Today we are going to investigate the motion of a dropped ball as it bounces on the floor.

1. To start with, sketch a graph of how you think the ball’s height will change with time for five bounces.

2. How does your graph compare with your neighbor’s graph?

Now observe what happens when the ball is dropped.

Here is a graph of a bouncing soccer ball, where the ball’s height was measured with an ultrasonic motion sensor.

4. How does an actual graph of height vs time for a dropped ball compare to your prediction?

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Now let’s look at a graph of the ball around just the first two bounces.

5. What part of the ball’s physical motion corresponds to the graphs local maximum and minimum values?

6. Looking at the graph of *h*(*t*), answer the following questions by listing the time values or
intervals:

a) When, if ever, is ?

b) When, if ever, is ?

c) When, if ever, is ?

d) When, if ever, is undefined?

e) When, if ever, is increasing in value?

7. From your answers above, sketch a graph of below.

8. What are the units of ?

9. Physically, what does the derivative of *h*(*t*) represent?

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Velocity

Here is a graph of the velocity of the ball.

10. Compare your graph of velocity versus time to the above graph. If there are any differences between the two graphs, explain what the differences are and why you drew your graph the way you did.

Acceleration

Acceleration is defined as the derivative of the velocity with respect to time, *a *= d*v*/d*t*.
11. From looking at the velocity graph,list the time values or intervals:

a) Where the acceleration is positive?

What motion of the ball does this correspond to?

b) Where the acceleration is approximately constant?

What motion of the ball does this correspond to?

c ) Where the acceleration is undefined (if ever)?

What motion of the ball does this correspond to?

13. What are the units of the acceleration?

14. Physically, what does the acceleration mean in terms of the ball’s motion.

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Here is a graph of the acceleration of the ball.

Now let’s compare the three graphs:

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From the graph of height vs time when the ball is in hitting the floor, it looks like the derivative is undefined. If it is undefined when the ball is in contact with the floor, that would mean the velocity is undefined.

16. Looking at the velocity graph, is it undefined when the ball is in contact with the floor?

From the graph of velocity vs time when the ball is in hitting the floor, it looks like the derivative is undefined. If it is undefined when the ball is in contact with the floor, that would mean the acceleration is undefined.

Here is what is happening. The following graphs shows the ball’s height and velocity around the time it is in contact with the floor. For the height graph, what looked like a function with a sharp corner (and an undefined derivative) is actually a function with a smooth corner that has a derivative.

For the velocity graph, what looked like a function that was vertical (with an undefined derivative) is actually a function that has a positive slope and therefore, a derivative.

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Appendix 1 Bouncing Ball Data

Time Height Time Height (sec) (m) (sec) (m)

### Instructor Notes:

Learning Outcomes:

Upon completion of this module the students should be able to: Explain the difference between velocity and acceleration,

Read a position graph to determine where the velocity is increasing, zero or decreasing, Construct a velocity graph from a position graph,

Construct an acceleration graph from a velocity graph and Identify regions where the derivative of a function is undefined.

Equipment: ball

3. Initial height, parabolic shape between bounces, maximum height after each bounce decreases 5. local maximum – highest point after bounce

Local minimum – when ball hits the floor.

6. a) When, if ever, is ? *t* = 0.4, *t* = 0.75, *t* = 1.125, *t *= 1.375
b) When, if ever, is ? 0.4 < *t* < 0.75, and 1.125 < *t* < 1.375

c) When, if ever, is ? 0 < *t* < 0.4, 0.75 < *t* < 1.125, and 1.375 < *t *< 1.575

d) When, if ever, is undefined? It looks like the derivative may be undefined at *t* = 0.4
and *t* = 1.125 because it looks like a sharp corner occurs at that point.

e) When, if ever, is increasing in value? 0 < *t *< 0.4, 0.75 < *t* < 1.125, and
1.375 < *t* < 1.575

f) When, if ever, is decreasing in value? 0.4 < *t *< 0.75, 1.125 < *t* < 1.375
7.

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8. m/s

9. velocity of ball

11.

a) Where the acceleration is positive? A short time interval around *t* = 0.4 seconds and a short
time inteval around *t* = 1.125 seconds.

What motion of the ball does this correspond to? When the ball is in contact with the floor.

b) Where the acceleration is approximately constant? All other times.

What motion of the ball does this correspond to? When the ball is traveling up or traveling down.

c ) Where the acceleration is undefined (if ever)? It is always defined.

13. m/s2

14. rate of change of the velocity

16. no, the velocity graph is not vertical, there are no sharp corners 17. No, there is not a sharp corner

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