Impulse and Momentum - University of Michigan



Impulse and Momentum

Introduction

Newton’s Second Law of Motion was originally written in terms of the “quantity of motion,” today called momentum. It can be rewritten to state that the change of momentum of a system is equal to the sum of the impulses applied to the system by forces external to it: .[pic]

What does that sentence mean in practice? How does impulse change the momentum of an object? What happens to the momentum of a system when no external forces act on it? In this laboratory you’ll investigate these questions using both probes of force and motion and digitized videos.

Textbook Reading

Halliday, Resnick, and Walker: Chapter 10, Sections 10-2, 10-5 (pages 216, 225, 226)

Overview

This laboratory includes two investigations. The first uses the air track, glider, motion and force probes that were used in the Energy laboratory. The momentum of the glider before and after its collision with the bumper is calculated. The force the bumper exerts on the glider is measured and the impulse the bumper delivers to the glider is also calculated. These calculations will allow you to compare the momentum change of the glider with the impulse the bumper gives it.

The second part uses the World in Motion software you first used to analyze the motion of the toy airplane. This time you will analyze video clips of collisions. You’ll analyze the collisions of billiard balls and hockey pucks and determine whether or not momentum and kinetic energy are conserved.

Objectives

* To explore the relationship between impulse and momentum change in one dimension.

* To develop an understanding of impulse as the integral of force over the time interval of the interaction.

* To use measurements of the position of an object in two dimensions to calculate its momentum.

* To examine the momentum of a system moving in two dimensions to see if momentum is conserved.

Pre-lab Questions

[pic]

1. A 300-g glider was moving at –0.57 m/s toward the left (negative direction). After colliding with the end of the air track its speed was 0.46 m/s toward the right. What is the magnitude and sign of the change in its momentum?

2. The graph shows the variation of the force exerted on the glider as a function of time. Find the impulse delivered to the glider by the bumper by finding the area under the curve. How closely does the impulse match the change in momentum?

3. A 0.150 kg ball was moving with a velocity of vx = –0.75 m/s,

vy = –1.50 m/s before being hit. After being struck its velocity was vx( = +1.75 m/s, vy( = –0.50 m/s. On the graph below draw arrows that represent the ball’s momentum before and after the collision. Draw an arrow representing the change in momentum of the ball.

Calculate the change in momentum by subtracting the components of the initial and final momentum, then find the magnitude and direction of the momentum change.

[pic]

| | | |

| |x-component |y-component |

| | | |

|p (before) | | |

| | | |

|p( (after) | | |

| | | |

|Δp = p( – p | | |

Magnitude of [pic]

|Δp| = ___________ kg m/s

Angle θ given by tan θ = Δpy /Δpx

θ = _________ (

Glider Collisions Name______________________________

Equipment Partner _____________________________

* Air track with glider, extra masses, flag

* ULI with motion detector and force probe

* File Impulse

Predictions

[pic]

1. In the space to the right draw motion diagrams for a glider that moves at constant speed to the left, the collides with a bumper and moves to the right at a slightly slower speed.

2. Draw vectors representing the momentum of the glider just before and just after the collision. Subtract these two vectors graphically to obtain a vector that represents the change in momentum, Δp.

3. According to the impulse-momentum theorem, J = Δp. Draw a vector representing J.

[pic]

4. Repeat these three steps for a glider that moves with the same velocity, hits the same bumper, but, rather than rebounding, stops at the bumper. If Δp and J have different magnitudes in this case, be sure that your arrows are of different length.

Equipment

The air track, probes, and glider should be set up as shown at the right. The force probe should be adjusted so that the knife edge on the glider strikes the center of the elastic cord bumper. When viewed from above the motion detector and force probe should point along the center of the track and the flag on the glider should be perpendicular to the motion detector. When the glider is resting against the elastic cord the flag should be about 45 cm from the motion detector.

Push the glider gently and observe its motion. Make sure that the track is level and that the glider moves without sticking and bounces smoothly off the bumper. Start LoggerPro and open the file Impulse. Gently push the glider and click on Collect to take data. Make sure that the motion detector “sees” the glider from its collision with the bumper until it is at least 1.0 m from the detector. If the graphs are noisy carefully adjust the location of the motion detector (or ask your instructor for help).

Checking the force probe

Check that the switch on the top of the force probe is on 10 N. Zero the probe.

Impulse and Momentum: Quantitative

In this section you will investigate the relationship between momentum change and impulse for three trials.

1. The “bare” glider colliding with the elastic band bumper.

2. The glider moving with the same speed, but having two 50-g masses attached.

3. The glider with added masses colliding with a soft clay bumper.

Gently push the unweighted glider from the far end of the track at about 0.5 m/s and record a single collision with the bumper.

Displaying glider momentum

To find the change in momentum you need to know the glider’s momentum before and after the collision. As you did in the energy lab, modify the Momentum column by changing the mass of the glider from the preset value (0.500) to the measured mass of your glider. When you change the mass of the glider in later parts of the experiment, be sure to change the definition of momentum in LoggerPro.

Finding the impulse

The impulse, J, is defined as the integral of the external force on the system over the collision time[pic]

You can let the .program do this integral, the area under the force-time curve. First select the region of the force-time graph that you want to analyze. To make the selection, position the mouse at the left-hand end of the region. Press and hold the left mouse button as you slide the mouse to the right-hand end of the region. Release the button. You should see black vertical bars at each end. To calculate the integral, click on the Integrate icon,[pic] The value of the integral is displayed in a small box.

Does the size of the integral make sense? Since the force-time curve is almost triangular in shape, you can estimate its area as ½ (base)(height). If the computer gives an obviously wrong answer you have probably selected the wrong region. Recheck your work.

Experimental results

In trial three you should remove the bumper from the force probe. Replace it with a cone of clay that is about 1.5 cm long and 1 cm in diameter. The knife edge on the cart should hit the center of the cone and stick in the clay. Be sure to remove the clay and replace the bumper when you have completed the experiment.

Report your results in the table below. Also hand in printed copies of the three graphs you record. Be sure to label your graphs.

| | | | | | | | | |

|Trial |Glider |Initial |Final |Δp |J |Fmax (N) |Collision time|Percent error |

| |Mass (kg) | | |(kg m/s) |(N s) | |Δt (s) |100%( |

| | | | | | | | ||Δp – J|/Δp |

| | | | | | | | | | | |

| | |vi (m/s) |pi |vf (m/s) |pf | | | | | |

| | | |(kg m/s) | |(kg m/s) | | | | | |

| | | | | | | | | | | |

|1 | | | | | | | | | | |

| | | | | | | | | | | |

|2 | | | | | | | | | | |

| | | | | | | | | | | |

|3 | | | | | | | | | | |

Questions

1. When the mass of the glider was increased (assuming that the glider’s speed was the same) was the change in the glider’s momentum changed? If so, by what percentage amount?

Was the impulse increased? If so, by what amount?

2. Compare trials 1 and 2. Did the bumper exert a greater force, or did the collision occur over a longer time (or both)?

3. Describe the difference in the impulse delivered by the bumper when the glider stopped in the clay as opposed to when it bounced off the bumper. (How did the impulse change? Did the maximum force change, the time interval change, or both? In what way did it change? Be specific.)

Collisions in Two Dimensions

Equipment

* World in Motion program

* Video clips

Introduction

Force, velocity, momentum, and impulse are all vectors. In one dimension, of course, this means that the sign is important. In more than one dimension the vector nature of these quantities means that they require more careful attention. While the motion and force detectors are fine for one-dimensional motion, a video clip provides a good record of motion in two dimensions. You will analyze two collisions, one of a pair of billiard balls, the second a pair of pucks on an air hockey table. Your task is to find out if the two objects act as a system in which momentum is conserved. You will also find out if kinetic energy is conserved in these collisions.

Analyzing a collision

After quitting LoggerPro, click on World in Motion. Your instructor will tell you which video files to open. Open the correct clip. According to the textbook, under what conditions is the momentum of a system conserved?_______________________________________________________

For the system of the two objects, either billiard balls or hockey pucks, using the textbook criterion, would you expect the momentum to be conserved? Give reasons for your prediction. ______________________________________________

The balls and pucks are extended objects. You must mark the same point on each object in each frame. The reflection of the light is a good point to mark on the billiard balls. The center of each puck is clearly marked. Begin by marking the location of one ball or puck (use the large brown puck) on each video frame. When you have reached the last frame click on the button Data Set 1 (lower right-hand corner). The video clip will return to the first frame and you should mark the location of the second ball or puck (the small yellow puck) on each frame.

When you have marked all the frames click on the Graphs button. On the Data analysis choices menu choose Option 3. On Direction of Motion menu choose both objects moving in the X and Y directions, with Y horizontal. On the Plots menu choose momentum and energy versus time. Click on the Next button.

Click on the Graph>> button to display the first graph, the momentum of object 1. The graphs will be cluttered. To view only one graph at a time, open the Graph Tools menu and click on Jog. Click on Jog again to see the second graph; again to see the third. To see the momentum of object 2, click on the Graph>> button.

Use your judgment to find best values for the momenta of the two objects in both the x- and y-directions before they hit and near the end of the clip. Subtract the components to find the momentum change Δp of each. In this collision, the system is the two objects. To find out if the momentum of the system is conserved, find Px = p1x + p2x and Py = p1y + p2y before the collision and at the end of the clip. If momentum is conserved, then the change in momentum of the system should equal zero (or at least be much smaller than either the initial or final momentum).

| | | | |

|Quantity |Ball 1 |Ball 2 |System |

| | | | | | | |

| |p1x (kg m/s) |p1y (kg m/s) |p2x (kg m/s) |p2y (kg m/s) |Px (kg m/s) |Py (kg m/s) |

| | | | | | | |

|pInitial | | | | | | |

| | | | | | | |

|pFinal | | | | | | |

| | | | | | | |

|Δp | | | | | | |

Within the percent differences, is the momentum of the system conserved? (Explain your reasoning) __________________________________.

The program calculates the kinetic energy of each object. Determine the kinetic energies of each ball before and after the collision. Add them to find the kinetic energy of the system. Report your results in the table below:

| | | | |

|Quantity |Ball 1 |Ball 2 |System |

| | | | |

|KInitial | | | |

| | | | |

|KFinal | | | |

| | | | |

|ΔK | | | |

Is the kinetic energy of the system conserved? Explain your conclusion _______________________.

Repeat the analysis for a video clip of a collision of two pucks.

| | | | |

|Quantity |Puck 1 |Puck 2 |System |

| | | | | | | |

| |p1x (kg m/s) |p1y (kg m/s) |p2x (kg m/s) |p2y (kg m/s) |Px (kg m/s) |Py (kg m/s) |

| | | | | | | |

|pInitial | | | | | | |

| | | | | | | |

|pFinal | | | | | | |

| | | | | | | |

|Δp | | | | | | |

Within the percent differences of your measurements, is the momentum of the system conserved? (Explain your reasoning) __________________________________.

| | | | |

|Quantity |Puck 1 |Puck 2 |System |

| | | | |

|KInitial | | | |

| | | | |

|KFinal | | | |

| | | | |

|ΔK | | | |

Is the kinetic energy of the system conserved? Explain your conclusion _______________________.

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