**LABORATORY V**

**MOMENTUM AND ENERGY CONSERVATION **

In this lab you will use *conservation of momentum and
energy *to analyze the motion resulting from interactions that
are difficult to analyze with force concepts. In particular, you
will explore the usefulness and applicability of the conservation
principles in describing the motion resulting from collisions.

In collisions, all the complication of the interaction occurs
in a very short period of time. Describing collisions in terms of
forces is remarkably difficult. Fortunately, conservation
principles* *can be used to relate the motion of an object
before the collision to that after the collision, without
worrying about the details of the collision process itself. This
works for all scales -- from colliding galaxies to colliding cars
and colliding electrons.

You will first explore collisions on air tracks, to reduce the complication of the friction interaction. In the final problem, you will explore collisions in a situation where the friction interaction plays a major role.

**OBJECTIVES:**

Successfully completing this laboratory should enable you to:

* Understand the principles of conservation of energy and of momentum as a means of describing the behavior of systems.

* Practice the application of the energy and momentum conservation principles to real systems.

*** **Gain experience in making and testing quantitative
predictions about physical systems.

**PREPARATION:**

Read Fishbane, Gasiorowicz, and Thornton, Chapter 8, sections 1 through 4. You should also be able to:

* Analyze the motion of an object from spark tapes.

* Calculate the kinetic energy of a moving object.

* Calculate the gravitational potential energy of an object with respect to the earth.

* Calculate the total energy and total momentum of a system of objects.

**PROBLEM #1: PERFECTLY INELASTIC COLLISIONS ON AN AIR TRACK**

You have a summer job at NASA with a group designing a docking mechanism that would allow two space shuttles to connect with each other. The mechanism is designed for one shuttle to move carefully into position and dock with a stationary shuttle. Since the shuttles may be carrying different payloads and have consumed different amounts of fuel, their masses may be different when they dock. Your supervisor wants you to calculate the velocity of the docked shuttles as a function of the initial velocity of the moving shuttle and the masses of the shuttles. You decide to first complete the calculations for a perfectly inelastic collision, then check the system in the lab with air tracks and gliders.

When two objects have a perfectly inelastic collision, what is the final velocity of the objects as a function of the initial velocity of the moving object and their masses?

**EQUIPMENT**

You will use the same air tracks as in Laboratory IV. Special "ballast" masses can be added in a symmetric fashion to change the mass of the gliders.

In this problem, glider A is given an initial velocity towards a stationary glider B. A special accessory is available to get the gliders to stick together after the collision. A spark record can be made of the motion before and after the collision.

**PREDICTION**

Predict (quantitatively) the final velocities of the gliders as a function of the initial velocity of glider A and the masses of the two gliders.

Consider three cases: the masses of the gliders are
equal

(mA = mB), the moving glider has a larger mass than the
stationary glider (mA > mB), and the moving glider has a
smaller mass than the stationary glider (mA < mB). How do the
final velocities of the stuck-together gliders compare to one
another and to the initial velocity of glider A? Discuss both
magnitude and direction.

**METHOD QUESTIONS**

It is useful to have an organized problem-solving strategy such as on page 35 (and 240) in your text:

**1.** Draw a sketch which shows the situation before the
collision and after the collision. Draw velocity vectors on your
sketch.

**2.** Write down the momentum conservation equation for
this situation and identify all of the terms in the equation. Are
there any of these terms that you cannot measure with the
equipment at hand?

Write down the total energy conservation equation for this situation and identify all the terms in the equation. Are there any of these terms that you cannot measure with the equipment at hand?

Which conservation principle should you use to predict the final velocity of the stuck-together gliders, or do you need both equations? Why?

**3.** Predict the final velocity of the stuck-together
gliders as a function of the initial velocity of the moving
glider and the masses of the two gliders.

*You can now complete the prediction for this problem.*
What do you think the spark tape records will look like when the
gliders have equal masses (mA = mB)? When glider A has a larger
mass than glider B (mA > mB)? When glider A has a smaller mass
than glider B (mA < mB)?

**EXPLORATION**

Practice setting the glider into motion so that it does not wobble side to side on the track. Also, after the collision carefully observe to determine whether or not either glider wobbles significantly. Adjust your procedure to minimize wobbling.

Try giving the moving glider various initial velocities. Note qualitatively the outcomes. Keep in mind also that you want to choose an initial velocity that gives you a good spark record.

Try varying the masses of the gliders. Be sure to add the same amount of mass to each side of the glider. Also be sure the gliders still move freely over the track. What masses will you use in your final measurements?

DO NOT TOUCH ANYTHING METAL ON THE APPARATUS WHILE THE SPARK TIMER IS IN OPERATION! It operates at 10,000 volts and can give you a nasty shock.

You can also vary the spark timer rate in order to obtain more or fewer points per second. What spark rate gives you the best record? Make sure the spark timer cables are connected in such a way that you only get spark dots from the cart which is moving initially.

**MEASUREMENT**

Record the masses of the two gliders. What is the uncertainty of these measurements? Make a spark record of their collision. Examine your record and decide if you have enough positions to determine the velocities you need. Are there any peculiarities that you notice in the data that might suggest that the data are inappropriate?

Analyze your data as you go along (before making the next spark tape), so you can determine how many different spark tapes you need to make, and what the gliders' masses should be for each tape. Collect enough data to convince yourself and others of your prediction about how the final velocity of two stuck-together gliders depends on the initial velocity of the moving glider and the masses of the gliders.

**ANALYSIS**

Determine the velocity of the gliders (with uncertainty) before and after each collision from your spark records. Should you include all of the spark dots in your analysis? Why or why not?

Now use your Prediction equation to calculate each final velocity (with uncertainty) of the stuck-together gliders.

How do your measured and predicted values of the final velocities compare?

**CONCLUSION**

When a moving shuttle collides with a stationary shuttle and they dock (stick together), how does the final velocity depend on the initial velocity of the moving shuttle and the masses of the shuttles? State your results in the most general terms supported by the data.

Did your measurement agree with your prediction? Why or why not? What are the limitations on the accuracy of your measurements and analysis?

**PROBLEM #2: INELASTIC COLLISIONS**

**ON AN AIR TRACK**

You are still working for NASA with a group designing a docking mechanism that would allow two space shuttles to connect with each other. The mechanism is designed for one shuttle to move carefully into position and dock with a stationary shuttle. Since the shuttles may be carrying different payloads and have consumed different amounts of fuel, their masses may be different when they dock.

Your supervisor wants you to consider another possible outcome of a shuttle docking. This scenario might result in a "Houston we have a problem!" message from the astronauts, and you want to be prepared. In this case, either the pilot misses the docking mechanism or the mechanism fails to function. The shuttles then collide and bounce off each other. Your supervisor asks you to calculate the final velocity of both shuttles as a function of the initial velocity of the moving shuttle, the masses of both shuttles, and the energy efficiency of the collision. You decide to first complete the calculations for an inelastic collision, then check the system in the lab with air tracks and gliders.

When a moving object has an inelastic collision with a stationary object, how do the final velocities depend on the initial velocity of the moving shuttle, the masses of both shuttles, and the energy efficiency of the collision?

**EQUIPMENT**

You will use the same air tracks as in Problem #1, but glider A needs an elastic-string bumper.

As in Problem #1, additional "ballast" masses can be added to the gliders in a symmetric fashion in order to alter the mass of either glider. The gliders have a metal switch on top that can flip from one screw to another during a collision. This produces a vertical displacement in the spark record for the gliders, allowing you to distinguish "before collision" sparks from "after collision" sparks. Furthermore, the two gliders produce sparks at different heights on the spark paper.

**PREDICTION**

Predict (quantitatively) the final velocities of the two gliders as a function of the initial velocity of glider A, the masses of the two gliders, and the energy efficiency of the collision.

Consider three cases: the masses of the gliders are equal

(mA = mB), the moving glider has a larger mass than the
stationary glider (mA > mB), and the moving glider has a
smaller mass than the stationary glider (mA < mB). How do the
final velocities of the two gliders compare to one another and to
the initial velocity of glider A in each of the three cases?
Discuss both magnitude and direction.

**METHOD QUESTIONS**

It is useful to have an organized problem-solving strategy such as on page 35 (and 240) in your text:

**1.** Draw a sketch which shows the situation before the
collision and after the collision. Draw and label velocity
vectors on your sketch.

**2.** Write down the momentum conservation equation for
this situation and identify all of the terms in the equation.

Write down the energy conservation equation for this situation and identify all the terms in the equation. Write the expression for the energy dissipated in the collision in terms of the energy efficiency and the initial kinetic energy of the system (see Problem #3, Laboratory IV)

**3.** Solve the two equations for the final velocity of
glider A and glider B. You should have two equations that
determine how the final velocity of each glider depends on the
initial velocity of glider A, the masses of the two gliders, and
the energy efficiency of the collision.

*You can now complete the prediction for this problem.*
What do you think the spark tape records will look like when the
gliders have equal masses (mA = mB)? When glider A has a larger
mass than glider B (mA > mB)? When glider A has a smaller mass
than glider B (mA < mB)?

**EXPLORATION**

Your exploration for this collision problem will be similar to the one described for Problem #1: Perfectly Inelastic Collisions on an Air Track. If you have not done that problem, refer to its exploration section. If you have, then review your notes before you proceed.

Decide the best way to determine the energy efficiency of a collision between a glider and the elastic-string bumper of the second glider based on your results for Problem #1 in Laboratory IV.

Plan a procedure in which your spark record will provide
clear information on the motion of *both* gliders -- each
glider should leave spark dots on the paper, but at different
levels so you can tell them apart. Determine how to use the metal
switches mounted on the gliders to produce sparks at different
heights before and after the collisions. If you are having too
much trouble with this, consult with a group that has made a
successful spark record (or ask your lab instructor for help).
Your objective today is to study collisions, not switch and
glider design.

DO NOT TOUCH ANYTHING METAL ON THE APPARATUS WHILE THE SPARK TIMER IS IN OPERATION! It operates at 10,000 volts and can give you a nasty shock.

**MEASUREMENT**

Make the measurements necessary to determine the energy efficiency of the collision.

Record the masses of the two gliders. Make a spark record of their collision. Examine your record and decide if you have enough positions to determine the velocities you need. Are there any peculiarities that you notice in the data that might suggest that the data are inappropriate?

Analyze your data as you go along (before making the next spark tape), so you can determine how many different spark tapes you need to make, and what the gliders' masses should be for each tape. Collect enough data to convince yourself and others of your conclusion about how the final velocities of the gliders depend on the initial velocity of glider A, the masses of the gliders, and the energy efficiency of the collision.

**ANALYSIS**

Determine the velocity of the gliders before and after the collision from your spark records. Should you include all of the spark dots in your analysis? Why or why not?

For each collision you observed and recorded, calculate the total kinetic energy before and after the collision. Was a significant portion of it dissipated? Into what other forms of energy do you think the glider's initial kinetic energy is most likely to transform?

Now compare the final velocities to the initial velocity of glider A (the launched glider) using the quantitative predictions you made.

**CONCLUSION**

When a moving shuttle has an inelastic collision with a stationary shuttle, how do the final velocities depend on the initial velocity of the moving shuttle, the masses of the shuttles, and the energy efficiency of the collision? State your results in the most general terms supported by the data.

Did your measurement agree with your prediction? Why or why not? What are the limitations on the accuracy of your measurements and analysis?

Do the principles of conservation of energy and momentum constitute a useful set of tools for describing the motion of objects which experience an inelastic collision?

**PROBLEM #3:**

**COLLISIONS WITH FRICTION**

You have been hired as an assistant to the stunt coordinator on a new action movie about an earthquake hitting San Francisco. In one scene a bus is knocked over on its side on a street going down a steep hill. It stays at rest for just enough time for the passengers to escape. Then a small tremor starts the bus sliding down the hill. At the bottom of the hill the street becomes level. On the level part of the street is another bus which has also been knocked on its side and is still full of people. The exciting script calls for the first bus to slide down the hill and hit the second bus. For the stunt the movie company will install special bumpers on the buses so that the collision is essentially elastic, and the buses bounce apart.

To position the cameras correctly, the stunt coordinator asks you to calculate the final positions of the two buses from the known masses of the buses, the distance the first bus slides down the hill, the distance the first bus slides along the level road before it hits the second bus, and the coefficient of kinetic friction between a sliding bus and the road.

Before you decide to risk a million dollar stunt on your calculation, you decide to build a laboratory model to check your calculation (see Equipment section below).

What is the final position for each cylinder on the brass track after the collision?

**EQUIPMENT**

In this problem you will use the apparatus pictured below.

It is a brass track with two sections, an inclined section and a level section. A brass or aluminum cylinder can slide from rest down the incline, across the level portion of the track and

collide with another cylinder. The angle of the incline can be adjusted.

**PREDICTION**

Predict (quantitatively) the final position of each cylinder after they collide.

**METHOD QUESTIONS**

It is useful to have an organized problem-solving strategy such as on page 35 (and 240) in your text:

**1**. Draw a sketch (or several sketches) of the problem
that will help you visualize the situation before and after the
collision.

**2**. Write down the measurable and known quantities. Use
the result of Problem #4 from Laboratory IV as needed.

**3**. Draw appropriate force and momentum vector
diagrams.

**4.** Apply the energy and momentum conservation
equations to solve the problem.

**EXPLORATION**

Practice sliding the cylinder down the inclined portion of the track to find out what height results in a smooth transition over the bend in the track.

Find where you should place the target cylinder so you can get reproducible results. Determine which cylinder should be the target.

**MEASUREMENT**

Make the measurements that you need to answer the major prediction.

**ANALYSIS**

Compare your predicted results and your actual results. What is the accuracy and precision for each result?

**CONCLUSION**

Do your results seem reasonable? Are they reproducible?

What information would you need to calculate the final position of the two buses for the movie stunt?

[[ogonek]] **Check Your Understanding:** Estimate
reasonable values for the information you need, and solve the
problem for the movie stunt.

1. If a runner speeds up from 2 m/s to 8 m/s, the runner's *momentum*
increases by a factor of

(a) 64.

(b) 16.

(c) 8.

(d) 4.

(e) 2.

2. A piece of clay slams into and sticks to an identical piece of clay that is initially at rest. Ignoring friction, what percentage of the initial kinetic energy goes into changing the internal energy of the clay balls?

(a) 0%

(b) 25%

(c) 50%

(d) 75%

(e) There is not enough information to tell.

3. A tennis ball and a lump of clay of equal mass are thrown with equal speeds directly against a brick wall. The lump of clay sticks to the wall and the tennis ball bounces back with one-half its original speed. Which of the following statements is (are) true about the collisions?

(a) During the collision, the clay ball exerts a larger average force on the wall than the tennis ball.

(b) The tennis ball experiences the largest change in momentum.

(c) The clay ball experiences the largest change in momentum.

(d) The tennis ball transfers the most energy to the wall.

(e) The clay ball transfers the most energy to the wall.

4. A golf ball is thrown at a bowling ball so that it hits head on and bounces back. Ignore frictional effects.

a. Just after the collision, which ball has the largest momentum, or are their momenta the same? Explain using vector diagrams of the momentum before and after the collisions.

b. Just after the collision, which ball has the largest kinetic energy, or are their kinetic energies the same? Explain your reasoning.

5. A 10 kg sled moves at 10 m/s. A 20 kg sled moving at 2.5 m/s has

(a) 1/4 as much momentum.

(b) 1/2 as much momentum.

(c) twice as much momentum.

(d) four times the momentum.

(e) None of the above.

6. Two cars of equal mass travel in opposite directions with equal speeds on an icy patch of road. They lose control on the essentially frictionless surface, have a head-on collision, and bounce apart.

a. Just after the collision, their velocities are

(a) zero.

(b) equal to their original velocities.

(c) equal in magnitude and opposite in direction to their original velocities.

(d) less in magnitude and in the same direction as their original velocities.

(e) less in magnitude and opposite in direction to their original velocities.

b. In the type of collision described above,

(a) kinetic energy is conserved.

(b) momentum is conserved.

(c) both momentum and kinetic energy are conserved.

(d) neither momentum nor kinetic energy are conserved.

(e) the extent to which momentum and kinetic energy are conserved depends on the coefficient of restitution.

7. Ignoring friction and other external forces, which of the following statements is (are) true just after an arrow is shot from a bow?

(a) The forward momentum of the arrow equals that backward momentum of the bow.

(b) The total momentum of the bow and arrow is zero.

(c) The forward speed of the arrow equals the backward speed of the bow.

(d) The total velocity of the bow and arrow is zero.

(e) The kinetic energy of the bow is the same as the kinetic energy of the arrow.

**PHYSICS 1251 LABORATORY REPORT**

**Laboratory V**

Name and ID#: ______________________________________________________

Date performed: ________________ Day/Time section meets: ______________

Lab Partners' Names: ________________________________________________

__________________________________________________________________

__________________________________________________________________

Problem # and Title: _________________________________________________

Lab Instructor's Initials: ____________

Grading Checklist Points LABORATORY JOURNAL: PREDICTIONS

(individual predictions completed in journal before each lab session) LAB PROCEDURES

(measurement plan recorded, tables and graphs made as data is collected) PROBLEM REPORT:* ORGANIZATION

(clear and readable; section headings provided by you) DATA AND DATA TABLES

(clear and readable; units and assigned uncertainties clearly stated) RESULTS

(results clearly indicated; correct derivations with uncertainties indicated; scales, labels and uncertainties on graphs) CONCLUSIONS

(comparison to prediction & theory discussed; possible sources of uncertainties identified; attention called to experimental problems) TOTAL BONUS POINTS

(if all members of your group score 8 points or above)

* An "R" in the points column means to __rewrite
that section only__ and return it to your lab instructor within
two days of the return of the report to you -- no exceptions.