Conservation of Momentum

One of the most powerful laws in physics is the law of

momentum conservation. The law of

momentum conservation can be stated as follows.

The

above statement tells us that the total momentum of a

collection of objects (a

- 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 Lesson 2. 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 contact)

between two objects. If object 1 contacts object 2 for 0.050

seconds, then object 2 must be contacting 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

impulse-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 to and opposite of the momentum change of object

2. That is, the momentum lost by object 1 is equal to the

momentum gained by object 2. In most collisions 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 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 that

each possesses. 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

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 to and opposite of 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. A common physics lab involves the

dropping of a brick upon a cart in motion.

The dropped brick is at rest and begins

with zero momentum. The loaded cart (a cart with a brick on

it) is in motion with considerable momentum. The actual

momentum of the loaded cart can be determined using the

velocity (often determined by a ticker tape analysis) and

the mass. The total amount of momentum is 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

(often found from a 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.

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 was 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

line and collides in midair with the 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 or by a momentum change analysis. Useful

means of representing such analyses include a momentum table

and a vector diagram. Later in Lesson 2, we will use the

momentum conservation principle to solve problems in which

the after-collision velocity of objects is predicted.

Using motion detectors and carts on a low-friction track, one can

collect data to demonstrate the law of conservation of momentum. The

video below demonstrates the process.

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 Lesson 2. 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 contact)

between two objects. If object 1 contacts object 2 for 0.050

seconds, then object 2 must be contacting 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

impulse-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 to and opposite of the momentum change of object

2. That is, the momentum lost by object 1 is equal to the

momentum gained by object 2. In most collisions 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 that

each possesses. 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) isconserved. 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 to and opposite of 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. A common physics lab involves the

dropping of a brick upon a cart in motion.

The dropped brick is at rest and begins

with zero momentum. The loaded cart (a cart with a brick on

it) is in motion with considerable momentum. The actual

momentum of the loaded cart can be determined using the

velocity (often determined by a ticker tape analysis) and

the mass. The total amount of momentum is 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

(often found from a 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.

| BeforeCollisionMomentum | AfterCollisionMomentum | Changein Momentum |

DroppedBrick | 0 units | 14 units | +14 units |

LoadedCart | 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 was 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 goalline and collides in midair with the 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 or by a momentum change analysis. Useful

means of representing such analyses include a momentum table

and a vector diagram. Later in Lesson 2, we will use the

momentum conservation principle to solve problems in which

the after-collision velocity of objects is predicted.

**Watc It!**Using motion detectors and carts on a low-friction track, one can

collect data to demonstrate the law of conservation of momentum. The

video below demonstrates the process.

سامح محى الدين12/2/2012, 9:24 pm