Here's something that conservation of momentum problems have in common with conservation of energy problems.  You have to decide on a system and identify the initial and final states.  Here are some differences for conservation of momentum problems.

• Select a system for which the net external force on the system is 0.  That's the condition required for momentum to be conserved.  We say in that case that the system is isolated.
• Momentum is a vector; energy isn't.  Therefore, when solving a conservation of momentum problem, you have to write a conservation equation for each axis and use momentum components.

Now for some examples of 1-dimensional problems.

Example 1:  Two ice skaters push off against one another starting from a stationary position. The 45-kg skater acquires a velocity of 0.375 m/s to the right. What velocity does the 60-kg skater acquire?

Given: v1i = v2i = 0 v2f = +0.375 m/s m1 = 60 kg m2 = 45 kg Goal: Find v1f System:  ice skaters, forces that skaters exert on each other External forces: friction (assumed to be negligible) 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. 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, but that's an unnecessary symbol.)
Solution:
	We use upper-case P to denote the total momentum of the system and lower-case 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 0.60-kg glider traveling at 8.0 m/s on a level air track undergoes a head-on collision with a 0.20-kg 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: v1i = +8.0 m/s v2i = -4.0 m/s m1 = 0.60 kg m2 = 0.20 kg Goal: Find vf System:  gliders, forces that gliders exert on each other External forces: friction with track (assumed to be negligible) 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 vf 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 m1 + m2.  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: v1i = v v2i = 0 v1f  = v/2 m1 =m m2 = m/3 Goal: Find v2f System:  gliders, forces that gliders exert on each other External forces: friction with track (assumed to be negligible) 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.

Elastic and Inelastic Collisions

It's important to understand the conditions under which particular conservation laws can be used.  You already know the following conditions.

1.  The energy conservation law W_ext = DE_sys applies in general. When W_ext = 0, total mechanical energy (K + U) is conserved.

2.  Momentum is conserved when the net, external force on the system is zero.

There are other conservation laws less general than the ones above.  There are some situations, for example, when kinetic energy is conserved.  However, it's much more likely that kinetic energy isn't conserved.  Let's start with some example situations when it isn't conserved.

recoiling objects--Example 1 of the ice skaters is such a case.  In the initial state, the total kinetic energy of the system is 0, since the skaters aren't moving.  In the final state, both have kinetic energy.  Since kinetic energy is always positive, the initial K of 0 can never equal a positive final K.

sticking collisions--See Example 2 of the coupling gliders.  Let's check this problem for conservation of kinetic energy.

Kinetic energy is obviously lost in this collision.  This is generally true in collisions where the objects stick or are permanently deformed.  You might ask what happened to the missing energy, since it apparently didn't go into potential energy.  Non-conservative forces are usually involved in these situations, and that results in conversion of kinetic energy to thermal energy.

Interactions in which kinetic energy isn't conserved are termed inelastic.  It's important to realize that whether or not a collision is inelastic has nothing to do with whether momentum is conserved.  In both Examples 1 and 2, we used the law of conservation of momentum.  Yet both collisions were inelastic.

There are some interactions in which both momentum and kinetic energy are conserved.  An interaction in which kinetic energy is conserved is termed elastic.  The only truly elastic collisions are those between elementary particles, and not all of those are elastic.  In the macroscopic world, we can sometimes create situations that approximate those of elastic interactions.  On an air track or air table, for example, we can arrange to eliminate as much friction as possible.  Nevertheless, there will still be some thermal energy losses in the bumpers of the colliding gliders.