# Relativity Question

• Joules6626

## Homework Statement

A particle of rest mass m and kinetic energy 2mc^c strikes and sticks to a stationary particle of rest mass 2m. Find the rest mass M of the composite particle

## Homework Equations

E = mc^2 + KE
E^2 = (mc^2)^2 + (pc)^2
p = mv/sqrt(1-v^2/c^2)

## The Attempt at a Solution

For finding the initial velocity of the moving particle:
2mc^2 = KE = mc^2/sqrt(1-v^2/c^2) - mc^2
m's cancel
and when solving for v, you get v1 = sqrt(2/3)c

To find M, I tried using conservation of momentum and energy.
p1 = p2
mv1/sqrt(1-v1^2/c^2) + 2m*0 = Mv2/sqrt(1-v2^2/c^2)
I can't seem to find a way to make the equation only have one unknown.

Where's your conservation of energy equation?

E1 = E2
(mc^2)^2 + (cmv1/sqrt(1-v1^2/c^2))^2 = (Mc^2)^2 + (cMv2/sqrt(1-v2^2/c^2))^2

That's not quite right. You forgot the energy of the stationary mass. Also, to simplify the algebra, you might want to use E=γmc2 rather than breaking out the rest energy and momentum contributions separately.

so it would be
(mc^2)^2 + (cmv1γ1) + 2mc^2 = (Mc^2)^2 + (cMv2γ2)^2?
how would that give me a function of just v2 or M?

No, that's still not right. It doesn't work out unit-wise. You have quantities equal to E2, not E. Plus you're making it more complicated than it needs to be. You can calculate the total energy of the system before the collision just by adding up a few quantities you were given.

You have two equations and two unknowns (M and v2). Now it's just a bunch of algebra to solve for them.