special relativity

A projectile of rest mass M1, total energy E1, and momentum p1, is directed at a stationary target of mass m2. Find the the velocity of the centre-of-momentum frame .

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1. The total energy of a system is given by:

E = gamma m c^2

The total momentum is:

p = gamma m v

The ratio p/E is thus:

p/E = v/c^2

So, if we know the energy and momentum of a system, we know the velocity.

For the given system, the energy is:

E = E1 + m2 c^2

The momentum is

p = p1

Therefore:

v = c^2 p/E = c^2 p1/(E1 + m2 c^2)

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