed in exactly the same place each time and that it was dropped and not thrown.

The results are fairly reliable because we repeated each height five times so that anomalous results would become obvious. Very few of our results varied that much within each height and therefore our results are probably quite reliable. In order to make the results even more reliable more repeats could be carried out, however I do not think this is entirely necessary as the heights reached were always so close to the same percentage.

If I were to complete a similar experiment I would wish to find out if after a certain height the golf ball can no longer maintain the percentage height reached after a bounce, as I discovered with the ping-pong ball. I would also like to see if the same percentage of height is lost after a second, third, fourth or fifth bounce etc.