# Speed of a particle

1. Jan 29, 2005

### mike217

Can you please give me a few pointers on this problem.

A particle of rest mass m is accelerated to a kinetic energy K in a nuclear reactor. This particle is incident on a stationary target particle, also of rest mass m.

a) Show that the speed of the centre of mass (that is the speed of the frame in which the total momentum is zero) is $$\gamma v/(\gamma +1)$$ where v is the speed of the incident particle and $$\gamma =(1-v^2/c^2)^-1/2$$. Verify that this expression reduces to the usual one in the non relativistic case when v<<c.

b) If the collision is perfectly inelastic-show by using the conservation of energy and momentum that the mass M of the resulting composite is $$M=m\sqrt{2(\gamma +1)}$$ Verify that this reduces to the usual value in the non relativistic case v<<c.

For part 'a' I get by writing the conservation of momentum:

$$Pbefore=mv/\sqrt{1-v^2/c^2}$$ and $$Pafter=MV/\sqrt{1-V^2/c^2}$$ after that I am stuck on what to do.

For part 'b' I get,

$$mc^2/\sqrt{(1-v^2/c^2)} + mc^2=Mc^2/\sqrt{(1-V^2/c^2)}$$

Thank you.

2. Jan 30, 2005

### mike217

Hi,

Please you don't have to solve this question. I just need a few tips on how to proceed, i.e. which particle to take as one having zero momentum etc.

Thank you.