# Special Relativity: Collision of a Particle and Antiparticle

1. Feb 11, 2015

### szimmy

1. The problem statement, all variables and given/known data
A particle and its anti-particle are directed toward each other, each with rest energy 1,000 MeV. We want to create a new particle with rest energy 10,000 MeV and total energy 100,000 MeV. What must the speed of the particle and antiparticle be before the collision.

ERest0 = m0c2 = 1,000 MeV
ERestFinal = 10,000 MeV
ETotal = 100,000 MeV

2. Relevant equations
γx = 1 / √[1 - (vx2 / c2)]

Conservation of Energy:
EInitial = EFinal
Conservation of Momentum:
PInitial = PFinal

Total energy:
ETotal = √[(pc)2 + (ERest)2]
ETotal = ERestγ

Relativistic Momentum for the particles:
PInitial = m0 / c2 * [v1γ1 + v2γ2]

Solving for momentum in terms of the total and rest energies:
PFinal = 1 / c * √[(ET)2 - (ERestFinal)2]

3. The attempt at a solution
First I laid out conservation of energy.
EInitial = EFinal
ERest01 + γ2] = 100,000 MeV
100 = γ1 + γ2

Then I solved for the final momentum using the expression above which is derived from the total energy formula.
PFinal = 1 / c * √[(100,000)2 + (10,000)2]
PFinal = 1 / c * 99,498 MeV

The I setup conservation of momentum.
PFinal = PInitial
1 / c * 99,498 MeV = m0 / c2 * [v1γ1 + v2γ2]

1 / c * 99,498 MeV / [1,000 MeV / c2] = [v1γ1 + v2γ2]

99.498c = [v1γ1 + v2γ2]

This is where I get stuck. I don't know whether I should attempt to solve for one of the velocities or just in general which step to take next. I tried solving by setting one of the velocities to zero but I'm not sure if this is the correct way to do it.

If I substitute v2 = 0:
γ2 = 1

100 = γ1 + 1
99 = γ1
√[1-(v12 / c2)] = 1 / 99
v12 / c2 = 1 - (1 / 99)2
v1 = √[1 - (1 / 99)2] c
v1 = .99995c

Is this answer right? v1 = .99995c and v2 = 0c

Can someone please either verify my answer or give me a step in the right direction? Thank you.

Last edited: Feb 11, 2015
2. Feb 11, 2015

### szimmy

I actually figured it out, no need to reply

3. Feb 14, 2015

### JaneWolf

Wait what did you do? I'm stuck.

4. Feb 14, 2015

### vela

Staff Emeritus
It's generally better to work in terms of energy and momentum and then find velocities using $v = E/p$. It simplifies the algebra.