Experiment 6U
Experiment 6U: Conservation of Momentum in Explosions and in Inelastic Collisions
Purpose
The purpose of this experiment is to:
a) demonstrate conservation of momentum with two carts of varying masses pushing away from each other from rest
b) show that momentum is conserved in inelastic collisions and that kinetic energy is not conserved in inelastic collisions
Apparatus
Track with feet and end stops, blue & red PAScar carts, 250 g mass bars (2), five pattern cart picket fences (2), photogates (2), photogate mounting brackets (2), PASCO 850 Universal Interface, level, equal-arm balances.
Theory
Explosions[*]
When two carts push away from each other in an explosion and there is no net force on the system, the total momentum ptotal is conserved. In this case if the system is initially at rest, the total initial momentum pinitial of the two carts is zero and the final total momentum pfinal of the system is also zero.
pfinal = m₁v₁f + m₂v₂f = pinitial = 0 (1)
To add to zero, the final momenta of the two carts must then be equal in magnitude and opposite in direction to each other. The velocity v is a vector with both magnitude and direction. If we take the velocity (v₁f) to be in the positive direction, then v₂f must be in the negative direction. We can show this in a simple way by letting v stand for the magnitude of the speed and indicate the opposite direction with a minus sign. Then the speeds v₁f and v₂f are related by:
m₁v₁f + m₂(-v₂f) = 0 (2)
m₁v₁f = m₂v₂f (3)
Collisions In One Dimension
When two carts collide with each other and there is no net force on the system, the total momentum p is conserved regardless of the type of collision.
pinitial = pfinal (4)
If the carts stick together after the collision, we find that:
m₁v1i + m₂v₂i = (m₁+m₂)vf (5)
where v1i is the initial velocity of cart 1, v₂i is the initial velocity of cart 2, and vf is the final velocity of the two carts stuck together.
The kinetic energy KE is given by (remember v is the magnitude of the velocity):
KE = (1/2) mv²
In an elastic collision in which the two carts bounce off each other, there is no loss of kinetic energy. On the other hand, in a completely inelastic collision in which the two carts hit and stick to each other, there is a loss of total kinetic energy. The total initial and final kinetic energies in this case are:
KEinitial = (1/2) m₁(v1i)² + (1/2) m₂(v₂i)² (6)
KEfinal = (1/2) (m₁+m₂)(vf)² (7)
Part 1 Explosions
Preliminary Setup[*]
1. Install the feet on the track. Mount an end stop at each end. Level the track.
2. Put two photogates on photogate mounting brackets and attach the mounting brackets to the track. Position the photogates about 50 cm apart so both carts (plus extended plunger) can fit between them without having the picket fences on top block either photogate. The photogate should be at least 30 cm from each end so the magnets in the endstops will not interact with a cart until it is past the photogate. (See Figure 1.)
3. Put a five pattern picket fence into the slots on top of each cart with the 1 cm pattern (13 bars) at the top of each. Place the carts with the picket fences on the track. Adjust the height of the photogates so the 1 cm rows of the picket fences are lined up with the infrared beam of the photogates. Remove the carts from the track.
4. Copy Table 1 to your data sheet. Measure the total mass of each cart with its picket fence and record the values in kilograms in Table 1. Complete Table 1.
Table 1: Cart MassesBlue Cart with Fence
(kg) / Red Cart with Fence
(kg)
5. Connect the 850 Universal Interface to the computer and turn on the 850. Plug the photogate on the blue cart side into digital input 1 of the 850 and plug the photogate on the red cart side into digital input 2.
6. Put the carts end-to-end (blue cart left, red cart right) in the middle of the track. [The blue cart is the “collision cart” and the red cart is the “plunger” cart.] Make sure that the cart picket fences plus the width of the fully extended plunger do not block a photogate.
7. Open the Exp6U Conservation of Momentum in Explosions.cap Capstone file on your desktop.
Part I: Procedure
8. Copy Table 2 to your data sheet.
Table 2: Cart SpeedsTrial / System / Avg Blue Cart Speed (m/s) / Avg Red Cart Speed (m/s)
1 / carts + fence
2 / carts + fence
3 / red cart + 250 g
4 / red cart + 250 g
5 / red cart + 500 g
6 / red cart + 500 g
9. Cock the plunger of the red cart to the second notch. Push the plunger in and slightly upward. You may have to push in the small release pin above the plunger. You will feel two clicks and the number 2 will show on the plunger.
10. Position the carts at the midway point between the photogates so each cart moves about the same distance before reaching a gate. This way friction will slow each cart by about the same amount. Make sure the blue cart is on the left and the velcro side of the blue cart is not facing the red cart. (Figure 1)
11. With the carts at rest click “Record”.
12. Gently push the release pin for the plunger (see Figure 2) with your finger or a pencil. Try to push it vertically so you don't give a horizontal impulse to the carts.
13. Data recording will stop automatically. The data for “Average Cart 1 Speed” is for the blue cart, and the data for “Average Cart 2 Speed” is for the red cart. Record the average cart speeds in Table 2 on the Trial 1 line.
14. Repeat the process and record the results as speeds for Trial 2.
15. Compare the two runs. The speeds should be about the same. More importantly, they should differ by about the same amount. If they don't, you probably did not release the pin cleanly and should redo the runs.
16. Add one 250 g mass bar to the red cart and make two more runs, record this data as speeds for Trials 3 4.
17. Add both 250 g mass bars to the red cart and make two more runs, record this data as speeds for Trials 5 6.
Part II Inelastic Collisions
Preliminary Setup
Table 3: Cart MassesBlue Cart
with
Picket Fence (kg) / Red Cart
Only
(kg)
18. Check the track and level it if necessary.
19. Copy Table 3 to your data sheet. Remove the picket fence from the red cart. Re-measure the mass of the blue cart with its picket fence and record the value in kilograms. Measure and record the mass of the red cart without the picket fence.
20. Push the plungers on both carts all the way in. Put the carts on the track with the velcro on the carts facing each other. The velcro should make them stick them together when they collide. Make sure that the 1 cm (13 bars) pattern on the picket fence interrupts the photogate beam as the blue cart moves through the photogates.
21. Test the placement of the equipment to make sure that the collision of the carts will occur between the two photogates. Place the blue cart with the picket fence to the left of photogate #1 and the red cart near photogate #2. (See Figure 3) Push the blue cart towards the red cart. Make sure the blue cart completely passes photogate #1 before they collide and that they stick together in the space between the two photogates and continue on through photogate #2. Adjust the position of the photogates if necessary.
Part II: Procedure
Table 4: Part II Cart velocitiesMass / Initial Velocity / Final Velocity
Run / Blue Cart + Fence
Mass (kg) / Red Cart
Mass (kg) / Blue Cart
(m/s) / Red Cart
(m/s) / Blue & Red Carts
Together (m/s)
1 / 0.0
2
3
22. Copy Table 4 to your data sheet.
23. Arrange the two carts as before. Once the carts are at rest and in the required initial position, click “RECORD”. Then push the blue cart to give it an initial velocity and be sure to release it before it enters gate #1. After the stuck-together pair passes through gate #2, click “STOP” to end data recording.
24. If the run looks good, record the data in Table 4 in Part II Run 1. The data from “average cart 1 speed” is the initial velocity of the blue cart, and the data from “average cart 2 speed” is the final velocity of the blue and red carts together. Redo the run if you don’t have good data.
25. Add one 250 g mass bar to the red cart and repeat Procedures 23-24. Record the data in Table 4 as Part II Run 2.
26. Add both 250 g mass bars to the red cart and repeat Procedures 23-24. Record the data in Table 4 as Part II Run 3.
Before You Leave:
Exit the program.
DO NOT SAVE YOUR CHANGES.
Click “Discard” when the window pops up.
Lab Report
Part 1
1. Complete Table 5 using data from Tables 1 and 2. Calculate the % difference between the blue momentum PB and the red momentum PR for each run. Show sample calculations including units.
% difference= PB-PRPB+PR2 ×100%
Table 5: Part I ExplosionSystem / Blue cart mass
(kg) / Red cart mass
(kg) / Avg Blue cart speed (m/s) / Avg Red cart speed (m/s) / PB
(kg m/s) / PR
(kg m/s) / % diff
1 / carts only
2 / carts only
3 / Red cart + 250 g
4 / Red cart + 250 g
5 / Red cart + 500 g
6 / Red cart + 500 g
Question 1:
An ice skater with a mass of 80 kg pushes off against a second skater with a mass of 32 kg. Both skaters are initially at rest.
a. What is the total momentum of the system before they push off?
b. What is the total momentum of the system after they push off?
c. Which skater has more momentum right after the push off?
d. If the larger skater moves off with a speed of 3 m/s, what is the corresponding speed of the smaller skater?
e. Which skater has more kinetic energy right after the push off?
Part II
Complete Table 6 for each run. Show your calculations:
2. Calculate the momentum PB and PR of each cart before the collision.
3. Calculate the total initial momentum Pinitial of the two carts before the collision.
4. Calculate the total momentum Pfinal of the joined carts after the collision.
5. Calculate the percent (%) difference between the total momentum of the carts before and after the collision using the following equation.
6. % difference=Pfinal-PinitialPfinal+Pinitial2×100%
7. Calculate the total kinetic energy of the carts before and after the collision. (Equations 6 7)
8. Calculate the percentage of kinetic energy that is lost in the collision: ΔKE/ KEinitial × 100% where ΔKE = KEinitial - KEfinal
Table 6: Part II Inelastic CollisionsRun / PB
(kg m/s) / PR
(kg m/s) / Pinitial
(kg m/s) / Pfinal
(kg m/s) / % diff / KEinitial
(J) / KEfinal
(J) / % Loss
in KE
1
2
3
Question 2:
a. Is kinetic energy conserved in an inelastic collision?
b. How about in an elastic collision?
Question 3:
a. Is momentum conserved in an inelastic collision?
b. How about in an elastic collision?
Question 4:
A 5 kg fish swimming at a speed of 1 m/s swallows an absent-minded 1 kg fish at rest.
a. What is the initial total momentum of the system?
b. What is the final total momentum of the system?
c. What is the speed of the larger fish after this lunch?
d. What is the initial total kinetic energy of the system?
e. What is the final total kinetic energy of the system?
f. Calculate the % loss in kinetic energy of the system.
32h
[*] In this lab, bold face type is used to connote vectors.
[*] (See Experiment 4U and Appendix I for more details)