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Elastic Collision

Objective

To study the laws of conservation of momentum and energy in an elastic collision.

Introduction

If no net external force acts on a system of particles, the total linear momentum of the system

cannot change. This is known as the law of conservation of linear momentum. In other words,

if the net external force on the system is zero, the total linear momentum of a system of

particles before and after a collision is equal. If the total kinetic energy of the system is also

conserved during the collision then it is called an elastic collision.

In this lab, you will study the properties of elastic collision in one dimension. Figure 1 shows

the experimental set up. A glider of mass m1, moving with speed v1i, collides with another

glider of mass m2, moving with speed v2i. Immediately after the collision, the first glider

moves with speed v1f and the second glider with speed v2f. You will determine the speeds

using motion sensors.

Figure 1

Flag

Elevation Screw USB Interface

Motion Sensor 2

v1i v2i

i

Motion Sensor 1

Glider 1 , mass m1

Glider 2 , mass m2

Air Track Air Blower

Rubber Band

Blade

Flag

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If friction may be neglected, then in the collision of two gliders the total linear momentum of

the system is conserved. That is,

m1 1i + m2 2i = m1 1f + m2 2f

Note that the velocities are vector quantities. In an elastic collision, the total kinetic energy of

the system before and after the collision is also conserved. That is,

2

22

2

11

2

22

2

11 v2

1v

2

1v

2

1v

2

1ffii mmmm

Exercise 1: m1 ≈ m2 and v2i = 0

In this exercise, you are provided with two gliders of almost equal masses on an air track.

You will keep one of the gliders at rest before collision, and verify the conservation laws of

momentum and energy.

1. Measure the masses of the gliders m1 and m2 using a triple-beam balance. Record the values in your report.

2. Download the file collision1.ds from the link in phys101 homepage and save it in the desktop. This file has been preconfigured for optimum experimental parameters such

as position and time scales.

3. From the Start button, go to All Program DataStudio English. This will open DataStudio. Click on Open Activity, navigate to the folder desktop and to the file

collision1.ds, and open it. This will open the position-time graph.

4. Level the air track as follows: turn on the air blower, and then adjust the elevation screws at the ends of the track until glider 2 either remains stationary at the mid-point

of the track or moves slowly back and forth about this point. Leave this setting fixed

for the duration of the experiment.

5. Make sure the range switch of the motion sensor is set to work with the (short range) option.

6. As shown in Figure 1, bring the glider 1 closer to the motion sensor 1 while the glider 2 stays at the center of the air track. Gently push the glider 1 away from the sensor 1

and click the Start button at the top left corner in DataStudio. Click the Stop button

when the glider 2 reaches the motion sensor 2, and switch off the air blower. It is

important to push the glider gently to obtain good data.

7. Your data will look like Figure 2.

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Figure 2

8. Choose the appropriate straight line portions of the data by pressing the left button of the mouse and drawing a rectangle as shown. Right click in the graph area and choose

Fit and then select Linear Fit. The magnitude of the slope of the fit gives the

corresponding speed. Draw the rectangle as close as possible to the location of the

collision and select just enough data points to obtain the speeds immediately before

Glider 1 comes to rest

Glider 1 is at rest

Glider 1 is moving away from Sensor 1 Positive slope

Collision occurs here

Glider 2 is at rest

Glider 2 is moving toward Sensor 2 Negative slope

Sensor starts sensing glider’s motion here

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and immediately after the collision. Why do you think this step is important? See

Figure 3 for a close-up version of one of the graphs after fitting. You may avoid the

point right at the instant of collision.

9. Record the values in your report, with the correct sign for velocities and momentum. Choose the direction to the left (moving away from the sensor 1) as positive.

10. Calculate the total momentum of the system before and after the collision and their percent difference. Record the results in your report.

11. Do the same for the total kinetic energy of the system. Record the results in your report.

Figure 3

Exercise 2: m1 ≈ m2 and v2i ≠ 0

In this exercise, you will study the collision between two gliders of almost equal mass,

moving in the same direction.

1. Delete all the data runs from the DataStudio file and get it ready for Exercise 2.

2. Switch on the air blower and bring the glider 1 closer to the motion sensor 1 and glider 2 to the middle of the air track. Gently push the glider 1 towards glider 2 and

glider 2 towards motion sensor 2 such that they can collide well before reaching

sensor 2 (v1i and v2i both positive with v2i < v1i). Click the Start button at the top left

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corner in DataStudio and click the Stop button when the glider 2 reaches sensor 2,

and switch off the air blower.

3. Fit the appropriate straight line portions of the data, as you did in Exercise 1, to determine the speeds of the gliders just before and just after the collision.

4. Record the values in your report, with the correct signs for velocity and momentum. Choose the direction to the left (moving away from the sensor 1) as positive.

5. Calculate the total momentum of the system before and after the collision and their percent difference. Record the results in your report.

6. Do the same for the total kinetic energy of the system. Record the results in your report.

Exercise 3: Impulse

Consider the collision between two gliders. They experience equal and opposite force (t)

that varies with time during the collision. The impulse of the force is defined as

dtt)(F J

Where the integration is over the duration of the collision. Therefore, impulse is a measure of

both the magnitude and the duration of the collision force, and is given by the area under the

Force versus time curve. Note also that the impulse is a vector quantity. That is, it has a

direction as well as magnitude.

Momentum of a body changes if a net force acts on it. Therefore, the momentum of each

glider changes due to the collision. It can be shown that the change in momentum of a glider

is the impulse of the force on that glider. That is

p J

Since the total momentum of the two-glider system is conserved, the change in momentum of

glider 1 should be equal and opposite to the change in momentum of glider 2. That is, the

impulse on glider 1 is equal in magnitude and opposite in direction of the impulse on glider 2.

In this exercise, you will determine the impulse during the collision between two gliders of

different mass from force versus time graph. You will also verify the force exerted between

them is equal in magnitude but opposite in direction at each moment, and thus the impulse on

the gliders are also equal and opposite to each other.

1. Download the file collision3.ds from the link in phys101 homepage and save it in the desktop. This file has been preconfigured for optimum experimental parameters such

as force and time scales.

2. Replace the two gliders used in Exercise 1 and 2 by two different gliders of unequal masses and with force sensors mounted on it (provided to you separately, ask the

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instructor if you can’t find them). Make sure only the bumpers of the force sensors

will be in contact when the gliders collide (see Figure 4).

Figure 4

3. Plug the cable of the force sensors 1 and 2 to the USB interface module, after unplugging the motion sensors. You will be prompted to include force sensor in the

graph. Click No button. Let the cable of the force sensor rest on the table. This step is

important, in each experimental run, to minimize the unwanted force exerted by the

cable on the glider.

4. From the Start button, go to All Program DataStudio English. This will open DataStudio. Click on Open Activity. Click No to saving the current activity. Navigate

to the folder collision3.ds on the desktop. This will open the Force Graph windows.

5. Press the zero button on each force sensor to zero the force sensor. This step is very important prior to each experimental run.

6. Switch on the air blower and bring the gliders closer to each other and the bumpers facing each other. Gently push the gliders toward each other while clicking the Start

button at the top left corner in DataStudio. Click the Stop button when the gliders get

separated after collision, and switch off the air blower.

7. Your data will look like Figure 5, after adjusting the x and y scales to zoom the area where collision occurred.

Bumpers

s

Big Glider

- Small Glider

Force

Sensors

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Figure 5

8. Expand the region that corresponds to the collision. Notice the large value of force acting for a very short duration of the collision. The impulse is given by the area

under the force versus time curve. Use the Statistics tool to find the area under the

curve.

9. Record the values of the impulse from each force sensor in your report.

10. Write a conclusion based on the result of step 9.

Statistics Tool