Guide 91a. Solving Conservation of
Momentum Problems in One
Dimension
The law of conservation of momentum is the second of the
major conservation laws that we'll use this year. The law in its most
general form is written in a way that looks similar to the law of
conservation of energy.
In all of the problems that you will do with conservation of
momentum in this course, you will be able to define the system in such a way
that
.
In that case, the law of conservation of momentum can be written as follows.
We will simplify the notation by letting upper case P
represent the net momentum of the system. Therefore, we have for an isolated
system the following.
In two dimensions, the vector equation becomes two component
equations:
P_{ix}
= P_{fx}
P_{iy} = P_{f}_{y}
For 1dimensional problems, we omit the x or y subscript.
The general plan of 1dimensional conservation of
momentum problem is similar to that of a conservation of energy problem with
these exceptions:

One selects the system so
that the net, external force on the system is 0.
 Momentum is treated as a vector. Therefore, direction
matters.

Now for some examples.
Example 1:
Two ice skaters push off against one another starting from a stationary
position. The 45kg skater acquires a velocity of 0.375 m/s to the right.
What velocity does the 60kg skater acquire?
Given:
v_{1i} = v_{2i} = 0
v_{2f} = +0.375 m/s
m_{1} = 60 kg
m_{2} = 45 kg
Goal:
Find v_{1f} System: ice skaters,
forces that skaters exert on each other
External forces: friction (assumed to be
negligible), normal and weight add to 0
Initial state: skaters motionless ready to push apart
Final state: skaters moving in opposite directions 

The skaters exert
internal forces on each other, but the external forces of normal and
weight balance. We're assuming no friction. Actually, if we
look at the situation immediately after they push off and haven't
separated far, then any influence due to friction will be minimal.
Therefore, we've selected the system so that F_{net,ext}
= 0.
In the diagram, note that:
 +x is to the right.
 The skaters are denoted 1 and 2 to
distinguish them.
 The initial and final velocities
have double subscripts. One subscript is for the object and the
other is for the state. (The textbook inserts a comma between
the subscripts.)

Solution: 

We use uppercase P to
denote the total momentum of the system and lowercase p to denote the
momentum of a single object. The total momentum is the sum of the
individual momenta of the objects in the system. 
The total initial
momentum of the system is 0, because the skaters are motionless. 
Substitute the definition
of momentum
. 
Solve for the unknown. 
Substitute values and
units and reduce. The sign of the result is negative, indicating
motion to the left as expected. The magnitude of the velocity of
skater 1 is less than skater 2. This is also expected, because
skater 1 has the greater mass. 
Example 2: A 0.60kg
glider traveling at 8.0 m/s on a level air track undergoes a headon
collision with a 0.20kg mass traveling toward it at 4.0 m/s. The two
gliders stick in the collision. What is the velocity of the combined
gliders after the collision?
Given:
v_{1i} = +8.0 m/s
v_{2i} = 4.0 m/s
m_{1} = 0.60 kg
m_{2} = 0.20 kg
Goal:
Find v_{f} System: gliders,
forces that gliders exert on each other
External forces: friction with track (assumed to be negligible),
normal and weight add to 0
Initial state: gliders moving toward each other before
collision
Final state: gliders moving together after collision 

We have to be careful to
assign the correct signs to the initial velocities.
There is only one final velocity, so a
single subscript is all that is needed on that velocity.
The direction of v_{f} isn't
given, but we're guessing it's to the right. The sign of the final
result will tell us whether that choice is correct. 
Solution: 

We proceed as in the last
example setting the total initial and final momenta equal to each other
and then writing each as the sum of the momenta of the objects in the
system. 
The mass of the combined
gliders is m_{1} + m_{2}. They have the same final
velocity. 
Solve for the unknown. 
Substitute values and
units and reduce. The sign of the result is positive, indicating
motion to the right as we had guessed. 
Example 3. A glider with mass m
and speed v moving along an air track collides with a stationary cart
with a mass m/3. After the collision the first cart has a speed v/2.
What is the velocity of the second cart?
Given:
v_{1i} = v
v_{2i} = 0
v_{1f }= v/2
m_{1} =m
m_{2} = m/3
Goal:
Find v_{2f} System:
gliders, forces that gliders exert on each other
External forces: friction with track (assumed to
be negligible), normal and weight add to 0 Initial state: gliders before collision
Final state: gliders after collision 

In order to solve this problem, we
need to be given both masses and 3 velocities. That leaves the 4th
velocity as the only unknown. Later, we'll see how to solve
problems like this when both of the final velocities are unknown. 
Solution: 

After the collision, both
gliders are moving to the right. Glider 1 is slower than glider 2.
Therefore, the gliders separate after the collision. 
Go on to read about
elastic and
inelastic collisions.
