Conservation of Momentum

One of the basic laws of Physics is the law of conservation of momentum. This states:

For an isolated system, the total momentum of all objects
interacting with each other remains constant regardless of the forces that
interact between the objects.

Or more simply stated,

The total momentum in a system before an event must equal
the total momentum in the system after the event.

Mathematically, we write this as


For example, if an object moving strikes an object at rest, the object at rest will gain momentum and the object moving will lose momentum. The total momentum in the system will remain the same, and the second object gains the same amount of momentum that the first object lost. We can use several examples that you might be familiar with to demonstrate this.

  • A person standing in a rowboat next to a dock steps off onto the dock. As the person moves forward, the boat will begin to move in the opposite direction. The initial momentum in the system was zero (both boat and person at rest) so the sum of the mass times velocity of the person plus the mass times velocity of the boat after must also equal zero.
  • An astronaut at rest in space throws a wrench away. The astronaut will begin to move away from the wrench with the same momentum that the wrench has, but in the opposite direction (remember, momentum is a vector with magnitude and direction.)
  • Two ice skaters at rest push against each other and begin to move backwards. Since the initial momentum of the two was zero, the sum of their final momentums must also be zero. In other words, if one skater weighs twice the weight of the other skater, the heavier skater will move with only half the speed of the other such that both have the same momentum in opposite directions and the total momentum is still zero.

We'll do a lot of examples in class!

This would be a good time to talk about a little nomenclature that we will be using. Our basic symbols are:

p – momentum

m – mass

v - velocity

Using subscripts allows us to denote which object we are talking about. For example, given two objects with two different masses or velocities, we would refer to those as m1 and m2 or v1 and v2. Finally, we will use ‘ as a superscript to denote what happens after an event takes place. For example, an objects momentum before an event takes place can be expressed as pbef or m1v1.  After an event takes place, we would express its momentum as paft or p’ or m1v1’.

Often, it is useful to see what someone else has to say on the topic; Check out these
lessons from "The Physics Classroom" (Glenbrook High School). Each
contains some practice problems at the end of the lesson.

Newton's Third Law

Momentum Conservation

Isolated Systems

For  Practice Problems,

From the University of Oregon (it's gross but fun!):

Also, try: Giancoli Multiple Choice Practice Questions (Select "Practice Problems) and try some.