Conservation of Momentum - … · They will use this information to verify the law of conservation of momentum. LEVEL Physics ... POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS …

Conservation of Momentum

430 Laying the Foundation in Physics

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Conservation of Momentum Using PASCOTM Carts and Track to Study

Collisions in One Dimension

OBJECTIVE Students will collide two PASCOTM carts on a track to determine the momentum before and after a collision. They will use this information to verify the law of conservation of momentum.

LEVEL Physics

NATIONAL STANDARDS UCP.2, UCP.3, A.1, B.4, B.5

TEKS 2(A), 2(B), 2(C), 2(D), 2(E), 2(F), 4(B), 4(C), 5(A), 5(C), 5(D)

CONNECTIONS TO AP I. Newtonian mechanics, D. Systems of particles, linear momentum, 2. Impulse and momentum, 3. Conservation of linear momentum, collisions

TIME FRAME 90 minutes

MATERIALS (For a class of 30 working in groups of 5)

6 Vernier LabPro® or PASCO interface devices 6 computers with Vernier Logger Pro® data

collection software or graphing calculators 12 ring stands and/or clamps to mount

photogates

6 PASCOTM tracks 12 PASCO collision carts and bar masses

(4 magnetic, 2 plunger carts, 4 non-magnetic carts with Velcro®)

12 photogates or 12 motion detectors masking tape 12 3" × 5" index cards

TEACHER NOTES In this activity the students will investigate the loss in momentum during the interaction of two PASCO carts in three situations: an inelastic collision, an elastic collision, and the recoil of two carts away from each other. The loss in momentum is found by subtracting the total momentum after the interaction from the total momentum before the interaction. This calculation assumes that the total momentum will be conserved if there are no dissipative forces such as friction.

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©2007 Laying the Foundation, Inc. All rights reserved. Visit: www.layingthefoundation.org

14 Conservation of Momentum

Laying the Foundation in Physics 431

This activity is written in such a way that the students in each group will spend the entire time taking data at only one station, and then share the data acquired with the other groups. If you can set up six lab stations, you should set up two inelastic stations, two elastic stations, and two recoil stations. Encourage the students in each group to run as many different runs as they can by varying the masses of each cart in each run. The data tables are designed to display 5 runs, but students can always run more runs and attach another page to the lab report.

Running the groups in this way serves several purposes. First, it allows the students to remain at one station and run many runs on one type of interaction, saving equipment and rotation time. Since the data at all of the lab stations are obtained in similar ways, the students are not missing out on any procedural learning. Second, it encourages the students to obtain, record, and communicate their data in such a way that other students can easily read and analyze it.

Although the instructions included here are written using photogates, the speeds of the carts before and after the interactions can also be measured by placing a motion detector at each end of the track and obtaining the speed from the position vs. time and/or velocity vs. time graphs.

PASCO carts, tracks, photogates, and motion detectors can be obtained from www.pasco.com.

Vernier probes and software cane be obtained from www.vernier.com.

POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS AND SAMPLE DATA

DATA AND OBSERVATIONS

Mass of one empty cart: 0.500 kg

Mass of one “black bar” mass: 0.500 kg

Length of card: 0.126 m

Data Table 1: Inelastic Collision

Run

Mass of Incident Cart,

m1 (kg)

Mass of Target Cart,

m2 (kg)

Velocity v1 of m1 Before

Collision (m/s)


Collision (m/s)

Velocity v1´ of m1 After Collision

(m/s)


(m/s)

1 0.500 0.500 0.185 0.000 0.088 0.088

2 1.000 0.500 0.214 0.000 0.129 0.129

3 1.500 0.500 0.159 0.000 0.110 0.110

4 0.500 1.000 0.241 0.000 0.073 0.073

5 0.500 1.500 0.232 0.000 0.045 0.045

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Data Table 2: Elastic Collision

Run


m1 (kg)


m2 (kg)

Velocity v1 of m1 Before Collision

(m/s)


Collision (m/s)


(m/s)


(m/s)

1 0.500 0.500 0.223 0.000 0.000 0.213

2 1.000 0.500 0.244 0.000 0.075 0.278

3 1.500 0.500 0.175 0.000 0.054 0.237

4 0.500 1.000 0.245 0.000 –0.039 0.137

5 0.500 1.500 0.266 0.000 –0.089 0.103

Data Table 3: Recoil

Run Mass of Plunger

Cart, m1 (kg)

Mass of Second Cart,

m2 (kg)


Collision (m/s)


Collision (m/s)


(m/s)


(m/s)

1 0.500 0.500 0.000 0.000 0.202 –0.197

2 1.000 0.500 0.000 0.000 0.113 –0.207

3 1.500 0.500 0.000 0.000 0.062 –0.175

4 1.500 1.000 0.000 0.000 0.106 –0.158

5 1.000 1.500 0.000 0.000 –0.158 0.106

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ANALYSIS

In the tables below, calculate the momentum for each cart before and after the collision or recoil. Indicate the velocity of any cart which reverses its direction by using a negative sign.

Momentum for Inelastic Collision

Run

Momentum p1 of m1 Before

Collision (kg⋅m/s)



Total Momentum

pT of the System Before


Momentum p1´ of m1

After Collision (kg⋅m/s)

Momentum p2´ of m2


Total Momentum

pT´of the System After


1 0.093 0.000 0.093 0.044 0.044 0.088

2 0.214 0.000 0.214 0.129 0.065 0.194

3 0.239 0.000 0.239 0.165 0.055 0.220

4 0.121 0.000 0.121 0.037 0.073 0.110

5 0.116 0.000 0.116 0.023 0.068 0.091

Momentum for Elastic Collision

Run





Total Momentum



Momentum p1´ of m1


Momentum p2´ of m2


Total Momentum



1 0.112 0.000 0.112 0.000 0.107 0.107

2 0.244 0.000 0.244 0.075 0.139 0.214

3 0.263 0.000 0.263 0.081 0.119 0.200

4 0.123 0.000 0.123 –0.020 0.137 0.117

5 0.133 0.000 0.133 –0.045 0.155 0.110

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Momentum for Recoil

Run Momentum

p1 of m1 Before Recoil

(kg⋅m/s)

Momentum p2 of m2

Before Recoil (kg⋅m/s)

Total Momentum

pT of the System


Momentum p1´ of m1

After Recoil (kg⋅m/s)

Momentum p2´ of m2


Total Momentum


Recoil (kg⋅m/s)

1 0.000 0.000 0.000 0.101 –0.099 0.002

2 0.000 0.000 0.000 0.113 –0.104 0.009

3 0.000 0.000 0.000 0.093 –0.088 0.005

4 0.000 0.000 0.000 0.159 –0.158 0.001

5 0.000 0.000 0.000 –0.158 0.159 0.001

CONCLUSION QUESTIONS

Using the data obtained in all three groups, answer the following questions.

1. In general, does the data collected for the inelastic collision seem to verify the law of conservation of momentum? Explain your answer and indicate which run of the inelastic collision best conserves momentum.

• In general, momentum is reasonably conserved, that is, the total momentum before the collision is nearly equal to the total momentum after the collision. The least amount of momentum loss occurs in Run 1, where only 0.005 kg·m/s is lost.

2. In general, does the data collected for the elastic collision seem to verify the law of conservation of momentum? Explain your answer and indicate which run of the elastic collision best conserves momentum.

• In general, momentum is reasonably conserved, that is, the total momentum before the collision is nearly equal to the total momentum after the collision. The least amount of momentum loss occurs in Run 1, where only 0.005 kg·m/s is lost.

3. In general, does the data collected for the recoil of the two carts seem to verify the law of conservation of momentum? Explain your answer and indicate which run of the recoil of the two carts best conserves momentum.

• In general, momentum is reasonably conserved, that is, the total momentum before the recoil is nearly equal to the total momentum after the recoil. The least amount of momentum loss occurs in Run 4, where only 0.001 kg·m/s is lost.

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4. List two sources of error, and explain how each affected the results of your experiments.

• One source of error would be friction between the cart and the track. Friction reduces the speed of the carts and contributes to the loss of momentum.

• Another source of error might be the slight variations in the timing of the photogates as the cards pass through them. We cannot guarantee they are perfectly consistent with each other in their timing.

5. The screenshot below represents the interaction of two carts in either an inelastic collision, elastic collision, or the recoil of the two carts. The target cart is initially at rest. Answer the questions that follow.

a. Which type of interaction does the data table represent? Check the appropriate answer below, and justify your answer.

_____ inelastic collision _____ elastic collision _____ recoil

• There is only one velocity after the collision, and it is less than the initial velocity of the incident cart, so we know that it is not two equal carts colliding elastically.

b. If the carts each have a mass of 0.500 kg, and the only available bar masses are 0.500 kg each, how is the mass most likely distributed in this interaction? Explain your answer.

• The velocity of the pair of carts after the collision is about half the initial velocity of the incident cart. This indicates that the mass has doubled after the collision. So, this is a collision between two carts of equal mass, perhaps each having a mass of 0.500 kg.

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√




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c. Assuming that your answer to part b is correct, how much momentum is lost in the interaction? Show your calculation in the space below.

• ( )1 1m kg m= = 0.500 kg 0.1846 = 0.0923s sbeforep m v ⎛ ⎞

⎜ ⎟⎝ ⎠

• ( ) ( )1 2m kg m= = 0.500 kg + 0.500 kg 0.0841 = 0.0841s safterp m m v ⎛ ⎞′+ ⎜ ⎟

⎝ ⎠

• kg m kg m kg m= 0.0923 0.0841 = 0.0082

s s sbefore afterp p− −




• There is only one velocity through each photogate, and the second velocity recorded by the photogate (Velocity 1 in the table) is smaller that the first velocity recorded (Velocity 2). Thus, it is not an inelastic collision, and must be a recoil interaction.

b. If the carts each have a mass of 0.500 kg, and the only available bar masses are 0.500 kg each, how is the mass most likely distributed in this interaction? Explain your answer.

• Since Velocity 1 (0.1128 m/s) is about half the value of Velocity 2 (0.2068), the mass moving at Velocity 1 must be about twice the mass moving at Velocity 2 for momentum to be reasonably conserved. Perhaps the mass moving at Velocity 1 is 1.000 kg, and the mass moving at Velocity 2 is 0.500 kg.

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• = 0beforep

• ( ) ( )1 1 2 2m m kg m= = 1.000 kg 0.1128 0.500 kg 0.2068 = 0.0094s s safterp m v m v ⎛ ⎞ ⎛ ⎞′ ′+ −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

• The difference between the total momentum before and after the recoil interaction is 0.0094 kg·m/s.

7. Consider the two screenshots A and B below, which represent two different elastic collisions. In each case, the target cart is initially at rest, and one cart is twice as massive as the other cart.

Screenshot A

Screenshot B

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Which screenshot represents an elastic collision in which the incident cart is more massive than the target cart? Explain your answer. • The incident cart is more massive in Screenshot B, since both carts continue forward through

photogate 2 after the collision.

8. The screenshot below represents the elastic collision between two carts of equal mass. Each cart has a mass of 0.500 kg.

a. On the axes below, sketch a graph of momentum p vs. time t for the incident and target carts. Be sure to indicate important values on both the horizontal and vertical axes.

• ( )1 1 1m kg m= = 0.500 kg 0.2228 = 0.1114s s

p m v ⎛ ⎞⎜ ⎟⎝ ⎠

• 1 1 1= = 0p m v′ ′

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Incident Cart:

Momentum vs. Time

Target Cart:

• 2 2 2= = 0p m v

• ( )2 2 2m kg m= = 0.500 kg 0.2133 = 0.1067s s

p m v ⎛ ⎞′ ′ ⎜ ⎟⎝ ⎠

Momentum vs. Time

b. Calculate the amount of momentum lost in this collision. Show your work in the space below.

• 1 2kg m kg m kg m= 0.1114 0.1067 = 0.0047

s s sp p ′− −

p(kg·m/s)

t(s)

0.1114 kg·m/s

0.300

0.1067 kg·m/s

p(kg·m/s)

t(s)

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0.300




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When two objects collide momentum is transferred between them. Momentum p is defined as the product of mass and velocity of an object (p = mv), and like velocity, momentum is a vector. The law of conservation of momentum states that in the absence of any external forces, the total momentum before a collision is equal to the total momentum after the collision. In this activity you will observe and determine the momentum transferred for an inelastic collision (in which the carts stick together), an elastic collision (in which the carts bounce off each other), and a recoil interaction (in which the carts “explode” apart).

PURPOSE In this activity you will determine the total momentum before and after three interactions of carts: an inelastic collision, an elastic collision, and a recoil interaction. You may be placed in a group to investigate just one of the interactions, and then asked to share your data with the other groups.

MATERIALS Vernier LabPro® or PASCO interface devices computer with Vernier Logger Pro® data

collection software or graphing calculator 2 ring stands and/or clamps to mount photogates

PASCOTM track 2 PASCO collision carts and bar masses 2 photogates or 2 motion detectors masking tape

2 3" × 5" index cards

PROCEDURE GROUP I – INELASTIC COLLISION

1. Place the two carts on the track. The ends with the Velcro® should be facing each other so that when the carts collide they will stick together. The carts will be referred to as the incident cart (first) and the target cart (hit by the incident cart).

2. Attach an index card of known length to each cart so that the card will pass through a photogate before and after the collision.

3. Mount the photogates above or beside the carts so that the attached index cards will pass through the photogates before and after the collision, as shown in Figure 1.




Mounted

Photogates

Index cards

PASCO track and carts

Figure 1

4. Connect the two photogates into the DIG/SONIC 1 and DIG/SONIC 2 ports on the LabPro interface. Open the Logger Pro data collection software, go to File, Open, Probes and Sensors, Photogate, Two Gate Timing. You should see a graph like the one in Figure 2 below.

Figure 2




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5. Measure the length of the card attached to each cart. A typical 3" × 5" index card has a length of 0.126 m.

6. Follow the instructions on the screen to calibrate the photogate to the length of the card mounted on the cart. If the computer knows the length of the card and the time it takes the card to pass through the photogate, it can calculate the average velocity of the cart as it passes through the photogate. Figures 3 and 4 show the pictures you will see as you calibrate the photogates.

Figure 3




Figure 4

7. Place one cart on the track between the two photogates, and the other cart on the track outside the photogates. Arrange the carts so that the incident cart will pass through the first photogate, collide with the target cart, and then the target cart will pass completely through the second photogate. When you are ready to collect data, click on the Collect button on the toolbar and roll the incident cart toward the first photogate. The cart should pass completely through the photogate and collide with the target cart. You may want to stop the carts as soon as the target cart passes through the photogate so that the second photogate will not record the velocity of the incident cart. However, if both carts pass through the photogate after the collision, we are only interested in the velocity recorded for the target cart passing through the photogate. Record the before and after velocities of the carts in the data table for inelastic collision on your student answer page.

8. Repeat the experiment for several more runs, adding various amounts of mass to each cart to see how the amount of mass affects the velocity and momentum of the carts before and after the collision. Remember to record the data in such a way that another lab group can understand how you have organized your data and use it to answer questions about your data.




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GROUP II – ELASTIC COLLISION

1. Place the two magnetic carts on the track so that they repel each other when they collide.

2. Follow steps 2–6 listed in the procedure for Group I.

3. Place one cart on the track between the two photogates and the other cart on the track outside the photogates. Arrange the carts so that the incident (first) cart will pass through the first photogate, collide with the target cart (the cart which is hit by the incident cart), and then the target cart will pass completely through the second photogate. Depending on the masses and speeds of the carts, the second cart may pass through the second photogate, or reverse its direction and pass back through the first photogate again.

4. When you are ready to collect data, click on the Collect button on the toolbar and roll the incident cart toward the first photogate so that it passes completely through it and collides with the target cart. Record the before and after velocities of the carts in the data table for elastic collision on your student answer page.

5. Repeat the experiment for several more runs, adding various amount of mass to each cart to see how the amount of mass affects the velocity and momentum of the carts before and after the collision. Remember to record the data in such a way that another lab group can understand how you have organized your data and use it to answer questions about your data.

GROUP III – RECOIL

1. Using one cart with a retractable “plunger” and another cart without a plunger, place the two carts on the track between the two photogates. Push the retractable plunger into the plunger cart so that it locks and does not pop out. Place the plunger end of the plunger cart up against the other cart so that when you tap the peg on the top of the plunger cart, the spring-loaded plunger pops out and pushes the two carts apart, each passing through a photogate.

2. Follow steps 2–6 listed in the procedure for Group I.

3. When you are ready to collect data, click on the Collect button on the toolbar and lightly tap the peg on the top of the plunger cart so that the spring-loaded plunger pops out and pushes the two carts apart causing each cart to pass through a photogate.

4. Record the velocity of the carts in the data table for recoil on your student answer page for the time just after they recoil away from each other.

5. Repeat the experiment for several more runs, adding various amount of mass to each cart to see how the amount of mass affects the velocity and momentum of the carts before and after the collision. Remember to record the data in such a way that another lab group can understand how you have organized your data and use it to answer questions about your data.



Name _____________________________________

Period ____________________________________




DATA AND OBSERVATIONS

After you have taken all of the data for your group, obtain the data taken by the other two groups and enter the results in the tables below.

Mass of one empty cart: _______________________ kg

Mass of one “black bar” mass: __________________ kg

Length of card: _______________ m

Data Table 1: Inelastic Collision

Run


m1 (kg)


m2 (kg)


Collision (m/s)


Collision (m/s)


(m/s)


(m/s)

1

2

3

4

5




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Data Table 2: Elastic Collision

Run


m1 (kg)


m2 (kg)


Collision (m/s)


Collision (m/s)


(m/s)


(m/s)

1

2

3

4

5

Data Table 3: Recoil

Run

Mass of Plunger Cart,

m1 (kg)

Mass of Second Cart,

m2 (kg)


Collision (m/s)


Collision (m/s)


(m/s)


(m/s)

1

2

3

4

5




ANALYSIS

In the tables below, calculate the momentum for each cart before and after the collision or recoil. Be sure to indicate and the velocity of any cart which reverses its direction with a negative sign.

Momentum for Inelastic Collision

Run





Total Momentum



Momentum p1´ of m1


Momentum p2´ of m2


Total Momentum



1

2

3

4

5

Momentum for Elastic Collision

Run





Total Momentum



Momentum p1´ of m1


Momentum p2´ of m2


Total Momentum



1

2

3

4

5




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Momentum for Recoil

Run

Momentum p1 of m1


Momentum p2 of m2


Total Momentum

pT of the System


Momentum p1´ of m1


Momentum p2´ of m2


Total Momentum


Recoil (kg⋅m/s)

1

2

3

4

5

CONCLUSION QUESTIONS

Using the data obtained in all three groups, answer the following questions.

1. In general, does the data collected for the inelastic collision seem to verify the law of conservation of momentum? Explain your answer and indicate which run of the inelastic collision best conserves momentum.

2. In general, does the data collected for the elastic collision seem to verify the law of conservation of momentum? Explain your answer and indicate which run of the elastic collision best conserves momentum.




3. In general, does the data collected for the recoil of the two carts seem to verify the law of conservation of momentum? Explain your answer and indicate which run of the recoil of the two carts best conserves momentum.

4. List two sources of error and explain how each affected the results of your experiments.







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b. If the carts each have a mass of 0.500 kg, and the only bar masses available are 0.500 kg each, how is the mass most likely distributed in this interaction? Explain your answer.








b. If the carts each have a mass of 0.500 kg, and the only bar masses available are 0.500 kg each, how is the mass most likely distributed in this interaction? Explain your answer.


7. Consider the two screenshots A and B below, which represent two different elastic collisions. In each case, the target cart is initially at rest, and one cart is twice as massive as the other cart.

Screenshot A




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Screenshot B

Which screenshot represents an elastic collision in which the incident cart is more massive than the target cart? Explain your answer.

8. The screenshot below represents the elastic collision between two carts of equal mass. Each cart has a mass of 0.500 kg.




a. On the axes below, sketch a graph of momentum p vs. time t for the incident and target carts. Be sure to indicate important values on both the horizontal and vertical axes.

Incident Cart:

Momentum vs. Time

Target Cart:

Momentum vs. Time

b. Calculate the amount of momentum lost in this collision. Show your work in the space below.

p(kg·m/s)

t(s)

p(kg·m/s)

t(s)


Documents

Conservation of Momentum - … · They will use this information to verify the law of conservation of momentum. LEVEL Physics ... POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS …