### Momentum

## Inelastic Collision

### description

Momentum is a concept developed by Isaac Newton that helped describe the dynamics of colliding bodies. The concept has been retained by relativity but transformed. In this section we shall see how the Newtonian concept of momentum breaks down at high speeds.

Newton's description of momentum was simply mass time velocity. The total momentum before and after a collision remains the same.

### animation

The animation portrays an inelastic collision between two spacecraft of identical mass. As seen from the reference frame of the grid the spacecrafts have the same speed but opposite velocities. The default is 0.5c. The velocities are vectorial quantities and cancel, so the total momentum of the system is zero. After the collision the speeds are now zero and so are the momenta. The total momentum is the same before and after the collision.

A flash of light generated from the centre of the grid is timed so the the two photons meet the nose of the spacecrafts at the edge of the grid. These events need to be maintained when viewed from another reference. The same collision can be witnessed from the yellow spacecraft's reference frame. In that frame the grid now approaching the yellow spacecraft at 0.5c. Note that the orange spacecraft approaches at not twice the speed, this would be 1.0c, but at the much lower speed of 0.8c. This is because in relativity velocities do not add together in a simple manner.

The Newtonian description is simple. The orange spacecraft should approaches the yellow spacecraft at twice the speed of the grid. On colliding with the yellow spacecraft the mass is now doubled so the velocity halves.

The relativity description is different and correct. Let us assign a mass of m to both spacecrafts. From the yellow spacecraft reference frame the momentum before the collision is m*0.8c + m*0.0c = m*0.8c. After the collision the total momentum is m*0.5c + m*0.5c. = m*c. This is more than the momentum before the collision. Simple Newtonian mechanics does not work at these speeds.

Try selecting a speed of 0.1c. You will see the Newton's concept of momentum conservation is almost correct, an error of 1%. The lower the speeds the better Newtonian mechanics can be applied. However the animations illustrate that Newtonian mechanics is wrong, and we need a better description.