Collisions in 2Dp. 1/7

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PRELAB: COLLISIONS IN TWO DIMENSIONS1. In the collision described in Prediction 1-1, what is the direction of the change in momentum

vector pr

D for the less massive puck? for the more massive puck? Explain how youdetermined your answers.

2. Make Prediction 1-2 in the space below. Explain the reasoning behind your prediction.

3. The diagram below shows the initial setup for the collision in Prediction 2-1. In the space tothe right of the diagram, sketch a diagram which shows the final momentum vector for eachpuck in the collision described in Prediction 2-1. Explain the reasoning behind your diagram.

4. Sketch the change in momentum vector for the slower puck in the collision of Prediction 2-1.Explain why the p

rD vector points in the direction you have shown.

Collisions in 2Dp. 2/7

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COLLISIONS IN TWO DIMENSIONSTopic: Momentum, impulse and vector addition

Overview: In this lab, you will observe collisions between two pucks as they travel on an airtable. The pucks are connected to a spark timer. As the pucks travel over a sheet of newsprint,the spark timer records the position of both pucks at regular time intervals. You can use theposition information from the marks on the paper to estimate the x- and y-components of thevelocity vector for each puck.

In class, you have probably studied momentum and used the concept to analyze collisions. In thislab, you will use momentum to analyze collisions and explore the limitations of the principle ofmomentum conservation.

Writing it up: Throughout this handout, you will be asked to answer questions, sketchgraphs and diagrams, and do calculations. Write these things in your lab notebook as you gothrough the experiment. Label each answer/graph/calculation/diagram so that you (or your labTA) can find things quickly. If you have any computer printouts (such as graphs), remember toaffix them to your lab notebook. After lab, write a short (<300 words) conclusion of theexperiment that summarizes what you did and the major findings of the experiment.

Safety/Equipment Tips

Shock hazard: It is possible to get an (unpleasant, but not dangerous) electric shock from theapparatus if you touch the metal part of one of the pucks while the spark timer isoperating.

Avoid getting shocked:

• Do not touch the metal part of the pucks when the spark generator is on!• Make sure both pucks are on the air table when the spark timer is on!• Turn the spark generator off after each run.

Protect the equipment:

• Press the foot pedal only while the pucks are moving. A series of sparks in oneposition will burn a hole in the specialty carbon paper and/or mar the smooth surfaceof the air table.

• Leave the pucks on the air table when the apparatus is idle (to avoid stretching thefragile air tubes).

Room setup notes (for TA’s)

Given the limited number of stations available for this experiment (five working air tables inOctober, 2003), students should stay at the air tables only when they are collecting data.Students can do the Investigations in either order, so you may want to have half the classwork on Investigation 1 while the other half works on Investigation 2.

Two or three stations should have unequal puck masses (for the elastic collision in Investigation1). The remaining two or three stations should have pucks with about equal masses andVelcro (for the completely inelastic collision in Investigation 2).

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Procedure:

Some preliminaries…

1. There should be a sheet of carbon paper (carbon side up) on the surface of the air table. (Ifthere isn’t, consult your TA). Place a sheet of newsprint on top of the carbon paper.

2. Turn on the air source. Check to make sure the table is level by placing the pucks in themiddle of the table. Ideally, the pucks should remain motionless. If the table is not level,level the table by adjusting the legs.

3. Set the spark timer to 30 Hz so that there is 1/30 of a second between the sparks.

Note: The two investigations in this experiment can be done in either order and the analysis ofthe data is essentially identical for the two Investigations. The questions in Investigation 1provide more guidance about how to analyze the data than do the questions in Investigation 2.

Investigation 1: A nearly elastic collision:

In elastic collisions (like those between billiard balls), the objects bounce off each other“cleanly” without sticking together at all. In this investigation, you will investigate changes invelocity and momentum before, during and after a nearly elastic collision.

1. Data taking throughout this lab takes a little coordination. You must release the pucks so thatthey have the velocities you want and then, as soon as the pucks are released, press down andhold the spark generator’s foot switch. (The spark generator only makes sparks when the footswitch is depressed). Just before the first puck hits the edge of the air table, release theswitch. To avoid confusing data, practice each run at least once with the sparkgenerator turned off.

Before tackling a two dimensional collision, it might be helpful to consider a one dimensionalexample first.

Prediction 1-1: Two pucks with different masses approach each other head-on, as shown below.The more massive puck is traveling faster than the less massive one before the collision. The twopucks collide elastically. Sketch a qualitative prediction of the movement of the pucks after thecollision. What is the direction of the change in momentum vector p

rD for the less massive puck?

What is the direction of the change in momentum vector for the more massive puck? Explainhow you determined your answers.

Prediction 1-2: Which puck will experience a larger change in velocity? Which puck willexperience a larger change in momentum? Explain your reasoning. (Keep in mind that velocityand momentum are both vector quantities).

2. Set up to test Prediction 1-1: Turn on the air source. Make sure the spark generator is off.(You will do a practice data run before turning on the spark generator). Start both pucks atopposite ends of the table. Push the pucks toward each other. Each puck should travel at low

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to moderate velocity. Press the foot switch just after you release the pucks. Hold the switchdown as the pucks travel across the table. Release the switch just before either puck hits theside of the air table. Once you are confident of your timing, turn on the spark generator andtake data to test Prediction 1-1. Turn the spark generator off. (Your collision does not have tobe exactly “head-on.” In practice, it is almost impossible to get a 1-dimensional collisionwith this apparatus.)

3. Pick up the newsprint and turn it over. (You should see a series of dots that traces the pathsof the two pucks). Place the newsprint into the frame of the measuring apparatus. Use themeasuring frame to record the x- and y- coordinates of one of the two pucks. Constructgraphs of x-position versus time and y-position versus time for that puck. The graph shouldhave 10 to 20 data points. (Note: Use Excel or some other spreadsheet program to make thegraph).

Q1-1: Is the x-component of the puck’s velocity constant before the collision? Is the y-component? Explain how you can tell from your graphs. Is this what you expected? Isthe puck’s velocity vector constant after the collision?

Q1-2: Determine the velocity vector for each puck before and after the collision. If you usethe definition of average velocity for your calculation, clearly show which two datapoints you used for the calculation. Explain why you chose those two points. Expressyour answers as vectors in component form (e.g. ]ˆ2.0ˆ1.3[,1 yxv initialpuck +=

r cm/spark).

Q1-3: Use the calculations you have done so far to find the change in velocity( ialfinal vvv int

rrr-=D ) of each puck as a result of the collision. Compare the result with

your prediction. Also, find the change in the momentum of each puck during thecollision. ( beforeafter ppp

rrr-=D ). Express your answers as vectors in component form.

Compare the results with your prediction. (Even though it is unlikely that yourexperiment produced the one-dimensional collision described in the Prediction, youshould still be able to make some legitimate and insightful comparisons).

A quantity called impulse may have been defined in lecture and/or in the textbook. It combinesthe applied force and the time interval over which it acts. In one dimension, for a constant forceF acting over a time interval tD , the magnitude of the impulse is

tFJ D=

As you can see, a large force acting over and short time and small force acting over a short timecan have the same impulse. Notice tFD that is the area of the rectangle, i.e., the area under theforce vs. time curve.

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In general, the impulse delivered by a force Fr

acting over the time interval from time 1t to time2t is a vector quantity defined by

Ú≡2

1

t

tdtFJ

rr

If the force Fr

in the integral equals the net force netFr

acting on the object, then the impulseequals the change in momentum of the object during the time interval from clock reading 1t toclock reading 2t .

pJrr

D=

This result is called the impulse-momentum theorem. You might notice the impulse-momentumtheorem is equivalent to Newton’s 2nd law. (Simply take the time derivative of both sides torecover Newton’s 2nd law).

Q1-4: Estimate the impulse delivered to each puck during the collision. Express youranswers as vectors. Explain how you determined your answer. (Yes, the answer tothis question is very easy).

You may have used the conservation of momentum principle to analyze collisions like the one inthis experiment. The principle simply states that the total momentum of all objects in a systembefore a collision equals the total momentum after the collision. While you have probably usedthe principle in homework problems, you may not be aware of its mathematical basis. Underwhat conditions does the principle apply? In order to find out, let’s take a careful look at itsderivation. Consider a collision between two objects. The argument starts will the impulse-momentum theorem. The impulse acting on each puck equals the change in momentum of thatpuck:

11 pJrr

D= and 22 pJrr

D=

where 1Jr

is the impulse delivered to puck 1 by the net force acting on puck 1 and 2Jr

is theimpulse delivered to puck 2 by the net force acting on puck 2. If the only force acting on puck 1during the collision is due to puck 2, the total impulse delivered to puck 1 is equal to the impulsedelivered by puck 2:

Ú Æ=2

1121

t

tdtFJ

rr

Similarly, if puck 1 is the only object exerting a force on puck 2:

Ú Æ=2

1212

t

tdtFJ

rr

The final step of the argument is to apply Newton’s 3rd law. At all instants during the collision,2112 ÆÆ -= FF

rr. Therefore, 21 JJ

rr-= and 21 pp

vvD-=D . Simple algebra gives the principle of

momentum conservation: 021 =D+D ppvv (i.e. there is no change in the total momentum 21 pp

vv+ of

the system).Q1-5: Is it possible that the net force experienced by puck 1 during the collision does not

equal the force puck 2 exerts on puck 1? Explain.Q1-6: In practice, you will find that the impulse delivered to puck 1 is not equal and

opposite to the impulse delivered to puck 2. Is this evidence that one of the

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assumptions in the derivation above is being violated during the collision? Or is thisevidence of inexact measurements? (Note: If you argue that the observed difference isdue to the violation of one of the assumptions of the derivation, clearly identify whichassumption is likely to have been violated and explain how the assumption mighthave been violated. If you argue that the observed difference is simply due to inexactmeasurements, back up your assertion with calculations! You will need toconvincingly show that the difference is most likely due to inexact measurements.)

Q1-7: Estimate the magnitude of average force exerted on puck 1 during the collision.Explain how you arrived at your estimate. How reliable is this estimate? Explain. Isthis estimate likely to be too low or too high? Explain.

Investigation 2: An inelastic collision

When two objects stick together after a collision, the collision is called a completely inelasticcollision. In this section, you will use two pucks with Velcro sides to examine a completelyinelastic collision.

Prediction 2-1: Two pucks approach each other at an angle, as shown below. The pucks sticktogether and travel together after the collision. Sketch a qualitative prediction of the movementof the pucks after the collision. Sketch and label the change in momentum vector p

rD for the

faster puck on your diagram. Explain how you determined your answer.

Prediction 2-2: Which puck will experience a larger change in velocity? Which puck willexperience a larger change in momentum? Explain your reasoning.

1. Put the Velcro bands around each puck.

2. Place the newsprint you used for Investigation 1 on the carbon paper. The dots you made inInvestigation 1 should visible. (Remember that the new carbon paper marks will appear onthe underside of the newsprint). Turn on the air source. (Make sure the spark generator is offfor the practice runs). Push the pucks toward each other. Start the pucks so that they willcollide. Press the foot switch just after you release the pucks. Hold down the foot switch asthe pucks travel, collide and then travel after the collision. Release the foot switch beforeeither puck hits the far edge of the table. Once you are confident of the timing, turn on thespark generator and take data. Note the general direction of each puck’s initial velocity, sothat you can identify which spark track goes with which puck when you analyze the data.

3. Pick up the newsprint and turn it over. (You should see a series of dots that traces the pathsof the two pucks). Place the newsprint into the frame of the measuring apparatus.

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Q2-1: Estimate the momentum of each puck just before the collision. Express your answeras a vector in component form. Explain how you arrived at your answer. If you usethe definition of average velocity for your calculation, clearly show which two datapoints you used for the calculation. Explain why you chose those two points.

Q2-2: Estimate the momentum of each puck just after the collision. Express your answer asa vector in component form. Explain how you arrived at your answer. If you use thedefinition of average velocity for your calculation, clearly show which two datapoints you used for the calculation. Explain why you chose those two points.

Q2-3: Use the results of the previous two questions to calculate the change in velocityvr

D for each puck during the collision. Express your answers as vectors.Q2-4: Calculate the change in momentum p

rD for each puck during the collision. Again,

express your answers as vectors.Q2-5: Compare the change in velocity for puck 1 with the change in velocity for puck 2.

Are they in the same direction? Do they have the same magnitude? Is this what youexpect? Explain.

Q2-6: Compare the change in momentum for puck 1 with the change in momentum for puck2. Are they in the same direction? Do they have the same magnitude? Is this what youexpect? Explain.

Q2-7: Estimate the magnitude of the average force exerted on puck during the collision.Explain how you arrived at your estimate. How reliable is this estimate? Explain. Isthis estimate likely to be too low or too high? Explain.