# Velocity Relationships in Bouncing Objects

#### Project PHYSLab 99 participant lab by Margaret Showalter

In this series of investigations, you will observe and analyze the motion of a ball bouncing on the floor and a cart rebounding from the bottom of an incline. There will be some similarities to these motions, and some differences. Your job is to determine and explain how these motions are similar and how they are different.

### Procedure

Set up your computer interface and software for data collection from an ultrasonic motion detector. You will need to collect distance-time data. From this you can calculate velocity and acceleration. If available, you many want to use an experiment file already configured to calculate v and a from d,t data.

##### Case I: Bouncing Ball

Attach the motion detector to a ring stand or other suitable device so that the detector can be positioned high enough over the floor for a ball to be dropped beneath it.

Collect data for the ball as it is dropped and bounces off of the floor. Take care not to get your hand in the way of the motion detector.

##### Case II: Cart on a Track

Make a ramp from a board, dynamics track, or other suitable object. Position the motion detector at the top of the ramp. Place a book, brick, or other heavy object at the end of the ramp to serve as a stop. Place a spring plunger cart on the ramp so that the plunger will hit the stop at the bottom of the ramp.

Start the cart from a position along the track, near the top, but not too close to the motion detector. Let go of the cart and begin collecting data. Do not push the cart. Simply release it and let it roll.

### Analysis

For each of the motions, create and view distance-time and velocity-time graphs. On each graph, label the regions where each of the following is occurring:

1. the initial drop of the ball or descent of the cart

2. the bounce

3. the ascent after the bounce

4. the ball/cart is at its maximum rebound height

5. the ball drops or the cart descends again

When running the experiment, you probably collected data for a series of bounces or rebounds. It is not necessary to label the five regions above for each of these segments. Simply label the first time that each of these occurs.

Carefully investigate the velocity-time graphs. Compare the slope of a rebounding ball or cart (labeled region 3) with the slope of the ball or cart as it falls again (labeled region 5). If available, use a curve fit tool to investigate the slopes of these regions of the graphs.

### Questions

1. For each of the 5 regions labeled on the graphs, explain what is going on and why the graph has that shape in that region.

2. In what ways are the motions of the bouncing ball and rebounding cart similar?

3. In what ways are the motions of the bouncing ball and rebounding cart different?

4. What quantity is represented by the slope of a velocity-time graph?

5. What do you notice about the slope of the velocity-time graph for the rebounding ball and the falling ball. Should the ball accelerate and decelerate at the same rate? Does it? Why or why not? (hint: Look at regions 3 and 5. Can you fit a single straight line through both of these regions?)

6. What do you notice about the slope of the velocity-time graph for the rebounding cart and the falling cart? Should the cart accelerate and decelerate at the same rate? Does it? Why or why not? (hint: Look at regions 3 and 5. Can you fit a single straight line through both of these regions?)

7. Draw a diagram showing the forces acting on a cart on a ramp. Use this diagram to explain your answer to the previous question.

### Teacher's Notes

The data was collected and the sample graphs were generated using a Vernier ULI, motion detector, and Logger Pro software. Vernier's Physics with Computers Experiment 2c file was used for data collection. However, other interfaces and/or software packages may be use. This lab can be used as an add on to Physics With Computers experiment 2.

If suitable equipment is available, you can use different cart systems with different amounts of friction. What you will see in the sample data that follows is one run with a relatively low friction Pasco dynamics cart on a Pasco dynamics track, and another run with a relatively high friction roller cart on a wooden board. There is a subtle difference in the rates of acceleration down the ramp and deceleration up the ramp for the low friction car. There is a much more dramatic difference in the rates of acceleration and deceleration for the higher friction cart and ramp system.

You could extend this lab by challenging students to quantitate the frictional force and calculate the coefficient of friction. You could also have students vary ramp height and/or mass (by adding mass to the cart) to determine whether or not either of these factors affects frictional force. Students might also think of other systems to study and compare (i.e. glider or air track, ball rolling down ramp).

The motions studied in this lab result from different objects (carts, balls) in different situations (free-falling, rolling on ramp). Thus students may need to be reminded not to attempt to make too many comparisons based on actual numbers, but rather to make qualitative comparisons about how the behaviors (and more specifically the changes in the numbers) are similar and different. For example, the ball and cart may not fall or roll through the same distance. Or, if multiple cart runs are performed as suggested above, the cart masses and ramp angles may not be identical. But trends in the motions can still be compared.

Sample Results

#### Case I Bouncing Ball

These are distance - time and velocity - time graphs for a ball which is dropped and allowed to bounce. The motion detector is positioned over the top of the ball and floor, thus the ball distance is increasing as the ball falls and is decreasing as the ball rebounds.

1 -- ball is dropped, v increases as it falls

2 -- ball decelerates rapidly when it hits the floor

3 -- ball is rebounding

4 -- velocity is zero when ball reaches maximum rebound height

5 -- ball falls again with increasing v

Note that the graph is one continuous straight line from regions 3 - 5, indicating that the rates of deceleration in the upward direction and acceleration in the downward direction are the same. Also note that the magnitude of this acceleration (slope) is approximately 9.8 m/s2.

#### Case II Cart on a Track (low friction)

These are distance - time and velocity - time graphs for a cart which is rolled down a ramp. The motion detector is positioned at the top of the ramp, thus the distance is increasing as the cart rolls down the ramp. The flat maxima on the d-t graph corresponds to the position of the cart at the bottom of the ramp. During this time, the spring was compressing and rebounding, but this was seen as zero displacement (and zero velocity) by the motion detector.

1 -- cart rolls down, v increases as it falls

2 -- cart decelerates rapidly when it hits the brick at the bottom of the ramp

3 -- cart is rebounding, travels back up the ramp

4 -- velocity is zero when cart reaches maximum rebound height

5 -- cart descends again with increasing v

Although it is not obvious from a visual inspection of the velocity graph, there is a slight change in slope (corresponding to a slight change in acceleration) for the cart as it goes up and down the ramp. This slight difference can be explained by friction.

When the cart is rolling down the ramp (region 5), gravity is accelerating the cart downward, but friction (which opposes motion) is pulling the cart upward. Thus the net accelerating force on the cart is gravity minus friction.

When the cart is rolling up the ramp (region 3), gravity and friction are both causing the cart to decelerate. Thus the net force decelerating the cart on the upward trip is larger than the net force accelerating the cart on the downward trip.

This difference can be seen more easily by fitting a straight line to regions 3 and 5. Unlike the previous case with the free-falling ball, here a straight line drawn through region 3 does not exactly fit region 5. The magnitude of the slope of region 3 (deceleration) is slightly larger than the magnitude of the slope of region 5 (acceleration).

#### Case II Cart on a Track (high friction)

These are distance - time and velocity - time graphs for a cart which is rolled down a ramp. The motion detector is positioned at the top of the ramp, thus the distance is increasing as the cart rolls down the ramp. Note the visible inflection of the velocity curve at label 4.

1 -- cart rolls down, v increases as it falls

2 -- cart decelerates rapidly when it hits the brick at the bottom of the ramp

3 -- cart is rebounding, travels back up the ramp

4 -- velocity is zero when cart reaches maximum rebound height

5 -- cart descends again with increasing v

In this case, the slope change from region 3 to 5 is much more obvious than in the previous low friction cart example. Here the slope change is dramatic enough to be readily spotted in the graph.

Again, fitting a line to these two regions shows a larger slope for region 3 (deceleration) than for region 5 (acceleration).

Although the two cart/ramp systems were completely different and thus it is not valid to directly compare numerical slopes between the two cases, these cases can be compared in terms of the percentage change in the slope. The higher friction cart data shows a much larger % difference in the acceleration and deceleration slopes than does the lower friction cart data. There is essentially no difference in the slopes for the free-falling ball.