How the Principle of Conservation of Momentum Works

In physics, the principle of conservation of momentum states that when you have an isolated system with no external forces, the initial total momentum of objects before a collision equals the final total momentum of the objects after the collision. In other words,

You may have a hard time dealing with the physics of impulses because of the short times and the irregular forces. But with the principle of conservation, items that are hard to measure — for example, the force and time involved in an impulse — are out of the equation altogether. This principle is simple but extremely useful.

You can derive the principle of conservation of momentum from Newton’s laws, what you know about impulse, and a little algebra.

Say that two careless space pilots are zooming toward the scene of an interplanetary crime. In their eagerness to get to the scene first, they collide. During the collision, the average force the second ship exerts on the first ship is F₁₂. Applying the impulse-momentum theorem to
the first ship gives you:

And if the average force exerted on the second ship by the first ship is F₂₁, you also know that

Now you add these two equations together, which gives you the resulting equation:

Distribute the mass terms and rearrange the terms on the right until you get the following:

This is an interesting result, because m₁v_i1 + m₂v_i2 is the initial total momentum of the two rocket ships (p_1i + p_i2) and m₁v_f1 + m₂v_f2 is the final total momentum (p_1f + p_2f) of the two rocket ships. Therefore, you can write this equation as follows:

If you write the initial total momentum as p_f and the final total momentum as p_i, the equation becomes

Where do you go from here? Both terms on the left include

so you can rewrite

as the sum of the forces involved,

multiplied by the change in time:

If you’re working with what’s called an isolated or closed system, you have no external forces to deal with. Such is the case in space. If two rocket ships collide in space, there are no external forces that matter, so by Newton’s third law,

In other words, when you have a closed system, you get the following:

This converts to

p_f = p_i

The equation p_f = p_i says that when you have an isolated system with no external forces, the initial total momentum before a collision equals the final total momentum after a collision, giving you the principle of conservation of momentum.