Momentum & Impulse
The Law of Momentum Conservation
Momentum Conservation Principle
One of the most powerful laws in physics is the law of momentum conservation. The law of momentum conservation can be stated as follows.
For a collision occurring between object 1 and object 2 in an isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2.
The above statement tells us that the total momentum of a collection of objects (a system) is conserved" - that is the total amount of momentum is a constant or unchanging value. This law of momentum conservation will be the focus of the remainder of this lesson. To understand the basis of momentum conservation, let's begin with a short logical proof.
Consider a collision between two objects - object 1 and object 2. For such a collision, the forces acting between the two objects are equal in magnitude and opposite in direction (Newton's third law). This statement can be expressed in equation form as follows.
The forces act between the two objects for a given amount of time. In some cases, the time is long; in other cases the time is short. Regardless of how long the time is, it can be said that the time that the force acts upon object 1 is equal to the time that the force acts upon object 2. This is merely logical; forces result from interactions (or touching) between two objects. If object 1 touches object 2 for 0.050 seconds, then object 2 must be touching object 1 for the same amount of time (0.050 seconds). As an equation, this can be stated as
Since the forces between the two objects are equal in magnitude and opposite in direction, and since the times for which these forces act are equal in magnitude, it follows that the impulses experienced by the two objects are also equal in magnitude and opposite in direction. As an equation, this can be stated as
But the impulse experienced by an object is equal to the change in momentum of that object (the impusle-momentum change theorem). Thus, since each object experiences equal and opposite impulses, it follows logically that they must also experience equal and opposite momentum changes. As an equation, this can be stated as
The above equation is one statement of the law of momentum conservation. In a collision, the momentum change of object 1 is equal and opposite to the momentum change of object 2. That is, the momentum lost by object 1 is equal to the momentum gained by object 2. In a collision between two objects, one object slows down and loses momentum while the other object speeds up and gains momentum. If object 1 loses 75 units of momentum, then object 2 gains 75 units of momentum. Yet, the total momentum of the two objects (object 1 plus object 2) is the same before the collision as it is after the collision; the total momentum of the system (the collection of two objects) is conserved.
A useful analogy for understanding momentum conservation involves a money transaction between two people. Let's refer to the two people as Jack and Jill. Suppose that we were to check the pockets of Jack and Jill before and after the money transaction in order to determine the amount of money which each possessed. Prior to the transaction, Jack possesses $100 and Jill possesses $100. The total amount of money of the two people before the transaction is $200. During the transaction, Jack pays Jill $50 for the given item being bought. There is a transfer of $50 from Jack's pocket to Jill's pocket. Jack has lost $50 and Jill has gained $50. The money lost by Jack is equal to the money gained by Jill. After the transaction, Jack now has $50 in his pocket and Jill has $150 in her pocket. Yet, the total amount of money of the two people after the transaction is $200. The total amount of money (Jack's money plus Jill's money) before the transaction is equal to the total amount of money after the transaction. It could be said that the total amount of money of the system (the collection of two people) is conserved; it is the same before as it is after the transaction.
A useful means of depicting the transfer and the conservation of money between Jack and Jill is by means of a table.
The table shows the amount of money possessed by the two individuals before and after the interaction. It also shows the total amount of money before and after the interaction. Note that the total amount of money ($200) is the same before and after the interaction - it is conserved. Finally, the table shows the change in the amount of money possessed by the two individuals. Note that the change in Jack's money account (-$50) is equal and opposite to the change in Jill's money account (+$50) .
For any collision occurring in an isolated system, momentum is conserved - the total amount of momentum of the collection of objects in the system is the same before the collision as after the collision. This is the very phenomenon which was observed in "The Cart and The Brick" lab. In this lab, a brick at rest was dropped upon a loaded cart which was in motion.
Before the collision, the dropped brick had 0 units of momentum (it was at rest). The momentum of the loaded cart can be determined using the velocity (as determined by the ticker tape analysis) and the mass. The total amount of momentum was the sum of the dropped brick's momentum (0 units) and the loaded cart's momentum. After the collision, the momenta of the two separate objects (dropped brick and loaded cart) can be determined from their measured mass and their velocity (found from the ticker tape analysis). If momentum is conserved during the collision, then the sum of the dropped brick's and loaded cart's momentum after the collision should be the same as before the collision. The momentum lost by the loaded cart should equal (or approximately equal) the momentum gained by the dropped brick. Momentum data for the interaction between the dropped brick and the loaded cart could be depicted in a table similar to the money table above.
BeforeCollision
Momentum / After
Collision
Momentum / Change in
Momentum
Dropped Brick / 0 units / 14 units / +14 units
Loaded Cart / 45 units / 31 units / -14 units
Total / 45 units / 45 units
Note that the loaded cart lost 14 units of momentum and the dropped brick gained 14 units of momentum. Note also that the total momentum of the system (45 units) was the same before the collision as it is after the collision.
Collisions commonly occur in contact sports (such as football) and racket and bat sports (such as baseball, golf, tennis, etc.). Consider a collision in football between a fullback and a linebacker during a goal-line stand. The fullback plunges across the goal line and collides in midair with linebacker. The linebacker and fullback hold each other and travel together after the collision. The fullback possesses a momentum of 100 kg*m/s, East before the collision and the linebacker possesses a momentum of 120 kg*m/s, West before the collision. The total momentum of the system before the collision is 20 kg*m/s, West (review the section on adding vectors if necessary). Therefore, the total momentum of the system after the collision must also be 20 kg*m/s, West. The fullback and the linebacker move together as a single unit after the collision with a combined momentum of 20 kg*m/s. Momentum is conserved in the collision. A vector diagram can be used to represent this principle of momentum conservation; such a diagram uses an arrow to represent the magnitude and direction of the momentum vector for the individual objects before the collision and the combined momentum after the collision.
Now suppose that a medicine ball is thrown to a clown who is at rest upon the ice; the clown catches the medicine ball and glides together with the ball across the ice. The momentum of the medicine ball is 80 kg*m/s before the collision. The momentum of the clown is 0 m/s before the collision. The total momentum of the system before the collision is 80 kg*m/s. Therefore, the total momentum of the system after the collision must also be 80 kg*m/s. The clown and the medicine ball move together as a single unit after the collision with a combined momentum of 80 kg*m/s. Momentum is conserved in the collision.
Momentum is conserved for any interaction between two objects occurring in an isolated system. This conservation of momentum can be observed by a total system momentum analysis and by a momentum change analysis. Useful means of representing such analyses include a momentum table and a vector diagram. Later in this lesson, we will use the momentum conservation principle to solve problems in which the after-collision velocity of objects is predicted
Check Your Understanding (to be completed on a separate sheet of paper)
Express your understanding of the concept and mathematics of momentum by answering the following questions.
1. Explain why it is difficult for a firefighter to hold a hose which ejects large amounts of high-speed water.
2. A large truck and a Volkswagen have a head-on collision.
a. Which vehicle experiences the greatest force of impact?
b. Which vehicle experiences the greatest impulse?
c. Which vehicle experiences the greatest momentum change?
d. Which vehicle experiences the greatest acceleration?
3. Miles Tugo and Ben Travlun are riding in a bus at highway speed on a nice summer day when an unlucky bug splatters onto the windshield. Miles and Ben begin discussing the physics of the situation. Miles suggests that the momentum change of the bug is much greater than that of the bus. After all, argues Miles, there was no noticeable change in the speed of the bus compared to the obvious change in the speed of the bug. Ben disagrees entirely, arguing that that both bug and bus encounter the same force, momentum change, and impulse. Who do you agree with? Support your answer.
4. If a ball is projected upward from the ground with ten units of momentum, what is the momentum of recoil of the Earth? ______Do we feel this? Explain.
5. If a 5-kg bowling ball is projected upward with a velocity of 2.0 m/s, then what is the recoil velocity of the Earth (mass = 6.0 x 10^24 kg).
6. A 120 kg lineman moving west at 2 m/s tackles an 80 kg football fullback moving east at 8 m/s. After the collision, both players move east at 2 m/s. Draw a vector diagram in which the before- and after-collision momenta of each player is represented by a momentum vector. Label the magnitude of each momentum vector. Before After
7. Would you care to fire a rifle that has a bullet ten times as massive as the rifle? Explain.
8. A baseball player holds a bat loosely and bunts a ball. Express your understanding of momentum conservation by filling in the tables below.
9. A Tomahawk cruise missile is launched from the barrel of a mobile missile launcher. Neglect friction. Express your understanding of momentum conservation by filling in the tables below.
Isolated System Notes
Total system momentum is conserved for collisions occurring in isolated systems. But what makes a system of objects an isolated system? And is momentum conserved if the system is not isolated? This is the focus of this part of Lesson 2.
A system is a collection of two or more objects. An isolated system is a system which is free from the influence of a net external force. There are two criteria for the presence of a net external force; it must be...
a force which originates from a source other than the two objects of the system
a force that is not balanced by other forces.
Check Your Understanding I
Concepts of Physics - Mr. Lawrence
Express your understanding of the concept and mathematics of momentum by answering the following questions.
1. Determine the momentum of a ...
- 60-kg halfback moving eastward at 9 m/s.
- 1000-kg car moving northward at 20 m/s.
- 40-kg freshman moving southward at 2 m/s.
2. A car possesses 20 000 units of momentum. What would be the car's new momentum if ...
- its velocity were doubled.
- its velocity were tripled.
- its mass were doubled (by adding more passengers and a greater load)
- both its velocity were doubled and its mass were doubled.
3. A halfback (m = 60 kg), a tight end (m = 90 kg), and a lineman (m = 120 kg) are running down the football field. Consider their ticker tape patterns below.
Compare the velocities of these three players. How many times greater is the velocity of the halfback and the velocity of the tight end than the velocity of the lineman?
Which player has the greatest momentum? Explain.
Vector Diagram
Greatest velocity change?Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
______
Velocity-Time Graph
Greatest velocity change?Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
______
Ticker Tape Diagram
Greatest velocity change?Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
Check Your Understanding Part II
Concepts of Physics - Mr. Lawrence
Express your understanding of the impulse-momentum change theorem by answering the following questions.
1. A 0.50-kg cart (#1) is pulled with a 1.0-N force for 1 second; another 0.50 kg cart (#2) is pulled with a 2.0 N-force for 0.50 seconds. Which cart (#1 or #2) has the greatest acceleration? Explain.
Which cart (#1 or #2) has the greatest impulse? Explain.
Which cart (#1 or #2) has the greatest change in momentum? Explain.
2. In a phun physics demo, two identical balloons (A and B) are propelled across the room on horizontal guide wires. The motion diagrams (depicting the relative position of the balloons at time intervals of 0.05 seconds) for these two balloons are shown below.
Which balloon (A or B) has the greatest acceleration? Explain.
Which balloon (A or B) has the greatest final velocity? Explain.
Which balloon (A or B) has the greatest momentum change? Explain.
Which balloon (A or B) experiences the greatest impulse? Explain.
3. Two cars of equal mass are traveling down Lake Avenue with equal velocities. They both come to a stop over different lengths of time. The ticker tape patterns for each car are shown on the diagram below.
At what approximate location on the diagram (in terms of dots) does each car begin to experience the impulse.
Which car (A or B) experiences the greatest acceleration? Explain.
Which car (A or B) experiences the greatest change in momentum? Explain.
Which car (A or B) experiences the greatest impulse? Explain.
4. The diagram to the right depicts the before- and after-collision speeds of a car which undergoes a head-on-collision with a wall. In Case A, the car bounces off the wall. In Case B, the car "sticks" to the wall.
In which case (A or B) is the change in velocity the greatest? Explain.
In which case (A or B) is the change in momentum the greatest? Explain.
In which case (A or B) is the impulse the greatest? Explain.
In which case (A or B) is the force which acts upon the car the greatest (assume contact times are the same in both cases)? Explain.
5. Rhonda, who has a mass of 60.0 kg, is riding at 25.0 m/s in her sports car when she must suddenly slam on the brakes to avoid hitting a dog crossing the road. She strikes the air bag, which brings her body to a stop in 0.400 s. What average force does the seat belt exert on her?
If Rhonda had not been wearing her seat belt and not had an air bag, then the windshield would have stopped her head in 0.001 s. What average force would the windshield have exerted on her?
6. A hockey player applies an average force of 80.0 N to a 0.25 kg hockey puck for a time of 0.10 seconds. Determine the impulse experienced by the hockey puck.
7. If a 5-kg object experiences a 10-N force for a duration of 0.1-second, then what is the momentum change of the object?
Check Your Understanding Part III
Concepts of Physics - Mr. Lawrence
Express your understanding of Newton's third law by answering the following questions.
1. While driving down the road, an unfortunate bug strikes the windshield of a bus. Quite obviously, a case of Newton's third law of motion. The bug hit the bus and the bus hit the bug. Which of the two forces is greater: the force on the bug or the force on the bus?
2. Rockets are unable to accelerate in space because ...
a. there is no air in space for the rockets to push off of.
b. there is no gravity is in space.
c. there is no air resistance in space.
d. ... nonsense! Rockets do accelerate in space.
3. A gun recoils when it is fired. The recoil is the result of action-reaction force pairs. As the gases from the gunpowder explosion expand, the gun pushes the bullet forwards and the bullet pushes the gun backwards. The acceleration of the recoiling gun is ...
a. greater than the acceleration of the bullet.
b. smaller than the acceleration of the bullet.
c. the same size as the acceleration of the bullet.
4. Would it be a good idea to jump from a rowboat to a dock that seems within jumping distance? Explain.
5. If we throw a ball horizontally while standing on roller skates, we roll backward with a momentum that matches that of the ball. Will we roll backward if we go through the motion of throwing the ball without letting go of it? Explain.
6. Suppose there are three astronauts outside a spaceship and two of them decide to play catch with the other woman. All three astronauts weigh the same on Earth and are equally strong. The first astronaut throws the second astronaut towards the third astronaut and the game begins. Describe the motion of these women as the game proceeds. Assume each toss results from the same-sized "push." How long will the game last?