UNIT 8: ONE-DIMENSIONAL COLLISIONS

Name ______________________ Date(YY/MM/DD) ______/_________/_______ St.No. __ __ __ __ __-__ __ __ __ Section__________

UNIT 8: ONE-DIMENSIONAL COLLISIONSApproximate Classroom Time: Three 100 minute sessions

In any system of bodies which act on each other, action and reaction, estimated by momentum gained and lost, balance each other according to the laws of equilibrium.

Jean de la Rond D'Alembert 18th Century

OBJECTIVES 1. To understand the definition of momentum and its vector nature as it applies to one-dimensional collisions.

2. To reformulate Newton's second law in terms of change in momentum, using the fact that Newton's "motion" is what we refer to as momentum .

3. To develop the concept of impulse to explain how forces act over time when an object undergoes a collision.

4. To use Newton's second law to develop a mathematical equation relating impulse and momentum change for any object experiencing a force.

5. To verify the relationship between impulse and momentum experimentally.

6. To study the forces between objects that undergo collisions and other types of interactions in a short time period.

7. To formulate the Law of Conservation of Momentum as a theoretical consequence of Newton's laws and to verify it experimentally.

© 1992-94 Dept. of Physics & Astronomy, Dickinson College Supported by FIPSE (U.S. Dept. of Ed.) and NSF. Modified for SFU by N. Alberding, 2005.

OVERVIEW25 min 1:30 Includes time to review unit 7 problems

In this unit we will explore the forces of interaction between two or more objects and study the changes in motion that result from these interactions. We are especially interested in studying collisions and explosions in which interactions take place in fractions of a second or less. Early investigators spent a considerable amount of time trying to observe collisions and explosions, but they encountered difficulties.

This is not surprising, since the observation of the details of such phenomena requires the use of instrumentation that was not yet invented (such as the high speed camera). However, the principles of the outcomes of collisions were well understood by the late seventeenth century, when several leading European scientists (including Sir Isaac Newton) developed the concept of quantity -of-motion to describe both elastic collisions (in which objects bounce off each other) and inelastic collisions (in which objects stick together). These days we use the word momentum rather than motion in describing collisions and explosions.

We will begin our study of collisions by exploring the relationship between the forces experienced by an object and its momentum change. It can be shown mathematically from Newton's laws and experimentally from our own observations that the integral of force experienced by an object over time is equal to its change in momentum. This time-integral of force is defined as a special quantity called impulse, and the statement of equality between impulse and momentum change is known as the impulse–momentum theorem.

Next you will study the one-dimensional interaction forces between two colliding and exploding objects. By combining the results of this study with the impulse-momentum theorem, you can prove theoretically that momentum ought to be conserved in any interaction. It can be verified experimentally that whenever an object explodes or whenever two or more bodies collide, the momentum of the bodies before the event and their momentum after the event remain the same as long as no external force acts on them. At the conclusion of this study you will have the opportunity to use video analysis to verify the Law of Conservation of Momentum.

When the Law of Conservation of Momentum and the impulse-momentum theorem are applied to the study of collisions between two or more objects, physicists can learn

Page 8-2 Workshop Physics II Activity Guide SFU 1057


about the interaction forces among them. There is no way to make direct measurements of the tiny forces of interaction between the various particles that are the fundamental building blocks of matter. Contemporary physicists working in accelerator laboratories bombard materials with tiny, rapidly moving particles and collect data on the momentum changes that occur during the collisions. This provides them with an indirect way to learn about fundamental forces of interaction. Those of you who will continue the study of physics will revisit these relationships between momentum changes and forces many times.

Workshop Physics II: Unit 8 – One-Dimensional Collisions Page 8-3Author: Priscilla Laws w/ David Sokoloff and Ronald Thornton

© 1992-94 Dept. of Physics & Astronomy, Dickinson College Supported by FIPSE (U.S. Dept. of Ed.) and NSF

SESSION ONE: MOMENTUM AND MOMENTUM CHANGE20 min 1:55

Defining MomentumIn this session we are going to develop the concept of momentum to predict the outcome of collisions. But you don't officially know what momentum is because we haven't defined it yet. Let’s start by predicting what will happen as a result of a simple one-dimensional collision. This should help you figure out how to define momentum to enable you to describe collisions in mathematical terms.

Figure 8-1: An impending collision between two unequal masses

It's early fall and you are driving along a two lane highway in a rented moving van. It is full of all of your possessions so you and the loaded truck have a mass of 4000 kg. You have just slowed down to 30 km/h because you're in a school zone. It's a good thing you thought to do that because a group of first graders is just starting to cross the road. Just as you pass the children you see a 1000 kg sports car in the oncoming lane heading straight for the children at about 120 km/h. What a fool the driver is! A desperate thought crosses your mind. You figure that you just have time to swing into the oncoming lane and speed up a bit before making a head-on collision with the sports car. You want your truck and the sports car to crumple into a heap that sticks together and doesn't move. Can you save the children or is this just a suicidal act? For simulated observations of this situation you can use two carts of different masses set up to stick together in trial collisions. You will need: • 1 cart of mass m • 1 cart of larger mass M • Velcro or clay to make collisions sticky • A level track or a smooth level surface



✍ Activity 8-1: Can You Stop the Car?(a) Predict how fast you would have to be going to completely stop the sports car. Explain the reasons for your prediction.

(b) Try some head on collisions with the carts of different masses to simulate the event. Describe some of your observations. What happens when the less massive cart is moving much faster than the more massive cart? Much slower? At about the same speed?

(c) Based on your intuitive answers in parts (a) and (b) and your observations, what mathematical definition might you use to describe the momentum (or motion) you would need to stop an oncoming vehicle travelling with a known mass and velocity?

Just to double check your reasoning, you should have come to the conclusion that momentum is defined by the vector equation

€

r p ≡m

r v [Eq. 8-1]

where the symbol ≡ means "defined as."

10 min 2:15Expressing Newton's Second Law Using MomentumOriginally Newton did not use the concept of acceleration or velocity in his laws. Instead he used the term "motion", which he defined as the product of mass and velocity (a quantity we now call momentum). Let's examine a translation from Latin of Newton's first two laws (with some parenthetical changes for clarity).



Newton’s First Two Laws of Motion 1,2

1. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed on it.

2. The (rate of) change of motion is proportional to the motive force impressed: and is made in the direction of the right line in which that force is impressed.

The more familiar contemporary statement of the second law is that the net force on an object is the product of its mass and its acceleration where the direction of the force and of the resulting acceleration are the same. Newton's statement of the law and the more modern statement are mathematically equivalent, as you will show.

✍ Activity 8-2: Re-expressing Newton's Second Law(a) Write down the contemporary mathematical expression for Newton's second law relating net force to mass and acceleration. Please use vector signs and a summation sign where appropriate.

(b) Write down the definition of instantaneous acceleration in terms of the rate of change of velocity. Again, use vector signs.

(c) It can be shown that if an object has a changing velocity and a constant mass then

€

md

r v

dt=

d(mr v )

dt . Explain why.



1 I. Newton, Principia Mathematica, Florian Cajori, Ed. (U. of Calif. Press, Berkeley, 1934). p. 13.

2 L.W. Taylor, Physics the Pioneer Science, Vol. 1 (Dover, New York, 1959). pp. 129-131

(d) Show that

€

r F = m

r a =

dr p

dt∑

(e) Explain in detail why Newton's statement of the second law and the mathematical expression

€

r F = d

r p

dt∑

are logically equivalent.

35 min 2:25Momentum Change and Collision Forces

What's Your Intuition? You are sleeping in your sister's room while she is away at college. Your house is on fire and smoke is pouring into the partially open bedroom door. The room is so messy that you cannot get to the door. The only way to close the door is to throw either a blob of clay or a super ball at the door – there's not enough time to throw both.

✍ Activity 8-3: What Packs the Biggest Wallop—A Clay Blob or a Super ball?Assuming that the clay blob and the super ball have the same mass, which would you throw to close the door – he clay blob (which will stick to the door) or the super ball (which will bounce back with almost the same velocity it had before it collided with the door)? Give reasons for your choice, using any notions you already have or any new concepts developed in physics such as force, momentum, Newton's laws, etc. Remember, your life depends on it!



Observing the Wallop!Let's check out your intuition by dropping a bouncy ball onto a scale and then dropping a dead ball of approximately the same mass onto the scale from the same height. We can associate the maximum force on the scale with the maximum force a thrown ball can exert on a door. We would like to investigate how the maximum force is related to the change in momentum of the ball in each case. To do these observations you'll need the following equipment:

• A small live ball (of mass m) • A small dead ball or blob of clay (also of mass m) • A platform scale • A metre stick

✍ Activity 8-4: Observing the Wallop of a Super Ball

m

(a) Drop the bouncy ball from a fixed height of about one metre onto the platform scale. What is the maximum force on the scale in newtons? Drop the dead ball in the same way. What is the maximum force on the scale in newtons? Record the two forces below. Also record the height of your drop. Note: If the scale reads in grams or kilograms, you will have to calculate the force in newtons by using the fact that g = 9.8 N/kg.

(b) How good was your intuition?

(c) What will happen to the maximum force if you increase the mass of the ball but allow it to collide with the scale with the same velocity? What will happen if you increase the magnitude of the velocity of the ball just before impact by dropping it from a greater height? Try it!

It would be nice to be able to use Newton's formulation of the second law of motion to find collision forces, but it is difficult to measure the rate of change of momentum during a rapid collision without special instruments.



However, measuring the momenta of objects just before and just after a collision is usually not too difficult. This led scientists in the seventeenth and eighteenth centuries to concentrate on the overall changes in momentum that resulted from collisions. They then tried to relate changes in momentum to the forces experienced by an object during a collision. In the next activity you are going to explore the mathematics of calculating momentum changes.

✍ Activity 8-5: Predicting Momentum ChangesWhich object undergoes the most momentum change during the collision with a door – the clay blob or the super ball? Explain your reasoning carefully.

Let's check your reasoning with some formal calculations of the momentum changes for both inelastic and elastic collisions. This is a good review of the properties of one-dimensional vectors. Recall that momentum is defined as a vector quantity that has both magnitude and direction. Mathematically, momentum change is given by the equation

€

Δr p = r

p f –r p i [Eq. 8-2]

where

€

r p i is the initial momentum of the object just before

and

€

r p f is its final momentum just after a collision.



✍ Activity 8-6: Calculating 1D Momentum Changes

(a) Suppose a dead ball (or clay blob) is dropped on a table and "sticks" in such a way that it has an initial momentum just

before it hits of p i= !p

i ! where ĵ is a unit vector pointing along

the positive y axis. Express the final momentum of the dead ball in the same vector notation.

€

r p f =

(b) What is the change in momentum of the clay blob as a result of its collision with the table? Use the same type of unit vector notation to express your answer.

€

Δr p =

(c) Suppose that a live ball (or a super ball) is dropped on a table and "bounces" on the table in an elastic collision so that its speed just before and just after the bounce are the same. Also suppose

that just before it bounces it has an initial momentum p i= !p

i !

where ĵ is a unit vector pointing along the positive y-axis. What is the final momentum of the ball in the same vector notation? Hint: Does the final p vector point along the +y or –y axis?

€

r p f =

(d) What is the change in momentum of the ball as a result of the collision? Use the same type of unit vector notation to express your result.

€

Δr p =

(e) The answer is not zero. Why? How does this result compare with your prediction? Discuss this situation.

(f) Using the balance, verify that the masses of the two balls are approximately the same and record the mass below. Using the measured value for mass of the balls and a calculated value for the velocity of each of the balls just before they hit the platform, you can calculate the momentum just before the collision for each of the balls. Also calculate the momentum of the balls just after the collision. Summarize the equations you are using and your results in the space below.



Reminder: î and ĵ represent unit vectors pointing along the x- and y-axes, respectively.

€

r v = (list the equation in terms of g and y)

€

r p = (list the equation in terms of m and v)

€

Δr p = (list the equation in terms of pi and pf )

Units Live Ball Dead Ball

mass, m

Height of Drop, y

Initial Velocity, v i

Final Velocity, v f

Final Momentum, p f

Momentum Change, Δp

20 min 3:00Applying Newton's Second Law to the Collision Process

The Egg TossSuppose somebody tosses you a raw egg and you catch it. In physics jargon, one would say (in a very official tone of voice) that "the egg and the hand have undergone an inelastic collision." What is the relationship between the force you have to exert on the egg to stop it, the time it takes you to stop it, and the momentum change that the egg experiences? You ought to have some intuition about this matter. In more ordinary language, would you catch an egg slowly or fast? For this consideration you may want to use:

• 3 or 4 raw eggs (in shell)

✍ Activity 8-7: Momentum Changes and Average Forces on an Egg: What's Your Intuition?(a) If you catch an egg of mass m that is heading toward your hand at speed v what is the magnitude of the momentum change that it undergoes?



(b) Does the total momentum change differ if you catch the egg more slowly or is it the same?

(c) Suppose the time you take to bring the egg to a stop is ∆t. Would you rather catch the egg in such a way that ∆t is small or large. Why?

(d) What do you suspect might happen to the average force you exert on the egg while catching it when ∆t is small?

Figure 8-2: Catching a punch instead of an egg

You can use Newton's second law to derive a mathematical relationship between momentum change, force, and collision times for objects. This derivation leads to the impulse-momentum theorem that we mentioned in the overview. Let's apply Newton's second law to the egg catching scenario.



✍ Activity 8-8: Force and Momentum Change (a) Sketch an arrow representing the magnitude and direction of the force exerted by your hand on the egg as you catch it.

(b) Write the mathematical expression for Newton's second law in terms of the net force and the time rate of change of momentum. (See Activity 8-2(e) for details.)

(c) Explain why, if

€

r F is a constant during the collision lasting a

time ∆t , then

€

dr p

dt=Δ

r p Δt

(d) Show that for a constant force

€

r F the change in momentum is

given by

!

"r p =

r

F "t. Note that for a constant force, the term

€

r F Δt

is known as the impulse given to one body by another.



SESSION TWO: IMPULSE, MOMENTUM AND INTERACTIONS25 min

Review of Homework AssignmentCome prepared to ask questions about the homework assignments on force problems involving free body diagrams problems like those in Unit 7 as well as problems on impulse and average force.

30 minThe Impulse-Momentum TheoremReal collisions, like those between eggs and hands, a Nerf ball and a wall, or a falling ball and a platform scale are tricky to study because ∆t is so small and the collision forces are not really constant over the time the colliding objects are in contact. Thus, we cannot calculate the impulse as F∆t. Before we study more realistic collision processes, let's redo the theory for a variable force. In a collision, according to Newton's second law, the force exerted on a falling ball by the platform at any infinitesimally small instant in time is given by

€

r F =

dr p

dt [Eq. 8-3]

To describe a general collision that takes place between an initial time ti and a final time tf , we must take the integral of both sides of the equation with respect to time. This gives

€

r F

ti

tf

∫ dt =d

r p

dtti

tf

∫ dt =r p f −

r p i = Δ

r p

[Eq. 8-4]

Impulse is a vector quantity defined by the equation

€

r J =

r F

ti

tf

∫ dt [Eq. 8-5]

By combining equations [8-4] and [8-5] we can formulate the impulse-momentum theorem in which

€

r J = Δ

r p [Eq. 8-6]

If you are not used to mathematical integrals and how to solve them yet, don't panic. If you have a fairly smooth graph of how the force F varies as a function of time, the



impulse integral can be calculated as the area under the F-t curve.

Let's see qualitatively what an impulse curve might look like in a real collision in which the forces change over time during the collision. In particular, let's play with a couple of objects that distort a lot during collisions:

• A Nerf ball • A dynamics cart with a spring plunger

✍ Activity 8-9: Observing Collision Forces That Change(a) Suppose the cart with the spring loaded plunger or a Nerf ball is barrelling toward a wall and collides with it. If friction is neglected, what is the net force exerted on the object just before it starts to collide?

(b) When will the magnitude of the force on the cart be a maximum?

(c) Watch the cart with its spring loaded plunger collide with a wall several times. Roughly how long does the collision process take? Half a second? Less? Several seconds?

(d) Remembering what you observed, attempt a rough sketch of the shape of the force the wall exerts on a moving object during a collision.



55 minVerification of the Impulse-Momentum TheoremTo verify the impulse-momentum theorem experimentally we must show that for an actual collision involving a single force on an object the equation

€

Δr p =

r F

ti

tf

∫ dt

holds, where the impulse integral can be calculated by finding the area under the curve of a graph of F vs. t.

Figure 8-3: Evaluating the impulse integral as the area under the graph of force as a function of time.

By using a computer it is possible to measure changes in force as a function of time during actual collisions. A force probe set-up is shown for the case of a dynamics cart colliding with a force probe in the illustration below.



Figure 8-4: Apparatus for measuring collision forces on a force probe mounted on a cart and an "immovable object "in the form of a wall. In your experiments the wall may be replaced by another obstacle.

Your task is to see if you can set up a collision situation that will allow you to monitor the change of force as a function of time during the collision so that you can calculate the impulse

€

r J =

r F

ti

tf

∫ dt

experienced by the cart. At the same time, you must take measurements to allow you to determine the momentum before and after the collision and hence the momentum change of the object. You can then determine whether or not the impulse associated with the collision is equal (within the limits of experimental uncertainty) to the momentum change of the cart. With a computer force detection system that is capable of taking 1000 or more force readings a second, there are many ways to set up a verification experiment with the equipment available in a typical introductory physics laboratory. For this experiment we will focus on studying very gentle collisions between a dynamics cart and a force probe, as shown in Figure 8-4.

If on wishes to measure velocity of an object at the same time as the force during a collision then two computer interfaces are needed because the data rates for collecting force and velocity data need to be very different.



Apparatus and supplies that you will need for this experiment include:

• A computer system with a force probe• A 5 N spring scale (for calibration)• A motion detection system• An electronic scale or balance (1000 g)• A ruler• A dynamics cart with a spring loaded plunger • Tape or a mounting system to secure the force probe to

the cart

Warning: Do not push or pull a force probe with more than 20 N.

Understanding How to Use and Calibrate a Force Detection System for Impulse StudiesOnly a small percentage of introductory physics students have done impulse experiments, because such experiments cannot be done without a force detection system that is capable of taking 1000 or more force readings a second. The system we will be using was developed at Tufts University for use with Macintosh Computers. It uses the Lab Pro Interface developed jointly at Dickinson College and Tufts University that is being distributed by Vernier Software Company in Portland, Oregon. Your instructor(s) may have created a set-up file for your use entitled Impulse Experiment. If so, you may want to use it instead of following the instructions below.

Set-Up for Impulse Measurements for Forces Under 1 g Before you start make sure the Computer is on and the LoggerPro interface is plugged into a USB port and is powered.

1. Plug the force probe into CH 1 of the LabPro interface.2. Load the LoggerPro software.3. Choose Data Collection under the Experiment menu and

specify the following options:“Collections” tab: Mode: Time Based Length: 0.25 seconds Sampling Speed: 5000 samples/s“Triggering” tab: Enable Triggering Start data collection when the CH1: Force is increasing across 0.3 N. Collect 10 points before trigger.Click “Done”.

4. After you have gone through all the set up procedures you should save this "experiment" file on your own disk giving it a name such as "Impulse Experiment."



The force probe has two tips: a plastic nib for pushing and a hook for pulling. The unused tip should be stored in the threaded hole on the adapter plate.

Calibration of the Force Probe for Gentle Collisions1. Clamp the Probe in the orientation needed to suit the needs

of your experimental design. THIS IS IMPORTANT! Check that the sign of the force is correct. Push on the force sensor nib. Pushes to the right should be positive and to the left negative. To reverse the sign of the force select Show All Interfaces under the Experiment menu and choose the drop-down menu for the CH1 Force Probe. Click “Reverse Direction” so that the sign of the force is reversed.

2. Pull on the probe with 5 N from the spring scale to check the calibration. If the reading is significantly different from –5 N (or +5 N), then follow the calibration procedure.

3. Dismiss the Force Sensor dialogue and return to the main graph screen to do your experiments.

4. Note: Since the force probe output voltages change a bit in time, it is important to zero it frequently when you are taking measurements. This is done by clicking the mouse on the “Zero” button in the upper right hand corner next to the “Collect” button.

5. If you think you might use the same probe again put a label on it and save the force probe calibration file.

✍ Activity 8-10: An Impulse Experiment(a) Describe the measuring techniques and calculation methods you are using to determine the change in momentum of your object as a result of a collision with the force detector.

(b) Do the experiment in which you determine both the velocity change and the impulse curve for the same gentle collision. Remember, the maximum forces between the cart and the probe must be less than 10N. After some practice, use the analyse feature associated with the LoggerPro software and figure out how to find the approximate value of the integral of the F vs. t curve for the time between the start and end times of your collision. Explain what you did and list the value of the integral below.



(c) Make a sketch of your impulse curve in the space below. The sketch should quantitatively show the duration of the interaction, the maximum force and the approximate shape of the force vs. time curve.

(d) Use the velocity data and the mass of the cart to calculate the change in momentum of the cart during the collision and give its value in the space below.

(e) Compare the change in momentum with the impulse, i.e., with the area under the F-t curve. Does the impulse-momentum theorem seem valid within the limits of experimental uncertainty for your collision?



SESSION THREE: NEWTON'S LAWS AND MOMENTUM CONSERVATION

20 min Review of Homework AssignmentCome prepared to ask questions about the homework assignment.

10 minPredicting Interaction Forces between ObjectsIn the last session we focused our attention on the change in momentum that an object undergoes when it experiences a force that is extended over time (even if that time is very short!). Since interactions like collisions and explosions never involve just one object, we would like to turn our attention to the mutual forces of interaction between two or more objects. As usual, you will be asked to make some predictions about interaction forces and then be given the opportunity to test these predictions.

✍ Activity 8-11: Predicting Interaction Forces(a) Suppose the masses of two objects are the same and that the objects are moving toward each other at the same speed so that

€

m1 = m2 and r v 1 =

r v 2

Predict the relative magnitudes of the forces between object 1 and object 2. Place a check next to your prediction!

____ Object 1 exerts more force on object 2.

____ The objects exert the same force on each other.




(b) Suppose the masses of two objects are the same and that object 1 is moving toward object 2, but object 2 is at rest.

€

m1 = m2 and r v 1 > 0,

r v 2 = 0

Predict the relative magnitudes of the forces between object 1 and object 2. Place a check next to your prediction!




(c) Suppose the mass of object 1 is much less than that of object 2 and that it is pushing object 2 which has a dead motor so that both objects move in the same direction at speed v.

€

m1 << m2 and r v 1 =

r v 2

Predict the relative magnitudes of the forces between object 1 and object 2. Place a check next to your prediction.


____ The objects exert the same force on each other




(d) Suppose the mass of object 1 is greater than that of object 2 and that the objects are moving toward each other at the same speed so that

€

m1 > m2 and r v 1 = −

r v 2





(e) Suppose the mass of object 1 is greater than that of object 2 and that object 2 is moving in the same direction as object 1 but not quite as fact so that

€

m1 > m2 and r v 1 >

r v 2







(f) Suppose the mass of object 1 is greater than that of object 2 and that both objects are at rest until an explosion occurs, so that

€

m1 > m2 and r v 1 =

r v 2 = 0





(g) Provide a summary of your predictions. What are the circumstances under which you predict that one object will exert more force on another object.

40 minMeasuring Mutual Forces of InteractionIn order to test the predictions you made in the last activity you can study gentle collisions between two force probes attached to carts. You can strap additional masses to one of the carts to increase its total mass so it has significantly more mass than the other. To make these observations you will need the following equipment:

• A computer set-up with software for measuring forces from 2 probes

• 2 force probes• A 5 N spring scale to calibrate the force probe.



• 2 dynamics carts• Masses to place on one of the carts to double or triple its

mass• OPTIONAL:

a compression spring for "explosions"

This apparatus should be set up as shown in the following diagram.

Figure 8-5: Set-up for reading two forces at once during a gentle collision or explosion.

Using the Computer System to Measure Two ForcesAn experiment file called Collisions has been prepared for these observations. Before these computer files can be used the two force probes must be zeroed and calibrated.

Set-up for the Measurement of Mutual Forces

SETUP1. Plug one force probe into CH 1 of the LabPro and the other

into CH 2.2. Load the Collisions experiment file into Logger Pro. You

should see two Force vs. Time axes, one for each force probe.3. Check to see if the probes are reading a reasonable value.

Hold the nib and pull on the hook using a spring scale. A 5-N pull using the spring scale should cause apx. 5 N to be displayed at the top of the display. If the force on the nib is to the right, then the force should be positive. A force to the left should read negative. If you’re unsure about the calibration, go to the calibration instructions at the end of this section. Don’t worry if the calibration is off by a small amount, (less than 1 N).

Note: Since the force probe output voltages change a bit in time, it is important to zero them frequently when you are taking measurements. This is done by clicking the mouse on the Zero box in the upper right hand corner of the graph screen.



✍ Activity 8-12: Measuring Slow Forces(a) Gently push the two carts together so that the force probe nibs touch. Watch the forces that are displayed. Are the forces opposite in sign? (If not adjust the sign of one probe.)

(b) Push the nibs of the two carts’ force probes together and move them back and forth. What do you observe about the size of the mutual forces?

Now that you're warmed up to this two force measurement technique go ahead and try some different types of gentle collisions between two carts of different masses and initial velocities.

✍ Activity 8-13: Measuring Collision Forces(a) Set up both sensors similar to the way they were set up in the impulse experiment. Choose a data rate of more than 1000 points per second and collect data for about 0.2 s. Trigger data collection when the force on one of the probes rises above 0.3 N. (Make sure you set the trigger on the probe sensing a positive force during the collision.) Use the two carts to explore various situations which correspond to the predictions you made about mutual forces. Your goal is to find out under what circumstances one object exerts more force on another object. Describe what you did in the space below and draw a sketch of at least one of your graphs of force 1 vs. time and force 2 vs. time.(Start by causing two carts of equal masses to collide. You should expect to see a two graphs of force vs. time corresponding to the two force probes. Then run more collisions with one or the other cart loaded with considerably more mass—use 250 g or more.)



(b) What can you conclude about forces of interactions during collisions? Under what circumstances does one object experience a different magnitude of force than another during a collision? How do the magnitudes and directions of the forces compare on a moment by moment basis in each case?

(c) Do your conclusions have anything to do with Newton's third law?

(d) How does the vector impulse due to object 1 acting on object 2 compare to the impulse of object 2 acting on object 1 in each case? Are they the same in magnitude or different? Do they have the same sign or a different sign?



Paul Hewitt, a well known physics teacher, who made a number of the drawings used in this Activity Guide, has a whimsical way of expressing the universality of Newton's third law in everyday life.

You can’t touchwithout being touched—Newton’s Third Law.

Figure 8-6: A "Hewitt drew it" cartoon on Newton's third law

35 minNewton's Laws and Momentum ConservationIn your investigations of interaction forces, you should have found that the forces between two objects are equal in magnitude and opposite in sign on a moment by moment basis for all the interactions you studied. This is of course a testimonial to the seemingly universal applicability of Newton's third law to interactions between ordinary masses. You can combine the findings of the impulse-momentum theorem (which is really another form of Newton's second law since we derived it mathematically from the second law) to derive the Law of Conservation of Momentum shown below.

Law of Conservation of Momentum

€

r P =

r P 1,i∑ +

r P 2,i =

r P 1,f +

r P 2,f = constant in time

[Eq. 8-7]

where 1 refers to object 1 and 2 to object 2 and i refers to the initial momenta and f to the final momenta.



✍ Activity 8-14: Deriving Momentum Conservation(a) What did you conclude in the last activity about the magnitude and sign of the impulse on object 1 due to object 2 and vice versa when two objects interact? (See Activity 8-13 (d).)

In other words, how does

€

r J 1 compare to

€

r J 2 ?

J ≡ F

ti

tf

dt

(b) Since you have already verified experimentally that the impulse-momentum theorem holds, what can you conclude about

how the change in momentum of object 1,

€

Δr p 1 , as a result of the

interaction compares to the change in momentum of object 2,

€

Δr p 2, as a result of the interaction?

J= ∆p

(c) Use the definition of momentum change to show that the Law of Conservation of Momentum ought to hold for a collision, so that

€

r P =

r P 1,i∑ +

r P 2,i =

r P 1,f +

r P 2,f = constant in time

In the next unit you will continue to study one- and two-dimensional collisions using momentum conservation. Right now you will attempt to verify the Law of Conservation of Momentum for a simple situation by using video analysis. To do this you will use a video movie entitled Magnetic Collision in which two carts interact at a distance, with one transferring momentum to the other. You may not be able to finish this in class, but you can complete the project for homework.



To demonstrate the magnetic collision you will need:

• A track • Two dynamics carts with magnets attached • Masses to place on the carts

✍ Activity 8-15: Verifying Momentum Conservation(a) Use the iSight camera and G-Cam to take a video of two carts (carts 1 and 2) of different masses colliding. Position the carts so that their magnetic bumpers cause them to bounce off each other. Use the same settings as you did last time with the projectile motion video. • Make a movie of a cart 1 colliding with cart 2.

Suggestion: Put a 250 g bar on cart 1 and and leave cart 2 without any extra mass. Send cart 1 toward a stationary cart 2. Feel free to try other variations...

• Analyse the frames to form a graph and data table of the velocities of cart1 before and after collision. Write down the initial and final velocities of the cart. Note: Since this is a horizontal 1D collision the y-coordinates are of no interest.

• Delete the video analysis of cart 1 and do the analysis to find the velocity of cart 2 after the collision. Record the final velocity of cart 2.

Initial velocity of cart 1: v1,i = _________

Final velocity of cart 1: v1,f = _________

Initial velocity of cart 2: v2,i = 0 (or ____________)

Final velocity of cart 2: v2,f = _________

(b) Use your data to calculate the momenta of carts 1 and 2 before the collision.

(c) Use the data to calculate the momenta of carts 1 and 2 after the collision.



(d) Within the limits of experimental uncertainty, does momentum seem to be conserved (i.e. is the total momentum of the two cart system the same before and after the collision)?



Calibrating the force probe:If you can apply a known force to the force probe you can calibrate it. Common ways to produce a known force are hanging a weight of a standard mass (a) or pulling on it with a spring scale marked in newtons (b,c or d). The mass is probably more accurate than a spring scale, but requires using a pulley. During the calibration, it is important that the force probe be held in the same orientation that it will be used in.

a)m

b) c) d)

1. Plug the force probe into CH1 (or CH2) input of the LabPro. Make sure the LabPro is powered and plugged into a USB port of the computer.

2. Zero the force probe while no force is applied by clicking on Zero next to the Collect button in the upper right-hand corner.

3. Under the Experiment menu choose Calibrate4. If there’s more than one sensor plugged in, select the channel having the sensor you’re going to calibrate. 5. Type in the value of the force that will be applied into the dialog box, While the force is being applied, click on Keep.6. You have to repeat this for another value of force.7. At this point the force probe is calibrated.8. If a second probe is connected to another channel you can repeat the procedure for it.

Note: Forces in opposite directions should have opposite signs. Normally forces to the left are are negative (a,b,c), whereas forces acting to the right (d) are positive. You can change the sign by either typing a negative value during the calibration procedure, or choosing Reverse Direction from the sensor’s drop-down menu (Setup..Show all interfaces).

UNIT 8 HOMEWORK

The homework following the Unit 8 sessions will include some review problems involving free body diagrams like those assigned in Unit 7. In addition, some simple problems will be assigned which involve the key concepts in our study of collisions: impulse, momentum change, and conservation of momentum.

UNIT 8 HOMEWORK AFTER SESSION ONE

Before Wednesday October 26th:

• Read Chapter 7 in the Textbook Understanding Physics

• Work Ch. 7 Problems 16, 17 & 18

• Work Supplemental Problem SP8-1

SP8-1) [20 pts] Hands-on inclined plane problem: Your table group is to take some very simple data and come up with a theoretical estimate of the angle with respect to the horizontal so that a 4-battery fan cart with its fan thrusting it up the track does not move. In order to do the calculations you will need to make some very simple measurements. Between session 1 and session 2 the following equipment will be available on the lecture demo table.

• a fan cart with 4 batteries • a spring scale • a balance or scale

Take whatever measurements you need in order to calculate the angle with respect to the horizontal through which a track should be tilted so the cart will not move when placed on the cart. The rule in making these measurements is that you must not place the cart on a track that is tilted up. Hint: You may want to review pages 7-39 and 7-40 in your Activity Guide and/or Examples in the text.

Note: If your group seems to have a correct analysis of this hands-on problem, the instructor may raise a track to the specified angle to demonstrate the viability of your solution.



UNIT 8 HOMEWORK AFTER SESSION TWO

Before Friday, October 28th:

• Work Supplemental Problems SP8-2 and SP8-3

SP8-2) [30 pts] In the classic film 2001: A Space Odyssey there is a scene in which the Aries 1B Lunar Vehicle is landing on the Moon. We would like you to examine a sequence of frames taken from the film and determine the thrust force exerted by Aries during its moon landing that allows it to land gently.

• Some data are provided by the film's scientific advisor*. Aries 1B was designed to use standard cryogenic propellent to provide thrust forces. It carries between 20 and 30 passengers and a crew. Aires has a total mass with fuel, passengers, and crew of about 4.5x105 kg.

• Given the mass and passenger capacity Aries probably has a length of about 30m. We know that the Moon has a gravitational acceleration constant which is 1/5.8 times that of the Earth.

(a) On the basis of a video analysis of the movie of Airies landing, what is the magnitude and

direction of the acceleration of Aires as it lands on the moon. Is it constant? What is the initial height of the centre of Aires above its location when it has landed on the surface of the Moon. What is the initial velocity? Explain how you determined the a,v, and y values at t=0s. (To do the video analysis use the video analysis ability of Logger Pro. Insert the movie entitled Moon_Landing.mp4). Note: The movie clip is 17 seconds recorded at the rate of two frame per second. You will need to calibrate using the length of Aires as a scale factor. When you select an origin, the location of Aries is calculated relative to that origin in each frame.(b) What is magnitude and direction of the gravitational force on Aires?(c) Draw a free body diagram showing the directions of the forces on Aires just above the surface of the Moon. Assume that the engine thrust force is vertical in the upward (positive direction). What is the other vertical force on Aires and in what direction is it? Are there any horizontal forces on Aires?(d) If Newton's Second Law still holds on the Moon, then what is the magnitude and direction of the net force on Aires? What is the magnitude and direction of the vertical thrust force provided by the Airies engines?

*Frederick I. Ordway, III, "Return Visit to 2001: A Space Odyssey," 1-78:supplementary text for 2110:A Space Odyssey, 3 laserdiscs, 149 min., 1988; The Criterion Collection, Santa Monica, CA.

30 m



UNIT 8 HOMEWORK AFTER SESSION TWO (CONTINUED)

• If necessary, refer to Chapter 10 Sections 10-3 and 10-4 in the Textbook before working problem SP8-3.

SP8-3) The force Fx acting on a 2.00 kg particle varies in time as shown in the figure below. Find (a) The impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the final velocity of the particle if it is initially moving along the x-axis with a velocity of -1.00 m/s.

0 1 2 3 4 5

F(N)

0

1

2

3

4

t(s)

UNIT 8 HOMEWORK AFTER SESSION THREE

Before Monday October 31st:

• Study for Midterm 2This exam will include a closed book concepts section covering basic ideas about graphs, motion, and force. The exam, as usual, be open to your problem sets, notes, and Activity Guide. It will cover the topics in Unit 4 Session 3 through Unit 8 Session 2. • Complete Unit 8 entries in the Activity GuideYou probably didn't have time to finish Session 3 in class.

• Work Ch. 7 Problems 21 & 30 on pages 248-249



Documents

UNIT 8: ONE-DIMENSIONAL COLLISIONS