Conservation of Conservation of MomentumMomentum

The Law of Action-Reaction The Law of Action-Reaction

RevisitedRevisited

An IntroductionAn Introduction

A collision is an interaction between two A collision is an interaction between two objects which have made contact objects which have made contact (usually) with each other. As in any (usually) with each other. As in any interaction, a collision results in a force interaction, a collision results in a force being applied to the two colliding being applied to the two colliding objects. Such collisions are governed by objects. Such collisions are governed by Newton's laws of motion. In the past, Newton's laws of motion. In the past, Newton's third law of motion was Newton's third law of motion was introduced and discussed. It was said introduced and discussed. It was said that...that...

... in every interaction, there is a pair ... in every interaction, there is a pair of forces acting on the two interacting of forces acting on the two interacting objects. The size of the force on the objects. The size of the force on the first object equals the size of the force first object equals the size of the force on the second object. The direction of on the second object. The direction of the force on the first object is the force on the first object is opposite to the direction of the force opposite to the direction of the force on the second object. Forces always on the second object. Forces always come in pairs - equal and opposite come in pairs - equal and opposite action-reaction force pairs.action-reaction force pairs.

Newton's third law of motion is Newton's third law of motion is naturally applied to collisions between naturally applied to collisions between two objects. In a collision between two objects. In a collision between two objects, both objects experience two objects, both objects experience forces which are equal in magnitude forces which are equal in magnitude and opposite in direction. Such forces and opposite in direction. Such forces cause one object to speed up (gain cause one object to speed up (gain momentum) and the other object to momentum) and the other object to slow down (lose momentum). slow down (lose momentum).

According to Newton's third law, the forces on According to Newton's third law, the forces on the two objects are equal in magnitude. While the two objects are equal in magnitude. While the forces are equal in magnitude and the forces are equal in magnitude and opposite in direction, the acceleration of the opposite in direction, the acceleration of the objects are not necessarily equal in objects are not necessarily equal in magnitude. In accord with magnitude. In accord with Newton's second law of motionNewton's second law of motion, the , the acceleration of an object is dependent upon acceleration of an object is dependent upon both force and mass. Thus, if the colliding both force and mass. Thus, if the colliding objects have unequal mass, they will have objects have unequal mass, they will have unequal accelerations as a result of the unequal accelerations as a result of the contact force which results during the collision. contact force which results during the collision.

Consider the collision between the club Consider the collision between the club head and the golf ball in the sport of head and the golf ball in the sport of golf. When the club head of a moving golf. When the club head of a moving golf club collides with a golf ball at rest golf club collides with a golf ball at rest upon a tee, the force experienced by the upon a tee, the force experienced by the club head is equal to the force club head is equal to the force experienced by the golf ball. Most experienced by the golf ball. Most observers of this collision have difficulty observers of this collision have difficulty with this concept because they perceive with this concept because they perceive the high speed given to the ball as the the high speed given to the ball as the result of the collision result of the collision

They are not observing unequal forces upon They are not observing unequal forces upon the ball and club head, but rather unequal the ball and club head, but rather unequal accelerations. Both club head and ball accelerations. Both club head and ball experience equal forces, yet the ball experience equal forces, yet the ball experiences a greater acceleration due to experiences a greater acceleration due to its smaller mass. In a collision, there is a its smaller mass. In a collision, there is a force on both objects which causes an force on both objects which causes an acceleration of both objects; the forces are acceleration of both objects; the forces are equal in magnitude and opposite in equal in magnitude and opposite in direction, yet the least massive object direction, yet the least massive object receives the greatest acceleration. receives the greatest acceleration.

Consider the collision between a moving Consider the collision between a moving seven-ball and an eight-ball that is at seven-ball and an eight-ball that is at rest in the sport of billiards. When the rest in the sport of billiards. When the seven-ball collides with the eight-ball, seven-ball collides with the eight-ball, each ball experiences an equal force each ball experiences an equal force directed in opposite directions. The directed in opposite directions. The rightward moving seven-ball rightward moving seven-ball experiences a leftward force which experiences a leftward force which causes it to slow down; the eight-ball causes it to slow down; the eight-ball experiences a rightward force which experiences a rightward force which causes it to speed up causes it to speed up

Since the two balls have equal masses, Since the two balls have equal masses, they will also experience equal they will also experience equal accelerations. In a collision, there is a accelerations. In a collision, there is a force on both objects which causes an force on both objects which causes an acceleration of both objects; the forces acceleration of both objects; the forces are equal in magnitude and opposite in are equal in magnitude and opposite in direction. For collisions between equal-direction. For collisions between equal-mass objects, each object experiences mass objects, each object experiences the same acceleration. the same acceleration.

Consider the interaction between a male and Consider the interaction between a male and female figure skater in pair figure skating. A female figure skater in pair figure skating. A woman (m = 45 kg) is kneeling on the woman (m = 45 kg) is kneeling on the shoulders of a man (m = 70 kg); the pair is shoulders of a man (m = 70 kg); the pair is moving along the ice at 1.5 m/s. The man moving along the ice at 1.5 m/s. The man gracefully tosses the woman forward through gracefully tosses the woman forward through the air and onto the ice. The woman receives the air and onto the ice. The woman receives the forward force and the man receives a the forward force and the man receives a backward force. The force on the man is equal backward force. The force on the man is equal in magnitude and opposite in direction to the in magnitude and opposite in direction to the force on the woman. Yet the acceleration of force on the woman. Yet the acceleration of the woman is greater than the acceleration of the woman is greater than the acceleration of the man due to the smaller mass of the the man due to the smaller mass of the woman.woman.

Many observers of this interaction Many observers of this interaction have difficulty believing that the man have difficulty believing that the man experienced a backward force. "After experienced a backward force. "After all," they might argue, "the man did all," they might argue, "the man did not move backward." Such observers not move backward." Such observers are presuming that forces cause are presuming that forces cause motion; that is a backward force motion; that is a backward force would cause a backward motion. This would cause a backward motion. This is a common misconception that has is a common misconception that has been addressed before in our class. been addressed before in our class.

Forces cause acceleration, not motion. Forces cause acceleration, not motion. The male figure skater experiences a The male figure skater experiences a backwards (you might say "negative") backwards (you might say "negative") force which causes his backwards (or force which causes his backwards (or "negative") acceleration; that is, the "negative") acceleration; that is, the man slowed down while the woman man slowed down while the woman sped up. In every interaction (with no sped up. In every interaction (with no exception), there are forces acting upon exception), there are forces acting upon the two interacting objects which are the two interacting objects which are equal in magnitude and opposite in equal in magnitude and opposite in direction. direction.

Collisions are governed by Newton's Collisions are governed by Newton's laws. The law of action-reaction laws. The law of action-reaction (Newton's third law) explains the (Newton's third law) explains the nature of the forces between the two nature of the forces between the two interacting objects. According to the interacting objects. According to the law, the force exerted by object 1 law, the force exerted by object 1 upon object 2 is equal in magnitude upon object 2 is equal in magnitude and opposite in direction to the force and opposite in direction to the force exerted by object 2 upon object 1.exerted by object 2 upon object 1.

Examples for you to tryExamples for you to try

Hint: Some of these questions Hint: Some of these questions could be seen again! could be seen again!

1. While driving down the road, Anna 1. While driving down the road, Anna Litical observed a bug striking the Litical observed a bug striking the windshield of her car. Quite windshield of her car. Quite obviously, a case of Newton's third obviously, a case of Newton's third law of motion. The bug hit the law of motion. The bug hit the windshield and the windshield hit the windshield and the windshield hit the bug. Which of the two forces is bug. Which of the two forces is greater: the force on the bug or the greater: the force on the bug or the force on the windshield?force on the windshield?

Answer 1Answer 1

TRICK Question! Each force is the TRICK Question! Each force is the same size. same size.

For every action there is an equal For every action there is an equal (yes equal) reaction. The fact that (yes equal) reaction. The fact that the bug splatters only means that it the bug splatters only means that it has a smaller mass that was unable has a smaller mass that was unable to withstand the larger acceleration to withstand the larger acceleration resulting from the interaction. resulting from the interaction.

2. Rockets are unable to 2. Rockets are unable to accelerate in space because ...accelerate in space because ...

A. there is no air in space for the A. there is no air in space for the rockets to push off of.rockets to push off of.

B. there is no gravity is in space.B. there is no gravity is in space.

C. there is no air resistance in C. there is no air resistance in space.space.

D. ...nonsense! Rockets do D. ...nonsense! Rockets do accelerate in space.accelerate in space.

Answer 2 Answer 2

Answer is DAnswer is D It is a common misconception that It is a common misconception that

rockets are unable to accelerate in rockets are unable to accelerate in space. The fact is that rockets do space. The fact is that rockets do accelerate. They are able to accelerate. They are able to accelerate due to the fact that they accelerate due to the fact that they burn fuel and push the exhaust in a burn fuel and push the exhaust in a direction opposite to the direction direction opposite to the direction they wish to accelerate. they wish to accelerate.

3. A gun recoils when it is fired. The recoil is 3. A gun recoils when it is fired. The recoil is the result of action-reaction force pairs. As the result of action-reaction force pairs. As the gases from the gunpowder explosion the gases from the gunpowder explosion expand, the gun pushes the bullet forwards expand, the gun pushes the bullet forwards and the bullet pushes the gun backwards. and the bullet pushes the gun backwards. The acceleration of the recoiling gun is ...The acceleration of the recoiling gun is ...

A. greater than the acceleration of the bullet.A. greater than the acceleration of the bullet.

B. smaller than the acceleration of the bullet.B. smaller than the acceleration of the bullet.

C. the same size as the acceleration of the C. the same size as the acceleration of the bullet. bullet.

Answer 3. Answer 3.

The answer is B.The answer is B. The force on the gun equals the force The force on the gun equals the force

on the bullet. Yet acceleration on the bullet. Yet acceleration depends on both force and mass. The depends on both force and mass. The bullet has greater acceleration due to bullet has greater acceleration due to the fact that it has a smaller mass. the fact that it has a smaller mass. Remember acceleration and mass Remember acceleration and mass are inversely proportional. are inversely proportional.

4. Why is it important that an 4. Why is it important that an airplane wing be designed so that it airplane wing be designed so that it deflects oncoming air downward? deflects oncoming air downward?

Answer 4 Answer 4

This can be explained by Newton’s This can be explained by Newton’s third law of motion. The more air that a third law of motion. The more air that a wing can push down, the more that the wing can push down, the more that the air is able to push the wing up. If air is able to push the wing up. If enough air is pushed downward, then enough air is pushed downward, then the reaction to this will result in the reaction to this will result in sufficient upward push on the wing and sufficient upward push on the wing and the plane to provide the lift necessary the plane to provide the lift necessary to elevate the plane off the ground. to elevate the plane off the ground.

5. Would it be a good idea to jump 5. Would it be a good idea to jump from a rowboat to a dock that seems from a rowboat to a dock that seems within jumping distance? Explain.within jumping distance? Explain.

Answer 5 Answer 5

NO! Don’t do this at home (at least, NO! Don’t do this at home (at least, not if you wish to dock the boat)! As not if you wish to dock the boat)! As you jump to reach the dock, the you jump to reach the dock, the rowboat pushes you forward (action), rowboat pushes you forward (action), and thus you push the rowboat and thus you push the rowboat backwards. You will indeed reach the backwards. You will indeed reach the dock; your rowboat will be several dock; your rowboat will be several feet away! feet away!

6. If we throw a ball horizontally 6. If we throw a ball horizontally while standing on roller skates, we while standing on roller skates, we roll backward with a momentum that roll backward with a momentum that matches that of the ball. Will we roll matches that of the ball. Will we roll backward if we go through the backward if we go through the motion of throwing the ball without motion of throwing the ball without letting go of it? Explain.letting go of it? Explain.

Answer 6Answer 6

The overall motion of a person who The overall motion of a person who merely goes through the motion of merely goes through the motion of throwing the ball (without letting go) throwing the ball (without letting go) will be “null.” Such a person will roll will be “null.” Such a person will roll backwards then forwards. Yet when backwards then forwards. Yet when finished the person will finish where finished the person will finish where she started. Recall the demonstration she started. Recall the demonstration of this in class. of this in class.

Astronaut Description Astronaut Description

We’ll look at an explanation We’ll look at an explanation then one more example. then one more example.

Imagine that you are hovering next Imagine that you are hovering next to the space shuttle in earth-orbit to the space shuttle in earth-orbit and your buddy of equal mass who is and your buddy of equal mass who is moving 4 m/s (with respect to the moving 4 m/s (with respect to the ship) bumps into you. If she holds ship) bumps into you. If she holds onto you, then how fast do the two of onto you, then how fast do the two of you move after the you move after the collisioncollision??

A question like this involves momentum A question like this involves momentum principles. In any instance in which two principles. In any instance in which two objects collide and can be considered objects collide and can be considered isolated from all other net forces, the isolated from all other net forces, the conservation of momentum principle conservation of momentum principle can be utilized to determine the post-can be utilized to determine the post-collision velocities of the two objects. collision velocities of the two objects. Collisions between objects are Collisions between objects are governed by laws of momentum and governed by laws of momentum and energy. energy.

When a collision occurs in an isolated When a collision occurs in an isolated system, the total momentum of the system, the total momentum of the system of objects is conserved. system of objects is conserved. Provided that there are no net external Provided that there are no net external forces acting upon the two astronauts, forces acting upon the two astronauts, the combined momentum of the two the combined momentum of the two astronauts before the collision equals astronauts before the collision equals the combined momentum of the two the combined momentum of the two astronauts after the collision. astronauts after the collision.

The mathematics of this problem is The mathematics of this problem is simplified by the fact that before the simplified by the fact that before the collision, there is only one object in motion collision, there is only one object in motion and after the collision both objects have and after the collision both objects have the same velocity. That is to say, a the same velocity. That is to say, a momentum analysis would show that all momentum analysis would show that all the momentum was the momentum was concentratedconcentrated in the in the moving astronaut before the collision. And moving astronaut before the collision. And after the collision, all the momentum was after the collision, all the momentum was the result of a the result of a single objectsingle object (the (the combination of the two astronauts) moving combination of the two astronauts) moving at an easily predictable velocity. Since at an easily predictable velocity. Since there is twice as much mass in motion there is twice as much mass in motion after the collision, it must be moving at after the collision, it must be moving at one-half the velocity. Thus, the two one-half the velocity. Thus, the two astronauts move together with a velocity astronauts move together with a velocity of 2 m/s after the collision. of 2 m/s after the collision.

7. Suppose there are three astronauts 7. Suppose there are three astronauts outside a spaceship and two of them decide outside a spaceship and two of them decide to play catch with the other woman. All to play catch with the other woman. All three astronauts weigh the same on Earth three astronauts weigh the same on Earth and are equally strong. The first astronaut and are equally strong. The first astronaut throws the second astronaut towards the throws the second astronaut towards the third astronaut and the game begins. third astronaut and the game begins. Describe the motion of these women as the Describe the motion of these women as the game proceeds. Assume each toss results game proceeds. Assume each toss results from the same-sized "push." How long will from the same-sized "push." How long will the game last? the game last?

Answer 7 Answer 7 The game will last two throws and one catch. The game will last two throws and one catch.

When astronaut #1 throws astronaut #2, the When astronaut #1 throws astronaut #2, the two astronauts will travel opposite directions two astronauts will travel opposite directions at the same speed (action-reaction). When at the same speed (action-reaction). When astronaut #3 catches astronaut #2, astronaut #3 catches astronaut #2, astronaut #2 will slow to half of her speed astronaut #2 will slow to half of her speed and move together with astronaut #3. Now and move together with astronaut #3. Now astronaut #1 is moving leftward with original astronaut #1 is moving leftward with original speed of astronaut #2 and #3 are moving speed of astronaut #2 and #3 are moving rightward at half of the original speed. When rightward at half of the original speed. When astronaut #3 pushes #2, the greatest speed astronaut #3 pushes #2, the greatest speed which #2 can have is half of the original which #2 can have is half of the original speed in the opposite direction. The game is speed in the opposite direction. The game is now over for astronaut #2 can never catch now over for astronaut #2 can never catch up with astronaut #1. up with astronaut #1.

Conservation of Conservation of Momentum Momentum

Momentum Conservation Momentum Conservation Principle Principle

One of the most powerful laws in One of the most powerful laws in physics is the law of physics is the law of momentummomentum conservation. The law of momentum conservation. The law of momentum conservation can be stated as conservation can be stated as follows.follows.

For a collision occurring between For a collision occurring between object 1 and object 2 in an object 1 and object 2 in an isolated systemisolated system, the total momentum , the total momentum of the two objects before the collision of the two objects before the collision is equal to the total momentum of is equal to the total momentum of the two objects after the collision. the two objects after the collision. That is, the momentum lost by object That is, the momentum lost by object 1 is equal to the momentum gained 1 is equal to the momentum gained by object 2. by object 2.

The above statement tells us that the The above statement tells us that the total momentum of a collection of total momentum of a collection of objects (a system) is conserved" - that is objects (a system) is conserved" - that is the total amount of momentum is a the total amount of momentum is a constant or unchanging value. This law constant or unchanging value. This law of momentum conservation will be the of momentum conservation will be the focus of the remainder of Lesson 2. To focus of the remainder of Lesson 2. To understand the basis of momentum understand the basis of momentum conservation, let's begin with a short conservation, let's begin with a short logical proof. logical proof.

Consider a collision between two Consider a collision between two objects - object 1 and object 2. For objects - object 1 and object 2. For such a collision, the forces acting such a collision, the forces acting between the two objects are equal in between the two objects are equal in magnitude and opposite in direction (magnitude and opposite in direction (Newton's third lawNewton's third law). This statement ). This statement can be expressed in equation form as can be expressed in equation form as follows.follows.

The forces act between the two objects for The forces act between the two objects for a given amount of time. In some cases, a given amount of time. In some cases, the time is long; in other cases the time is the time is long; in other cases the time is short. Regardless of how long the time is, short. Regardless of how long the time is, it can be said that the time that the force it can be said that the time that the force acts upon object 1 is equal to the time that acts upon object 1 is equal to the time that the force acts upon object 2. This is merely the force acts upon object 2. This is merely logical; forces result from interactions (or logical; forces result from interactions (or touching) between two objects. If object 1 touching) between two objects. If object 1 touches object 2 for 0.050 seconds, then touches object 2 for 0.050 seconds, then object 2 must be touching object 1 for the object 2 must be touching object 1 for the same amount of time (0.050 seconds). As same amount of time (0.050 seconds). As an equation, this can be stated asan equation, this can be stated as

Since the forces between the two Since the forces between the two objects are equal in magnitude and objects are equal in magnitude and opposite in direction, and since the opposite in direction, and since the times for which these forces act are times for which these forces act are equal in magnitude, it follows that equal in magnitude, it follows that the the impulsesimpulses experienced by the two experienced by the two objects are also equal in magnitude objects are also equal in magnitude and opposite in direction. As an and opposite in direction. As an equation, this can be stated asequation, this can be stated as

But But the impulse experienced by an object isthe impulse experienced by an object is equal to the change in momentum equal to the change in momentum of that object ( of that object (the the impusleimpusle-momentum change theorem-momentum change theorem). Thus, ). Thus, since each object experiences equal since each object experiences equal and opposite impulses, it follows and opposite impulses, it follows logically that they must also experience logically that they must also experience equal and opposite momentum equal and opposite momentum changes. As an equation, this can be changes. As an equation, this can be stated asstated as

The above equation is one statement The above equation is one statement of the law of momentum of the law of momentum conservation. In a collision, the conservation. In a collision, the momentum change of object 1 is momentum change of object 1 is equal and opposite to the equal and opposite to the momentum change of object 2. That momentum change of object 2. That is, the momentum lost by object 1 is is, the momentum lost by object 1 is equal to the momentum gained by equal to the momentum gained by object 2. object 2.

In a collision between two objects, one In a collision between two objects, one object slows down and loses momentum object slows down and loses momentum while the other object speeds up and gains while the other object speeds up and gains momentum. If object 1 loses 75 units of momentum. If object 1 loses 75 units of momentum, then object 2 gains 75 units of momentum, then object 2 gains 75 units of momentum. Yet, the total momentum of momentum. Yet, the total momentum of the two objects (object 1 plus object 2) is the two objects (object 1 plus object 2) is the same before the collision as it is after the same before the collision as it is after the collision; the total momentum of the the collision; the total momentum of the system (the collection of two objects) is system (the collection of two objects) is conserved. conserved.

A useful analogy for understanding momentum A useful analogy for understanding momentum conservation involves a money transaction between conservation involves a money transaction between two people. Let's refer to the two people as Jack and two people. Let's refer to the two people as Jack and Jill. Suppose that we were to check the pockets of Jill. Suppose that we were to check the pockets of Jack and Jill before and after the money transaction Jack and Jill before and after the money transaction in order to determine the amount of money which in order to determine the amount of money which each possessed. Prior to the transaction, Jack each possessed. Prior to the transaction, Jack possesses $100 and Jill possesses $100. The total possesses $100 and Jill possesses $100. The total amount of money of the two people before the amount of money of the two people before the transaction is $200. During the transaction, Jack transaction is $200. During the transaction, Jack pays Jill $50 for the given item being bought. There pays Jill $50 for the given item being bought. There is a transfer of $50 from Jack's pocket to Jill's pocket. is a transfer of $50 from Jack's pocket to Jill's pocket. Jack has lost $50 and Jill has gained $50. The money Jack has lost $50 and Jill has gained $50. The money lost by Jack is equal to the money gained by Jill. lost by Jack is equal to the money gained by Jill. After the transaction, Jack now has $50 in his pocket After the transaction, Jack now has $50 in his pocket and Jill has $150 in her pocket. Yet, the total amount and Jill has $150 in her pocket. Yet, the total amount of money of the two people after the transaction is of money of the two people after the transaction is $200. The total amount of money (Jack's money plus $200. The total amount of money (Jack's money plus Jill's money) before the transaction is equal to the Jill's money) before the transaction is equal to the total amount of money after the transaction. It could total amount of money after the transaction. It could be said that the total amount of money of the be said that the total amount of money of the system (the collection of two people) is conserved; it system (the collection of two people) is conserved; it is the same before as it is after the transaction.is the same before as it is after the transaction.

A useful means of depicting the A useful means of depicting the transfer and the conservation of transfer and the conservation of money between Jack and Jill is by money between Jack and Jill is by means of a table.means of a table.

The table shows the amount of money The table shows the amount of money possessed by the two individuals possessed by the two individuals before and after the interaction. It also before and after the interaction. It also shows the total amount of money shows the total amount of money before and after the interaction. Note before and after the interaction. Note that the total amount of money ($200) that the total amount of money ($200) is the same before and after the is the same before and after the interaction - it is conserved. Finally, interaction - it is conserved. Finally, the table shows the change in the the table shows the change in the amount of money possessed by the amount of money possessed by the two individuals. Note that the change two individuals. Note that the change in Jack's money account (-$50) is in Jack's money account (-$50) is equal and opposite to the change in equal and opposite to the change in Jill's money account (+$50) .Jill's money account (+$50) .

Virtual Lab Virtual Lab

Truck and Brick Lab Truck and Brick Lab

Collisions between objects are governed Collisions between objects are governed by laws of momentum and energy. When a by laws of momentum and energy. When a collision occurs in an isolated system, the collision occurs in an isolated system, the total momentum of the system of objects total momentum of the system of objects is conserved. Provided that there are no is conserved. Provided that there are no net external forces acting upon the net external forces acting upon the objects, the momentum of all objects objects, the momentum of all objects before the collision equals the momentum before the collision equals the momentum of all objects after the collision. If there are of all objects after the collision. If there are only two objects involved in the collision, only two objects involved in the collision, then the momentum lost by one object then the momentum lost by one object equals the momentum gained by the other equals the momentum gained by the other object object

The animation below portrays the collision The animation below portrays the collision between a 3.0-kg loaded cart and a 2-kg between a 3.0-kg loaded cart and a 2-kg dropped brick. It will be assumed that dropped brick. It will be assumed that there are no net external forces acting there are no net external forces acting upon the two objects involved in the upon the two objects involved in the collision. The only net force acting upon collision. The only net force acting upon the two objects (loaded cart and dropped the two objects (loaded cart and dropped brick) are internal forces - the force of brick) are internal forces - the force of friction between the loaded cart and the friction between the loaded cart and the droped brick. The before- and after-droped brick. The before- and after-collision velocities and momentum are collision velocities and momentum are shown in the data tables.shown in the data tables.

In the collision between the loaded cart and In the collision between the loaded cart and the dropped brick, total system momentum the dropped brick, total system momentum is conserved. Before the collision, the is conserved. Before the collision, the momentum of the loaded cart is 150 kg*cm/s momentum of the loaded cart is 150 kg*cm/s and the momentum of the dropped brick is 0 and the momentum of the dropped brick is 0 kg*cm/s; the total system momentum is 150 kg*cm/s; the total system momentum is 150 kg*cm/s. After the collision, the momentum kg*cm/s. After the collision, the momentum of the loaded cart is 90.0 kg*cm/s and the of the loaded cart is 90.0 kg*cm/s and the momentum of the dropped brick is 60.0 momentum of the dropped brick is 60.0 kg*cm/s; the total system momentum is 150 kg*cm/s; the total system momentum is 150 kg*cm/s. The momentum of the loaded cart-kg*cm/s. The momentum of the loaded cart-dropped brick system is conserved. The dropped brick system is conserved. The momentum lost by the loaded cart (60 momentum lost by the loaded cart (60 kg*cm/s) is gained by the dropped brick kg*cm/s) is gained by the dropped brick

1 kg cart and 2 kg brick 1 kg cart and 2 kg brick

In the collision between the cart and the In the collision between the cart and the dropped brick, total system momentum is dropped brick, total system momentum is conserved. Before the collision, the conserved. Before the collision, the momentum of the cart is 60 kg*cm/s and the momentum of the cart is 60 kg*cm/s and the momentum of the dropped brick is 0 kg*cm/s; momentum of the dropped brick is 0 kg*cm/s; the total system momentum is 60 kg*cm/s. the total system momentum is 60 kg*cm/s. After the collision, the momentum of the cart After the collision, the momentum of the cart is 20.0 kg*cm/s and the momentum of the is 20.0 kg*cm/s and the momentum of the dropped brick is 40.0 kg*cm/s; the total dropped brick is 40.0 kg*cm/s; the total system momentum is 60 kg*cm/s. The system momentum is 60 kg*cm/s. The momentum of the loaded cart-dropped brick momentum of the loaded cart-dropped brick system is conserved. The momentum lost by system is conserved. The momentum lost by the loaded cart (40 kg*cm/s) is gained by the the loaded cart (40 kg*cm/s) is gained by the dropped brick. dropped brick.

For any collision occurring in an For any collision occurring in an isolated systemisolated system, momentum is , momentum is conserved - the total amount of conserved - the total amount of momentum of the collection of objects momentum of the collection of objects in the system is the same before the in the system is the same before the collision as after the collision. This is collision as after the collision. This is the very phenomenon which was the very phenomenon which was observed in "The Cart and The Brick" observed in "The Cart and The Brick" lab. In this lab, a brick at rest was lab. In this lab, a brick at rest was dropped upon a loaded cart which was dropped upon a loaded cart which was in motion.in motion.

Cart and Brick Lab Cart and Brick Lab

Before the collision, the dropped brick had 0 units Before the collision, the dropped brick had 0 units of momentum (it was at rest). The momentum of of momentum (it was at rest). The momentum of the loaded cart can be determined using the the loaded cart can be determined using the velocity (as determined by the ticker tape velocity (as determined by the ticker tape analysis) and the mass. The total amount of analysis) and the mass. The total amount of momentum was the sum of the dropped brick's momentum was the sum of the dropped brick's momentum (0 units) and the loaded cart's momentum (0 units) and the loaded cart's momentum. After the collision, the momenta of momentum. After the collision, the momenta of the two separate objects (dropped brick and the two separate objects (dropped brick and loaded cart) can be determined from their loaded cart) can be determined from their measured mass and their velocity (found from the measured mass and their velocity (found from the ticker tape analysis). If momentum is conserved ticker tape analysis). If momentum is conserved during the collision, then the sum of the dropped during the collision, then the sum of the dropped brick's and loaded cart's momentum after the brick's and loaded cart's momentum after the collision should be the same as before the collision should be the same as before the collision. The momentum lost by the loaded cart collision. The momentum lost by the loaded cart should equal (or approximately equal) the should equal (or approximately equal) the momentum gained by the dropped brick. momentum gained by the dropped brick.

Momentum data for the interaction between Momentum data for the interaction between the dropped brick and the loaded cart could the dropped brick and the loaded cart could be depicted in a table similar to the money be depicted in a table similar to the money

table above.table above.

BeforeCollisionMomentum

AfterCollisionMomentum

Change inMomentum

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 Note that the loaded cart lost 14 units of momentum and the dropped units of momentum and the dropped brick gained 14 units of momentum. brick gained 14 units of momentum. Note also that the total momentum Note also that the total momentum of the system (45 units) was the of the system (45 units) was the same before the collision as it is after same before the collision as it is after the collision.the collision.

Collisions commonly occur in contact sports Collisions commonly occur in contact sports (such as football) and racket and bat sports (such as football) and racket and bat sports (such as baseball, golf, tennis, etc.). Consider (such as baseball, golf, tennis, etc.). Consider a collision in football between a fullback and a a collision in football between a fullback and a linebacker during a goal-line stand. The linebacker during a goal-line stand. The fullback plunges across the goal line and fullback plunges across the goal line and collides in midair with linebacker. The collides in midair with linebacker. The linebacker and fullback hold each other and linebacker and fullback hold each other and travel together after the collision. The fullback travel together after the collision. The fullback possesses a momentum of 100 kg*m/s, East possesses a momentum of 100 kg*m/s, East before the collision and the linebacker before the collision and the linebacker possesses a momentum of 120 kg*m/s, West possesses a momentum of 120 kg*m/s, West before the collision. The total momentum of before the collision. The total momentum of the system before the collision is 20 kg*m/s, the system before the collision is 20 kg*m/s, West (West (review the section on adding vectorsreview the section on adding vectors if if necessary). necessary).

Therefore, the total momentum of the Therefore, the total momentum of the system after the collision must also be 20 system after the collision must also be 20 kg*m/s, West. The fullback and the kg*m/s, West. The fullback and the linebacker move together as a single unit linebacker move together as a single unit after the collision with a combined after the collision with a combined momentum of 20 kg*m/s. Momentum is momentum of 20 kg*m/s. Momentum is conserved in the collision. A conserved in the collision. A vector diagramvector diagram can be used to represent this principle of can be used to represent this principle of momentum conservation; such a diagram momentum conservation; such a diagram uses an arrow to represent the magnitude uses an arrow to represent the magnitude and direction of the momentum vector for and direction of the momentum vector for the individual objects before the collision the individual objects before the collision and the combined momentum after the and the combined momentum after the collision. collision.

Now suppose that a medicine ball is thrown to a Now suppose that a medicine ball is thrown to a clown who is at rest upon the ice; the clown clown who is at rest upon the ice; the clown catches the medicine ball and glides together catches the medicine ball and glides together with the ball across the ice. The momentum of with the ball across the ice. The momentum of the medicine ball is 80 kg*m/s before the the medicine ball is 80 kg*m/s before the collision. The momentum of the clown is 0 m/s collision. The momentum of the clown is 0 m/s before the collision. The total momentum of the before the collision. The total momentum of the system before the collision is 80 kg*m/s. system before the collision is 80 kg*m/s. Therefore, the total momentum of the system Therefore, the total momentum of the system after the collision must also be 80 kg*m/s. The after the collision must also be 80 kg*m/s. The clown and the medicine ball move together as a clown and the medicine ball move together as a single unit after the collision with a combined single unit after the collision with a combined momentum of 80 kg*m/s. Momentum is momentum of 80 kg*m/s. Momentum is conserved in the collision.conserved in the collision.

Momentum is conserved for any Momentum is conserved for any interaction between two objects interaction between two objects occurring in an isolated system. This occurring in an isolated system. This conservation of momentum can be conservation of momentum can be observed by a total system observed by a total system momentum analysis and by a momentum analysis and by a momentum change analysis. Useful momentum change analysis. Useful means of representing such analyses means of representing such analyses include a momentum table and a include a momentum table and a vector diagram. Later in Lesson 2, we vector diagram. Later in Lesson 2, we will use the momentum conservation will use the momentum conservation principle to solve problems in which principle to solve problems in which the after-collision velocity of objects is the after-collision velocity of objects is predicted.predicted.

Examples Examples

1. Explain why it is difficult for a 1. Explain why it is difficult for a firefighter to hold a hose which ejects firefighter to hold a hose which ejects large amounts of high-speed water.large amounts of high-speed water.

Answer 1 Answer 1

The hose is pushing lots of water The hose is pushing lots of water (large mass) forward at a high speed. (large mass) forward at a high speed. This means that the water has a This means that the water has a huge forward momentum. In turn, huge forward momentum. In turn, the hose must have an equally large the hose must have an equally large backwards momentum, making it backwards momentum, making it difficult for firefighters to manage. difficult for firefighters to manage.

2. A large truck and a Volkswagen have a 2. A large truck and a Volkswagen have a head-on collision.head-on collision.

a. Which vehicle experiences the greatest a. Which vehicle experiences the greatest force of impact?force of impact?

b. Which vehicle experiences the greatest b. Which vehicle experiences the greatest impulse?impulse?

c. Which vehicle experiences the greatest c. Which vehicle experiences the greatest momentum change?momentum change?

d. Which vehicle experiences the greatest d. Which vehicle experiences the greatest acceleration?acceleration?

Answer 2 Answer 2

A , b, c, same answer for all A , b, c, same answer for all Both the Volkswagen and the large truck Both the Volkswagen and the large truck

encounter the same force, the same encounter the same force, the same impulse, and the same momentum change. impulse, and the same momentum change.

D. Acceleration is greatest for the D. Acceleration is greatest for the Volkswagen. While the two vehicles Volkswagen. While the two vehicles experience the same force, the experience the same force, the acceleration is greatest for the Volkswagen acceleration is greatest for the Volkswagen which has the smaller mass. If you find this which has the smaller mass. If you find this hard believe then read the next question hard believe then read the next question and its accompanying explanation. and its accompanying explanation.

3. Miles Tugo and Ben Travlun are riding in 3. Miles Tugo and Ben Travlun are riding in a bus at highway speed on a nice summer a bus at highway speed on a nice summer day when an unlucky bug splatters onto day when an unlucky bug splatters onto the windshield. Miles and Ben begin the windshield. Miles and Ben begin discussing the physics of the situation. discussing the physics of the situation. Miles suggests that the momentum change Miles suggests that the momentum change of the bug is much greater than that of the of the bug is much greater than that of the bus. After all, argues Miles, there was no bus. After all, argues Miles, there was no noticeable change in the speed of the bus noticeable change in the speed of the bus compared to the obvious change in the compared to the obvious change in the speed of the bug. Ben disagrees entirely, speed of the bug. Ben disagrees entirely, arguing that that both bug and bus arguing that that both bug and bus encounter the same force, momentum encounter the same force, momentum change, and impulse. Who do you agree change, and impulse. Who do you agree with? Support your answer with? Support your answer

Answer 3 Answer 3 Ben Travelun is correct. The bug and bus Ben Travelun is correct. The bug and bus

experience the same force, same impulse and experience the same force, same impulse and the same momentum change. This is contrary the same momentum change. This is contrary to popular (but false) belief which matches to popular (but false) belief which matches Miles’ statement. The bug has less mass and Miles’ statement. The bug has less mass and therefore more acceleration; occupants of the therefore more acceleration; occupants of the very massive bus do not feel the extremely very massive bus do not feel the extremely small acceleration. Furthermore, the bug is small acceleration. Furthermore, the bug is composed of a less hardy material and thus composed of a less hardy material and thus splatters all over the windshield. Yet the splatters all over the windshield. Yet the greater splatterability of the bug and the greater splatterability of the bug and the greater acceleration do not mean that the greater acceleration do not mean that the bug has a greater force, impulse, or bug has a greater force, impulse, or momentum change. momentum change.

4. If a ball is projected upward from 4. If a ball is projected upward from the ground with ten units of the ground with ten units of momentum, what is the momentum momentum, what is the momentum of recoil of the Earth? ____________ Do of recoil of the Earth? ____________ Do we feel this? Explain. we feel this? Explain.

Answer 4Answer 4

The earth recoils with 10 units of The earth recoils with 10 units of momentum. This is not felt by Earth’s momentum. This is not felt by Earth’s occupants. Since the mass of the occupants. Since the mass of the Earth is extremely large, the recoil Earth is extremely large, the recoil velocity of the Earth is extremely velocity of the Earth is extremely small and therefore not felt small and therefore not felt

5. If a 5-kg bowling ball is projected 5. If a 5-kg bowling ball is projected upward with a velocity of 2.0 m/s, upward with a velocity of 2.0 m/s, then what is the recoil velocity of the then what is the recoil velocity of the Earth (mass = 6.0 x 10^24 kg). Earth (mass = 6.0 x 10^24 kg).

Answer 5 Answer 5

Since the ball has an upward Since the ball has an upward momentum of 10 kg m/s, the Earth momentum of 10 kg m/s, the Earth must have a downward momentum must have a downward momentum of 10 kg m/s. To find the velocity of of 10 kg m/s. To find the velocity of the Earth, use the momentum the Earth, use the momentum equation p = m * v. This equation equation p = m * v. This equation rearranges to v = p/m. By rearranges to v = p/m. By substituting into this equation substituting into this equation

v = (10 kg m/s) / (6 x 10 v = (10 kg m/s) / (6 x 10 2424 kg). kg). V = 1.67 * 10 V = 1.67 * 10 -24-24 m/s (downward) m/s (downward)

6. A 120 kg lineman moving west at 2 6. A 120 kg lineman moving west at 2 m/s tackles an 80 kg football fullback m/s tackles an 80 kg football fullback moving east at 8 m/s. After the moving east at 8 m/s. After the collision, both players move east at 2 collision, both players move east at 2 m/s. Draw a vector diagram in which m/s. Draw a vector diagram in which the before- and after-collision the before- and after-collision momenta of each player is represented momenta of each player is represented by a momentum vector. Label the by a momentum vector. Label the magnitude of each momentum vector. magnitude of each momentum vector.

Answer 6 Answer 6

7. Would you care to fire a rifle that 7. Would you care to fire a rifle that has a bullet ten times as massive as has a bullet ten times as massive as the rifle? Explain.the rifle? Explain.

Answer 7 Answer 7

Absolutely not! In a situation like Absolutely not! In a situation like this, the target would be a safer this, the target would be a safer place to stand than the rifle. The rifle place to stand than the rifle. The rifle would have recoil velocity that is ten would have recoil velocity that is ten times larger than the bullet’s times larger than the bullet’s velocity. This would produce the velocity. This would produce the effect of “the rifle actually being the effect of “the rifle actually being the bullet.” bullet.”

8. A baseball player holds a bat 8. A baseball player holds a bat loosely and bunts a ball. Express loosely and bunts a ball. Express your understanding of momentum your understanding of momentum conservation by filling in the tables conservation by filling in the tables below. below.

Answer 8 Answer 8

A) + 40 (add momentum of ball and A) + 40 (add momentum of ball and bat)bat)

C) + 40 (momentum must be C) + 40 (momentum must be conserved) conserved)

B) + 30 (the bat must have 30 units B) + 30 (the bat must have 30 units of momentum in order in order for of momentum in order in order for the total to be +40)the total to be +40)

9. A Tomahawk cruise missile is 9. A Tomahawk cruise missile is launched from the barrel of a mobile launched from the barrel of a mobile missile launcher. Neglect friction. missile launcher. Neglect friction. Express your understanding of Express your understanding of momentum conservation by filling in momentum conservation by filling in the tables below. the tables below.

Answer 9 Answer 9

A) 0 (add the momentum of the A) 0 (add the momentum of the missile and launcher)missile and launcher)

C) 0 (the momentum is the same C) 0 (the momentum is the same after as it is before the collision) after as it is before the collision)

B) -5000 (the launcher must have -B) -5000 (the launcher must have -5000 units of momentum in order for 5000 units of momentum in order for the total to be zero)the total to be zero)