Impulse and Momentum

The impulse-momentum theorem relates impulse, the average force applied to an object times the length of time the force is applied, and the change in momentum of the object:

Here we will only consider motion and forces along a single line. The average force,, is the net force on the object, but in the case where one force dominates all others it is sufficient to use only the large force in calculations and analysis.

For this experiment, a dynamics cart will roll along a level track. Its momentum will change as it reaches the end of an initially slack elastic tether cord, much like a horizontal bungee jump. The tether will stretch and apply an increasing force until the cart stops. The cart then changes direction and the tether will soon go slack. The force applied by the cord is measured by a Force Sensor. The cart velocity throughout the motion is measured with a Motion Detector. Using Logger Pro to find the average force during a time interval, you can test the impulse-momentum theorem.

 

objectives

   Measure a cartís momentum change and compare to the impulse it receives.

   Compare average and peak forces in impulses.

 

Materials

Power Macintosh or Windows PC

dynamics cart and track

LabPro or Universal Lab Interface

clamp

Logger Pro

elastic cord

Vernier Motion Detector

Vernier Force Sensor

string

500-g mass

 

Preliminary questions

1.†† In a car collision, the driverís body must change speed from a high value to zero. This is true whether or not an airbag is used, so why use an airbag? How does it reduce injuries?

2.†† You want to close an open door by throwing either a 400-g lump of clay or a 400-g rubber ball toward it. You can throw either object with the same speed, but they are different in that the rubber ball bounces off the door while the clay just sticks to the door. Which projectile will apply the larger impulse to the door and be more likely to close it?

Procedure

1.†† Measure the mass of your dynamics cart and record the value in the data table.

2.†† Connect the Motion Detector to DIG/SONIC 2 of the LabPro or PORT 2 of the Universal Lab Interface. Connect the Force Sensor to Channel 1 of the interface (PORT 1 if you are using the ULI Force Probe). If your Force Sensor has a range switch, set it to 10 N.

3.†† Open the Experiment 20 folder from Physics with Computers. Then open the experiment file that matches the force sensor you are using. Logger Pro will plot the cartís position and velocity vs. time, as well as the force applied by the Force Sensor vs. time. Data are collected for 5 s, at a rate of 50 Hz.

4.†† If you are using the ULI Force Probe it is necessary to calibrate the sensor. Other Force Sensors can optionally be calibrated using the same procedure.

a.    Choose Calibrate from the Experiment menu. Click on the PORT 1 icon so the port is highlighted. Click .

b.    Remove all force from the Force Sensor. Enter a 0 (zero) in the Value 1 field. Hold the sensor vertically with the hook downward and wait for the reading shown for Input 1 to stabilize. Click . This defines the zero force condition.

c.    Hang the 500-g mass from the sensor. This applies a force of 4.9 N. Enter 4.9 in the Value 2 field, and after the reading shown for Input 1 is stable, click . Click to close the calibration dialog.

 

5.†† Place the track on a level surface. Confirm that the track is level by placing the low-friction cart on the track and releasing it from rest. It should not roll. If necessary, adjust the track.

6.†† Attach the elastic cord to the cart and then the cord to the string. Tie the string to the Force Sensor a short distance away. Choose a string length so that the cart can roll freely with the cord slack for most of the track length, but be stopped by the cord before it reaches the end of the track. Clamp the Force Sensor so that the string and cord, when taut, are horizontal and in line with the cartís motion.

7.†† Place the Motion Detector beyond the other end of the track so that the detector has a clear view of the cartís motion along the entire track length. When the cord is stretched to maximum extension the cart should not be closer than 0.4 m to the detector.

8.†† Click , then , to zero the Force Sensor.

9.†† Practice releasing the cart so it rolls toward the Motion Detector, bounces gently, and returns to your hand. The Force Sensor must not shift and the cart must stay on the track. Arrange the cord and string so that when they are slack they do not interfere with the cart motion. You may need to guide the string by hand, but be sure that you do not apply any force to the cart or Force Sensor. Keep your hands away from between the cart and the Motion Detector.

10.††† Click to take data; roll the cart and confirm that the Motion Detector detects the cart throughout its travel. Inspect the force data. If the peak is flattened, then the applied force is too large. Roll the cart with a lower initial speed. If the velocity graph has a flat area when it crosses the x-axis, the Motion Detector was too close and the run should be repeated.

11.††† Once you have made a run with good distance, velocity, and force graphs, analyze your data. To test the impulse-momentum theorem, you need the velocity before and after the impulse. Choose a time interval just before the bounce when the speed was approximately constant, and drag the mouse pointer across the interval. Click the Statistics button, , and read the average velocity. Record the value in your data table. In the same manner, determine the average velocity just after the bounce and record the value in your data table. Remove the floating boxes by clicking the gray close box in their upper right corners.

12.††† Now record the time interval of the impulse. There are two ways to do this. Use the first method if you have studied calculus and the second if you have not.

   Method 1: Calculus tells us that the expression for the impulse is equivalent to the integral of the force vs. time graph, or

†††††

On the force vs. time graph, drag across the impulse, capturing the entire period when the force was non-zero. Find the area under the force vs. time graph by clicking the integral button, . Record the value of the integral in the impulse column of your data table.

 

   Method 2: On the force vs. time graph, drag across the impulse, capturing the entire period when the force was non-zero. Find the average value of the force by clicking the Statistics button, , and also read the length of the time interval over which your average force is calculated. The number of points used in the average divided by the data rate of 50 Hz gives the time interval Dt. Record the values in your data table.

 

13.††† Perform a second trial by repeating Steps 10 Ė 12, record the information in your data table.

14.††††† Change the elastic material attached to the cart. Use a new material, or attach two elastic bands side by side.

15.††††† Repeat Steps 10 Ė 13, record the information in your data table.

Data Table

Mass of cart

kg

 

Trial

 

Final Velocity
vf

Initial Velocity
vi

Change of Velocity†† Dt

Average Force
F

Duration of Impulse
Dt

Impulse

Elastic 1

(m/s)

(m/s)

(m/s)

(N)

(s)

(Ns)

1

 

 

 

 

 

 

2

 

 

 

 

 

 

 

Elastic 2

 

 

 

 

 

 

1

 

 

 

 

 

 

2

 

 

 

 

 

 

 

Trial

Impulse†††††††††††
Ft


Change in momentum


% difference between Impulse and Change in momentum

Elastic 1

(Ns)

(kgm /s) or (Ns)

(Ns)

1

 

 

 

2

 

 

 

 

Elastic 2

 

 

 

1

 

 

 

2

 

 

 

 

Analysis

1.†† Calculate the change in velocities and record in the data table. From the mass of the cart and change in velocity, determine the change in momentum as a result of the impulse. Make this calculation for each trial and enter the values in the second data table.

2.†† If you used the average force (non-calculus) method, determine the impulse for each trial from the average force and time interval values. Record these values in your data table.

3.†† If the impulse-momentum theorem is correct, the change in momentum will equal the impulse for each trial. Experimental measurement errors, along with friction and shifting of the track or Force Sensor, will keep the two from being exactly the same. One way to compare the two is to find their percentage difference. Divide the difference between the two values by the average of the two, then multiply by 100%. How close are your values, percentage-wise? Do your data support the impulse-momentum theorem?

4.†† Look at the shape of the last force vs. time graph. Is the peak value of the force significantly different from the average force? Is there a way you could deliver the same impulse with a much smaller force?

5.†† Revisit your answers to the Preliminary Questions in light of your work with the impulse-momentum theorem.

6.†† When you use different elastic materials, what changes occurred in the shapes of the graphs? Is there a correlation between the type of material and the shape?

7.†† When you used a stiffer or tighter elastic material, what effect did this have on the duration of the impulse? What affect did this have on the maximum size of the force? Can you develop a general rule from these observations?

Extensions

1.†† Try other elastic materials, doing the same experiment.