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 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) 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 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
goal-line stand. The fullback plunges across the goal
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.
Theabove 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
Theabove 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) 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 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.
| | Before Collision 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 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 goal
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.
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 الدينالأحد 12 فبراير 2012, 9:24 pm