Find Rest Mass of Composite Particle After Collision

[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.

 

[vela]

Where's your conservation of energy equation?

 

[Joules6626]

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

 

[vela]

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.

 

[Joules6626]

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?

 

[vela]

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.