AP Physics 1 2016 Free Response Question 2

Question: 
2. There is a claim that a toy ball can “bounce perfectly elastically” on hard surfaces. A student disagrees collisions can be perfectly elastic and hypothesizes that the collisions are very close to being perfectly elastic for low-speed collisions and they deviate more from perfectly elastic if the collision speed is increased. 

(a) Design an experiment to test the hypothesis about collisions of the ball by using equipment that can be commonly found in a physics laboratory. 
Your procedure should include the following points:
(i) Specify physical quantities that should be measured. 
(ii) Specify the equipment that can be used for the measurements. 
(iii) Provide a detailed procedure that can help to test the student’s hypothesis.

(b) Describe how the student might present the experimental data in a graph or table. 
Explain how the data would be consistent with the student’s hypothesis.

(c) Assume the experiment data could not be conclusive because the graph shows that low-speed collisions are almost perfectly elastic, whereas high-speed collisions violate a physical principle.
(i) Provide an example of a graph that suggests high-speed collisions violate a physical principle. 
(ii) State a physical principle that appears to be violated. Explain briefly which aspect of the graph that indicates the physical principle is violated. 

Scoring guidelines: 

(a) For a plan in which the measured physical quantities can be used to compare the total mechanical energy before and after a collision. [1]

For a plan that measure pre- and post-collision positions and/or speeds that can be used for comparing total mechanical energies. [1]
For selecting correct lab equipment and writing appropriate measurement procedure. [1]
For a procedure that includes trials of different pre-collision positions that can result in a range of low speed to high speed. [1]

(b) For describing an appropriate graph or table could be used to test the hypothesis. [1] 
For describing a comparison of post-collision to pre-collision mechanical energy that is related to the elasticity of the collision. [1] 
For comparing collisions that are related to the ball moving at low-speed and high-speed. [1] For analyzing the hypothesis in terms of a slope, ratio, or other measured quantities. [1]

(c) (i) For drawing a graph that demonstrates that the low-speed (or low drop heights) collisions are almost perfectly elastic. [1] 
For drawing a graph that demonstrates a violation of one physical principle for higher-speed (or higher drop heights) collisions. [1] 
(ii) For specifying an aspect of the graph that shows a violation of the physical principle. [1] For correctly explaining why the graph shows a violation of the physical principle. [1]
(https://secure-media.collegeboard.org/digitalServices/pdf/ap/apcentral/ap16_physics_1_q2.pdf)

Possible answers: 

(a) The procedure may include measured quantities, equipment, and how student’s hypothesis as shown below: 

1. Release the ball from rest at 10 different drop heights (controlled variables). 

2. The bounce heights (measured quantities or dependent variables) are measured by using a meter rule and video camera (equipment). 

3. Ensure the meter rule to be vertical without tilting and there should be no draught throughout the experiment. 

4. The experiment should be repeated at least three times with the video camera to check the ratio of bounce height to drop height. 

(b) 
1. Graph: Plot a graph of the bounce height (h_f) against the drop height (h_i). 

2. Hypothesis: To test the hypothesis, we can compare the post-collision mechanical energy with pre-collision mechanical energy by using the ratio mgh_f/mgh_i (=h_f/h_i). 

3. For low speed (or low drop heights), the experimental data is close to the straight line h_f = h_i and the ratio h_f = h_i is close to 1. 

4. For high speed (or high drop heights), the experimental data should be clearly below the straight line h_f = h_i. 

Note: The reduction in bounce height can be explained by a greater loss in kinetic energy due to a greater air resistance on the faster-moving ball. 

(c) (i) Below is a graph that indicates nearly elastic behavior for low-speed collisions but appears to violate a basic physics principle for high-speed collisions. 

(ii) The value of the energy ratio is greater than 1.0 suggests a violation of conservation of mechanical energy. This is because the post-collision mechanical energy of the ball is greater than the pre-collision mechanical energy. 

Note: The apparent violation of conservation of mechanical energy can be further explained from a different perspective as follows. First, the ball moving at higher speed tends to experience a greater air resistance. This results in an energy loss because of a conversion of kinetic energy of the ball into thermal energy (rubbing of the ball with air molecules). Second, the ball that is dropped from a greater height tends to produce a louder sound. This is due to a conversion of kinetic energy of the ball into sound energy (vibration of the ground and ball). 

On the other hand, it is not true that the momentum of a ball is conserved during falling and bouncing. This is because there are external forces such as gravitational forces and air resistance. Furthermore, we can explain that the momentum of the (ball-earth) system is conserved throughout this time. However, there is an idealization in the sense that we have ignored the (external) gravitational force acting on the ball due to the sun or moon. 

Feynman’s insights or goofs?: 

In Feynman’s words, “[i]t is possible to make the colliding bodies from highly elastic materials, such as steel, with carefully designed spring bumpers, so that the collision generates very little heat and vibration. In these circumstances the velocities of rebound are practically equal to the initial velocities; such a collision is called elastic (Feynman et al., 1963, section 10–4 Momentum and energy).” Interestingly, Superball is a toy ball made from a type of synthetic rubber instead of steel. This bouncy ball was invented by Norm Stingley, a chemical engineer, in 1964. Stingley offered the bouncy ball to his employer, Bettis Rubber Company, California, but it was turned down because it might not be a profitable product. However, Wham-O agreed to work with Stingley on his idea and manufacture the bouncy balls in 1965. The bouncy balls have an elastic property in which they can bounce to about 90% (or up to 92%) of the drop height. 

On the other hand, Feynman adopts the term elastic instead of perfectly elastic to describe the collision in which the velocity of rebound is practically equal to the initial velocity. Similarly, some physics textbook authors define an elastic collision as a collision in which the total kinetic energy is unchanged (e.g. Kleppner & Kolenkow, 2014). However, there is always a conversion of kinetic energy of the ball into thermal energy, elastic potential energy, and sound energy in a collision. Importantly, there are four types of collisions based on what happens during the collision and the total kinetic energy of the system. The elasticity of the collision can be related to the ratio of the relative velocities of the two colliding objects after and before the collision: k = (v_2f – v_1f) / (v_1i – v_2i). Thus, we can define four types of collision as shown below: 

Type of (elastic) collision	Description	Elasticity
Super-elastic	Final kinetic energy > Initial kinetic energy (Conversion of other forms of energy into kinetic energy: e.g. explosion)	k > 1
Perfectly elastic	Final kinetic energy = initial kinetic energy (total kinetic energy is conserved)	k = 1
inelastic	Final kinetic energy < initial kinetic energy (Conversion of kinetic energy into heat etc.)	1 > k
Perfectly inelastic	Final kinetic energy < initial kinetic energy (both objects stick together)	k = 0

Alternatively, we can write ½m₁v_1i² + ½m₂v_2i² = ½m₁v_1f² +½m₂v_2f² + Q 

For a super-elastic collision:      Q = –ve 

For a perfectly elastic collision: Q = 0 

For an inelastic collision:           Q = +ve 

Note: 
During a British Broadcasting Corporation interview, Feynman (1994) explains that “[w]hen a ball comes down and bounces, it shakes irregularly some of the atoms in the floor, and when it comes up again, it has left some of those floor atoms moving, jiggling. As it bounces, it is passing its extra energy, extra motions to little patches on the floor... (p. 127).”

References: 

1. Feynman, R. P. (1994). No Ordinary Genius: The Illustrated Richard Feynman. New York: W. W. Norton & Company.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Kleppner, D., & Kolenkow, R. (2014). An Introduction to Mechanics (2nd ed.). Cambridge: Cambridge University Press.