California Standard Addressed
2. The laws of conservation of energy and momentum provide a way to predict and describe the movement of objects. As a basis for understanding this concept:
2. d. Students know how to calculate momentum as the product mv.
2. e. Students know momentum is a separately conserved quantity different from energy.
2. g. Students know how to solve problems involving elastic and inelastic collisions in one dimension by using the principles of conservation of momentum and energy.
Student Performance Outcomes
Use conservation of momentum principles to solve problems with linear elastic and inelastic collisions.
Use conservation of energy principles to solve problems with linear elastic collision principles. / DO NOT WRITE ON THIS HANDOUT
Engage
/ A Newton’s Cradle is a toy shown on the left. When one ball is lifted and released, it will collide with the balls at the bottom, exerting a force.1. How many balls will move on the right, and approximately how fast will it (they) move after the collision. Support your thinking.
A common attraction at an amusement park is bumper cars.
Imagine bumper car A chases down bumper car B as the picture on the right shows.
Predict what will happen during and after the collision. /
2. Draw an arrow on the diagram to show the direction of the net force on car A during the collision.
The length of the arrow should represent the size of the force. / 3. Draw an arrow on the diagram to show the direction of the net force on car B during the collision.
The length of the arrow should represent the size of the force. / 4. What will happen to car A during the collision?
(Speed up, slow down, move at the same speed...)
/ 5. What will happen to car B during the collision?
(Speed up, slow down, move at the same speed...)
6. What do you predict the speed of each car will be after the collision?
/ 7. What will be transferred from car A to car B during the collision?
Share your answers with your lab group. Come to consensus with your answers and share them with the class.
Explore
You will now have an opportunity to test your ideas about collisions using dynamics carts.
Push cart A so that it gently collides with a stationary cart B of equal mass.Predictions / Observations and inferences
Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. / / What happened to the velocity of cart A (include directions in your description)?
Draw an arrow on the diagram to show the direction of the net force on cart B during the collision. / / What happened to the velocity of cart B (include directions in your description)?
What will happen to the cart A during the collision? / Based on your observations Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. /
What will happen to cart A after the collision?
What will happen to the cart B during the collision? / Based on your observations Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. /
What will happen to the cart B after the collision?
Push carts A and B so that cart A “chases” cart B and collides with it gently.
Predictions / Observations and inferences
Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. / / What happened to the velocity of cart A (include directions in your description)?
Draw an arrow on the diagram to show the direction of the net force on cart B during the collision. / / What happened to the velocity of cart B (include directions in your description)?
What will happen to the cart A during the collision? / Based on your observations Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. /
What will happen to cart A after the collision?
What will happen to the cart B during the collision? / Based on your observations Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. /
What will happen to the cart B after the collision?
Push carts A and B so that carts have the same initial speeds but in opposite directions.
Predictions / Observations and inferences
Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. / / What happened to the velocity of cart A (include directions in your description)?
Draw an arrow on the diagram to show the direction of the net force on cart B during the collision. / / What happened to the velocity of cart B (include directions in your description)?
What will happen to the cart A during the collision? / Based on your observations Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. /
What will happen to cart A after the collision?
What will happen to the cart B during the collision? / Based on your observations Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. /
What will happen to the cart B after the collision?
Review
Suppose we have a system consisting of object (particle) A and object B.
Assume that these particles collide.
Newton's Third Law says that if object A exerts a force (of any type) on object B (FAB), then B exerts an equal and opposite force on object A (-FAB). So:
-FAB = FAB
where the negative sign indicates that the forces are in opposite directions.
Newton's Third Law also insists that these two forces occur at exactly the same time.
This means that the impulse (force times time) that A exerts on B is equal and opposite to the impulse that B exerts on A.
ImpulseAB = FAB •t = -FAB•t = -ImpulseAB
Finally, since the impulses are equal and opposite, B's change in momentum is equal and opposite to A's change in momentum:
Change in momentum of B = ImpulseAB = FAB •t = -FAB•t = -ImpulseAB= -Change in momentum of A
This is conservation of momentum - whatever momentum is lost by A must be gained by B, and vice versa.
This means that even though the momentum of A and B can increase or decrease (or change direction), the total (vector) momentum of both A and B must remain constant.
8. Does the total momentum of a system change when the carts interact?
9. Compare the momentum of any of the systems prior to an interaction to the momentum after the interaction.
Law of Conservation of Momentum
The total momentum of a system remains constant.
In these examples, the interacting objects bounced off of one another. This type of interaction is defined to be an elastic collision.
When objects collide and they stick to one another, the interaction is defined to be an inelastic collision.
The carts you used have the ability to stick to one another when arranged appropriately. You will now conduct the same experiments with the carts arranged so that they will have inelastic collisions. Make predictions, conduct the tests and record your observations.
Push cart A so that it gently collides with a stationary cart B of equal mass.Arrange the carts so that they will stick together after the collision.
Predictions / Observations and inferences
Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. / / What happened to the velocity of cart A (include directions in your description)?
Draw an arrow on the diagram to show the direction of the net force on cart B during the collision. / / What happened to the velocity of cart B (include directions in your description)?
What will happen to the cart A during the collision? / Compare the momentum of the linked system after the collision to the momentum of the individual carts prior to the collision.
What will happen to the cart B during the collision?
Predict the velocity of the linked system A+B to the initial velocity of cart A.
Push carts A and B so that cart A “chases” cart B and collides with it gently.
Arrange the carts so that they will stick together after the collision.
Predictions / Observations and inferences
Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. / / What happened to the velocity of cart A (include directions in your description)?
Draw an arrow on the diagram to show the direction of the net force on cart B during the collision. / / What happened to the velocity of cart B (include directions in your description)?
What will happen to the cart A during the collision? / Compare the momentum of the linked system after the collision to the momentum of the individual carts prior to the collision.
What will happen to the cart B during the collision?
Predict the velocity of the linked system A+B to the initial velocities of cart A and B.
Push carts A and B so that carts have the same initial speeds but in opposite directions.
Arrange the carts so that they will stick together after the collision.
Predictions / Observations and inferences
Draw an arrow on the diagram to show the direction of the net force on cart A during the collision. / / What happened to the velocity of cart A (include directions in your description)?
Draw an arrow on the diagram to show the direction of the net force on cart B during the collision. / / What happened to the velocity of cart B (include directions in your description)?
What will happen to the cart A during the collision? / Compare the momentum of the linked system after the collision to the momentum of the individual carts prior to the collision.
What will happen to the cart B during the collision?
Predict the velocity of the linked system A+B to the initial velocities of carts A and B.
Tables like the ones below are used to demonstrate the conservation of momentum in
elastic and inelastic collisions. We will use a simplified version once we begin with calculations in connection with the conservation of momentum.
Cart A Initial MomentumpAo = mA•vAo
(kg•m/s) / Cart B Initial Momentum
pBo = mB•vBo
(kg•m/s) / Elastic
Collision / Cart A final Momentum
pAo = mA•v’A
(kg•m/s) / Cart B final Momentum
pBo = mB•v’B
(kg•m/s)
Sum of Momenta
∑po = pAo +pBo
(kg•m/s) / Sum of Momenta
∑p’ = pA’+ pB’
(kg•m/s)
Cart A Initial Momentum
pAo = mA•vAo
(kg•m/s) / Cart B Initial Momentum
pBo = mB•vBo
(kg•m/s) / Inelastic
Collision / Combined Cart A + B final Momentum
pA+B’ = (mA+mB)•v’
(kg•m/s)
Sum of Momenta
∑po = pAo +pBo
(kg•m/s)
When using these tables it is possible to isolate missing terms to make predictions.
Explain
10. When objects interact in a collision, what is exerted on each object?
11. Describe the magnitude of the force exerted on each object in a collision.
12. Do the forces exerted in an interaction between objects result in identical accelerations? If not, why not?
13. In an ideal system, when objects interact, a property may be transferred from one object to another. What may be transferred?
14. How is the value for momentum determined?
15. Is momentum a vector or scalar quantity?
16. ∆p = 0. This is a statement of
17. What is a collision?
18. What is an elastic collision?
19. Is momentum conserved in an elastic collision?
20. What expanded equation can be used to describe the momentum values in an elastic collision?
21. What is an inelastic collision?
22. What expanded equation can be used to describe the momentum values in an inelastic collision?
Elaborate
In real world systems, fiction acts as a dissipative force making it difficult to observe theoretical results. Internet applets such as the ones found at:
Allow for precise calculations of momenta in elastic and inelastic collisions. Please conduct the following trials.
Trial 1 - Elastic
/ Use the settings illustrated on the left.Mass 1 = 1kg
Mass 2 = 1kg
Velocity 1 = 1.00 m/s
Velocity 2 = 0 m/s
You can stop the simulation at any point to obtain values for the data section of the table below.
BEFORE AFTER
m1 v1 / m2 v2 / m1 v1 / m2 v2m1v1 + m2v2 = m1v1 + m2v2
Trial 2 - Elastic
/ Use the settings illustrated on the left.Mass 1 =1.0 = 1.0 kg
Mass 2 = 1.0 = 1.0 kg
Velocity 1 = 2.0 m/s
Velocity 2 = 0.5 m/s
You can stop the simulation at any point to obtain values for the data section of the table below.
BEFORE AFTER
m1 v1 / m2 v2 / m1 v1 / m2 v2m1v1 + m2v2 = m1v1 + m2v2
Trial 3 - Elastic
/ Use the settings illustrated on the left.Mass 1 =1.0 = 1.0 kg
Mass 2 = 1.0 = 1.0 kg
Velocity = 1.0 m/s
Velocity = -1.0 m/s
You can stop the simulation at any point to obtain values for the data section of the table below.
BEFORE AFTER
m1 v1 / m2 v2 / m1 v1 / m2 v2m1v1 + m2v2 = m1v1 + m2v2
Trial 1 - Inelastic
/ Use the settings illustrated on the left.Mass 1 = 1.0 = 1.0 kg
Mass 2 = 1.0 = 1.0 kg
Velocity = 1.0 m/s
You can stop the simulation at any point to obtain values for the data section of the table below.
BEFORE AFTER
m1 v1 / m2 v2 / m1 v1 / m2 v2m1v1 + m2v2 = m1v1 + m2v2
Trial 2 - Inelastic
/ Use the settings illustrated on the left.Mass 1 =1.0 = 1.0 kg
Mass 2 = 1.0 = 1.0 kg
Velocity = 2.0 m/s
Velocity = 0.5 m/s
You can stop the simulation at any point to obtain values for the data section of the table below.
BEFORE AFTER
m1 v1 / m2 v2 / m1 v1 / m2 v2m1v1 + m2v2 = m1v1 + m2v2
Trial 3 - Inelastic
/ Use the settings illustrated on the left.Mass 1 =1.0 = 1.0 kg
Mass 2 = 1.0 = 1.0 kg
Velocity = 1.0 m/s
Velocity = -1.0m/s
You can stop the simulation at any point to obtain values for the data section of the table below.
BEFORE AFTER
m1 v1 / m2 v2 / m1 v1 / m2 v2m1v1 + m2v2 = m1v1 + m2v2
23. How did the initial momentum compare to the final momentum in each trial?
24. Summarize the Law of Conservation of Momentum.
Evaluate
Impulse
25. A 0.9 kg bat strikes a 0.14 kg ball so that the ball acquires a velocity of 43 m/s.
a) What is the momentum acquired by the baseball?
b) If the collision occurred over a time of 0.002 seconds, what was the average force delivered to the ball?
c) What was the average force delivered to the bat?
d) What was the average acceleration experienced by the ball during the collision?
e) What was the average acceleration experienced by the bat?
26. A 0.34 kg tennis racquet strikes a 0.057 kg tennis ball so that the ball acquires a momentum of 1.55 kg•m/s.
a) What is the velocity of the tennis ball after the collision?
b) If the average force exerted on the ball during the collision is 222 N, what was the duration of the collision (time)?
c) What was the average force exerted on the tennis racquet in the collision?
d) What was the average acceleration experienced by the ball?
e) What was the average acceleration experienced by the racquet?
Elastic Collisions
27. A cue ball (0.17 kg) strikes an “8” ball (0.16 kg) elastically on a pool table. The cue ball has an initial velocity of 0.2m/s, and a velocity after the collision of 0.006 m/s.
a) What is the initial momentum of the cue ball?
b) What is the final momentum of the cue ball?
c) What is the final momentum of the “8” ball?
d) What is the final velocity of the “8” ball?
28. A bowling ball (6.2 kg) strikes a bowling pin (0.9kg) elastically on a bowling alley. The bowling ball has an initial velocity of 1.2 m/s, and the bowling pin has a final velocity of 2.1 m/s.
a) What is the initial momentum of the bowling ball?
b) What is the final momentum of the bowling pin?
c) What is the final momentum of the bowling ball?
d) What is the final velocity of the bowling ball?
Inelastic Collisions
29. A Cesna 150 plane with passenger (800. kg) collides inelastically with a Canadian goose (10.9 kg). The plane has an initial velocity of 45 m/s and the Canadian goose has an initial velocity of -25m/s.
a) What is the initial momentum of the plane?
b) What is the initial momentum of the goose?
c) What is the total momentum of the system?
d) What is the final velocity of the plane + goose object?
30. A 0.22 kg Sharp-shinned hawk dives on a 0.04kg Mocking bird and catches it mid-air. The initial velocity of the Mocking bird was 5.00 m/s, and the final velocity of the Sharp-shinned hawk carrying its prey is 9.00 m/s.
a) What was the final momentum of the hawk + mocking bird object?
b) What was the momentum of the mocking bird before the collision?
d) What was the momentum of the hawk prior to the collision?
e) What was the velocity of the hawk prior to the collision?
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