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Conservation of Linear Momentum: Formula and Examples

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  • 0:01 Momentum Requires…
  • 1:18 Momentum Is Conserved…
  • 4:32 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

The law of conservation of momentum tells us that the amount of momentum for a system doesn't change. In this lesson, we'll explore how that can be true even when the momenta of the individual components does change.

Momentum Requires External Forces

Even objects at rest are technically 'moving,' but you don't see it because the movement is on a microscopic scale. For example, your computer is a solid piece of equipment, and it may be just sitting there on your table, but all of the molecules that make up the materials of your computer are vibrating around inside. The same is true for your shoes, your coffee cup, a tennis ball, and any other solid object.

The reason these vibrations, or forces, don't cause the object to move is because they are internal forces, and an external force is needed to get an object moving. When you pick up your computer, you cause it to move, just like a tennis racket moves the tennis ball when the two collide.

We look to Newton's third law of motion to explain why this is true. This law states that whenever an object exerts a force on another object, the second object exerts an equal and opposite force on the first. In other words, the forces cancel each other out.

It would be like you sitting inside your car and trying to get it moving by pushing on the dashboard. You're inside the car, so you and the car are part of the same system. You have to push from the outside of the car to get it to move, just like the tennis racket has to hit the ball from the outside to send it flying.

Momentum Is Conserved in a System

Momentum is mass in motion, and we can apply our understanding of outside forces here as well. Momentum can only occur when there is an outside force or impulse, not from within the system itself. This important concept is called the law of conservation of momentum. It describes how when there are no external forces, the momentum of a system doesn't change.

In equation form, momentum = mass * velocity. To increase the momentum of an object, you need to increase its mass, its velocity, or both. This also means that different objects can have the same momentum. Say, for example, that one object is twice as massive as the other, but the second object has twice the velocity. For both objects, the product of mass and velocity is the same, so the momentum for both objects is the same.

When this happens to two objects that are part of the same system, there is no net momentum, so we say that it is conserved. This means that there is no change in the overall quantity. If the momenta of the objects are the same magnitude but opposite in direction, there will be no change in the net momentum of the system. They cancel each other out, just like internal forces.

Let's look at an example to see how this works. Though they are two separate objects, a cannonball and a cannon are actually part of the same system. If it helps, you can think of the cannon as your car and the cannonball as you sitting inside it. As the cannonball is fired, it travels out of the cannon, while the cannon recoils a bit in response in the opposite direction. The velocity of the cannonball is greater, and we see this because the cannonball travels very fast out of the cannon. The cannon moves only a little bit, so there is small velocity, but the mass of the cannon is much greater than the cannonball.

In this situation, the momentum of each ends up being the same because the product of mass and velocity of each is the same. And, because the momentum of each occurs in opposite directions, they cancel each other out, meaning for the overall cannon-cannonball system, there is no net change in momentum. The momentum before the firing of the cannonball is zero because neither the cannon nor the cannonball is moving. However, the momentum after the firing is also zero because the momentum of the cannon is equal and opposite to the momentum of the cannonball!

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