# Working with relativity 1641 The recoiling gun revisited

In elementary textbooks, one of the first examples of conservation of momentum is the 'recoiling gun'.

From the conservation of momentum, the momen- momentum = MV

tum of the recoiling gun is equal and opposite to the momentum of the bullet.

Classical calculation

Mass is a measure of resistance to change of motion, or inertia. There is only 'one kind' of mass (It so happens that gravitational attraction is proportional to inertial mass.)

The speeds of the bullet and the gun recoil are inversely proportional to their masses.

For example, if the mass of the gun is 40 times the mass of the bullet, the initial speed of the bullet will be 40 times that of the recoil.

speed of the bullet mass of the gun recoil speed of the gun mass of the bullet

### When the classical formula fails

We can see that this formula will not work at very high velocities. For example, if we were to have a 'super-gun' recoiling with a speed of V = 0.5 c, the bullet would, according to the classical calculation, have a speed of 20 c — in breach of all the principles of special relativity!

Relativistic calculation for the imaginary super-gun:

Example

A super-gun fires a bullet with speed 0.999 c and recoils with speed 0.5 c. Calculate the ratio of the mass of the gun to the mass of the bullet, and compare it to the ratio of the speed of the bullet to the recoil speed of the gun. Going back to Table 15.1 we find

Velocity v Gamma factor y

Inserting values for the velocities and the y factor into Equation (16.10),