# Elastic Collision Tutorial

__Key concepts:__

**Elastic collision**

**Conservation of Momentum**

**Conservation of Energy**

From billiard balls to atoms in the air, elastic collisions are found all over the place. Let us try to figure out how they work!

**Question:**

"A 2 kg ball is moving with 5m/s. It collides elastically with another ball that is stationary and has a mass of 20 kg. Determine the speeds of the two balls after the collision." To get full marks, we need the following:

A sketch/cartoon showing the two balls before, during and after the collision.

The total momentum before the collision.

The total momentum after the collision.

The total kinetic energy before the collision

The total kinetic energy after the collision.

Let's get started with 1.

A sketch/cartoon showing the two balls before, during and after the collision.

To make the drawing correctly we have to remember what an "elastic collision" means. In short:

__A collision is " elastic" if the objects simply bounce off of each other when they touch. The "kinetic energy" is CONSERVED! __

An everyday example could be a soccer ball hitting your forehead.

The other case is that of "inelastic" collisions:

__A collision is " inelastic" if the objects fuse together when they crash into each other. The "kinetic energy" is NOT CONSERVED! __

An everyday example could be a soft snowball colliding with your face.

With this in mind, it is easy to make the drawing correctly.

*Before:*

*During:*

*After:*

**TIP: Always make a drawing when solving any physics problem. It makes it easier to figure out what to do in the beginning and easier to spot mistakes in your calculations! **

In this problem we have defined the symbols:

They represent the mass of the first ball, the mass of the second ball and the speed of the first ball respectively. Also, we have let

define the final speeds of the two balls. These are the two quantities we want to find.

Let us get ready to solve part 2.

2. The total momentum before the collision.

The momentum before the collision can be calculated from:

Think of "momentum" as meaning "the total amount of stuff that is moving". Let's look at the "before" drawing. Clearly only the mass **m1** is moving at the speed **v1**. The other mass, **m2** is stationary, so

Great! Now we can tackle 3.

3. The total momentum after the collision.

Here, we can use something called "** Conservation of Momentum**". This principle basically says that "The amount of stuff moving around" in the beginning is the same as "The amount of stuff moving around" in the end. Mathematically, we say that:

This is a good start!

In the equation above, we only know the masses and the initial speed. We need to find both of the final speeds. To do this, we need to think about 4.

4. The total kinetic energy before the collision

Remember that kinetic energy of a single object is defined as:

So the total kinetic energy before the collision must be

Of course the second mass has no energy, because it is not moving initially. By plugging in the numbers we have, we find: