# Newton's Law of Restitution: Definition & Use

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Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

What is Newton's law of restitution and how is it used? Discover what the velocities before and after a collision tell you about that collision. See what problems you can solve by taking a quiz.

## What Is Newton's Law of Restitution?

When two things hit each other, we have a lot of wonderful mathematical equations to make sense of the situation. And many of those laws were derived by Sir Isaac Newton. Newton's law of restitution says that when two objects collide, their speeds after the collision depend on the material from which they are made.

This measure of the nature of the materials colliding is represented by a number called the coefficient of restitution. This coefficient is equal to the relative speed between the two objects after the collision, divided by the relative speed between the two objects before the collision:

This is a useful measure, because it tells you something about the 'bounciness,' or elasticity, of the collision. A collision where no overall kinetic (movement) energy is lost is called a perfectly elastic collision, and these are the most bouncy. They have a coefficient of restitution of e = 1. A collision where the maximum kinetic (movement) energy is lost is called a perfectly inelastic collision, and this is where the two objects stick together and move together after the collision. They have a coefficient of restitution of e = 0. And a number in between is... in between. It's an inelastic collision where some kinetic energy is lost. Most real-life collisions are in between.

With this equation, you always take the smaller number away from the larger, so it never comes out as a negative number. The coefficient of restitution is always positive.

## Collision Between Two Moving Objects

When two objects collide, we can use the equation previously mentioned to calculate the coefficient of restitution. Or if you're given the coefficient of restitution, you can use it to find a missing velocity.

For example, let's say one car rear-ends another on a nearby highway. Car A was going at 70 miles per hour, and car B was going at 50 miles per hour. After the collision, Car A is going 40 miles per hour in the same direction, and car B is going 60 miles per hour. What is the coefficient of restitution between the two cars?

Well, we need to figure out the relative speed between the two cars. That's just the difference of the two velocities. Knowing that leads to an expanded equation:

The two cars are going in the same direction, so both have positive velocities. Let's plug the numbers into the equation:

So the coefficient of restitution is 1. That means that it is an elastic collision.

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