


Conservation of Momentum
If we combine Newton’s Third Law with what we know about
impulse, we can derive the important and extremely useful law of
conservation of momentum.
Newton’s Third Law tells us that, to every action, there
is an equal and opposite reaction. If object A exerts
a force F on object B,
then object B exerts a force –F on
object A. The net force exerted between
objects A and B is
zero.
The impulse equation, , tells us that if the
net force acting on a system is zero, then the impulse, and hence
the change in momentum, is zero. Because the net force between the
objects A and B that
we discussed above is zero, the momentum of the system consisting
of objects A and B does
not change.
Suppose object A is a cue
ball and object B is an eight ball
on a pool table. If the cue ball strikes the eight ball, the cue
ball exerts a force on the eight ball that sends it rolling toward the
pocket. At the same time, the eight ball exerts an equal and opposite
force on the cue ball that brings it to a stop. Note that both the
cue ball and the eight ball each experience a change in momentum.
However, the sum of the momentum of the cue ball and the momentum
of the eight ball remains constant throughout. While the initial
momentum of the cue ball, , is not the same as its
final momentum, , and the initial momentum
of the eight ball, , is not the same as its
final momentum, , the initial momentum
of the two balls combined is equal to the final momentum of the
two balls combined:
The conservation of momentum only applies to systems that
have no external forces acting upon them. We call such a system
a closed or isolated system: objects within the system may
exert forces on other objects within the system (e.g., the cue ball
can exert a force on the eight ball and vice versa), but no force
can be exerted between an object outside the system and an object
within the system. As a result, conservation of momentum does not apply
to systems where friction is a factor.
Conservation of Momentum on SAT II Physics
The conservation of momentum may be tested both quantitatively
and qualitatively on SAT II Physics. It is quite possible, for instance,
that SAT II Physics will contain a question or two that involves
a calculation based on the law of conservation of momentum. In such a
question, “conservation of momentum” will not be mentioned explicitly,
and even “momentum” might not be mentioned. Most likely, you will
be asked to calculate the velocity of a moving object after a collision
of some sort, a calculation that demands that you apply the law
of conservation of momentum.
Alternately, you may be asked a question that simply demands
that you identify the law of conservation of momentum and know how
it is applied. The first example we will look at is of this qualitative
type, and the second example is of a quantitative conservation of momentum
question.
Example 1

Although the title of the section probably gave the solution
away, we phrase the problem in this way because you’ll find questions
of this sort quite a lot on SAT II Physics. You can tell a question
will rely on the law of conservation of momentum for its solution
if you are given the initial velocity of an object and are asked
to determine its final velocity after a change in mass or a collision
with another object.
Some Supplemental Calculations
But how would we use conservation of momentum to find
the speed of the toy truck after the apple has landed?
First, note that the net force acting in the x direction
upon the apple and the toy truck is zero. Consequently, linear momentum
in the x direction
is conserved. The initial momentum of the system in the x direction
is the momentum of the toy truck,.
Once the apple is in the truck, both the apple and the
truck are traveling at the same speed, . Therefore, . Equating and , we find:
As we might expect, the final velocity of the toy truck
is less than its initial velocity. As the toy truck gains the apple
as cargo, its mass increases and it slows down. Because momentum
is conserved and is directly proportional to mass and velocity,
any increase in mass must be accompanied by a corresponding decrease
in velocity.
Example 2

Questions involving firearms recoil are a common way in
which SAT II Physics may test your knowledge of conservation of
momentum. Before we dive into the math, let’s get a clear picture
of what’s going on here. Initially the cannon and cannonball are
at rest, so the total momentum of the system is zero. No external
forces act on the system in the horizontal direction, so the system’s
linear momentum in this direction is constant. Therefore the momentum
of the system both before and after the cannon fires must be zero.
Now let’s make some calculations. When the cannon is fired,
the cannonball shoots forward with momentum (10 kg)(100 m/s)
= 1000 kg · m/s. To keep the total momentum of the
system at zero, the cannon must then recoil with an equal momentum:
Any time a gun, cannon, or an artillery piece releases
a projectile, it experiences a “kick” and moves in the opposite
direction of the projectile. The more massive the firearm, the slower
it moves.
