# Linear collisions

What do we call a collision in which the total kinetic energy is the same, both before and after the collision?

## Linear collisions

Two carriages, about to collide... We know that the carriages each have a momentum, p, that we calculate by multiplying the mass and the velocity. And here's something else important. The carriages also have energy due to their motion, not to be confused with their momentum. These are different things.

Energy due to motion - kinetic energy - is calculated with this formula. Energy: 'E' equals mass: 'm' times velocity: 'v' to-the-power-of-two divided by 'two' The letter E has a small subscript letter, k, for kinetic. This shows that we mean kinetic energy. The unit for energy is joules. Momentum and kinetic energy are different things, but both are important when objects collide.

Momentum is conserved when two objects collide. What about kinetic energy. Is that conserved too? In fact this varies. Let's look at a few examples.

Here are two carriages, equal in size and mass. They both weigh zero-point-five kilograms. Oh. Here comes a collision. Let's pause before that happens.

One of the carriages is moving with a velocity of four meters per second. And for the kinetic energy, let's look at the formula. The mass is: Point five kilograms ... times the velocity: four meters-per-second... Squared Divided by two That's four joules.

This is the kinetic energy for the carriage on the left. The carriage on the right is not in motion; so it has no kinetic energy. That's zero. Let's start the film again. The carriages collide...

See? One carriage stops completely, while the other takes off at... four meters per second. What is the kinetic energy now? The carriage on the left has a kinetic energy of zero, and the carriage on the right is moving at four meters per second.

It has a kinetic energy of -- four joules. We just calculated that. In this collision the total kinetic energy is the same, four joules, before and after the collision. We call it an elastic collision. Let's do another experiment.

Now the carriage on the right is replaced with a slightly smaller one - of zero point four kilograms. Here comes the large carriage at four meters per second. We already calculated, that means a kinetic energy of four joules. They collide... Oh, now the carriages both took off in the same direction.

The large one with a velocity of two meters per second; the smaller one moving a bit faster, two point five meters per second. What's the total kinetic energy? For the large carriage, the kinetic energy is: 'Zero-point-five' times 'two-squared'... divided by two The small carriage has a kinetic energy of: 'Zero-point-four' times 'two-point-five, squared'... over two This equals '1' plus '1.25' joules This is the total kinetic energy of the carriages...

which we can round off to two-point-three joules. This means that the kinetic energy is much smaller after the collision than before. The kinetic energy beforehand was four joules. A certain amount of kinetic energy is lost! We call that an inelastic collision.

Where does the energy go? It's converted... perhaps to heat or sound. We hear a sound when the carriages hit each other. If the objects are deformed during the collision, this requires energy too.

But what's this? It's the half kilo carriages again. But something is different... Ah, magnetic contacts! The carriages will stick together when they collide.

What happens then? Let's look. The left carriage comes in at... four meters per second. They collide...

Yes, as expected, they stick to each other, and now function as one single object... which is moving at... two meters per second. Half the velocity that the left one had before the collision. If we calculate the kinetic energy for the two carriages combined, it's...

2 joules. Since the kinetic energy has decreased - from 4 joules to 2 joules - it is an inelastic collision. But when the objects are stuck together... after the collision we then call it a... perfectly inelastic collision.

Are there more examples of perfectly inelastic collisions? Oh yes. A meteorite hits a moon, and becomes part of it. That's a perfectly inelastic collision. A rifle bullet embeds itself and stays in a wooden target: ...

Perfectly inelastic collision. So we have three important concepts when it comes to collisions. Elastic collision, when the kinetic energy is conserved. Inelastic collision, when the kinetic energy has decreased. And then there's that special kind of inelastic collision: when two objects are stuck together after they collide.

That's a perfectly inelastic collision.