# Special Relativity: Collision of a Particle and Antiparticle

## Homework Statement

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

## Homework 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]

## 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:

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I actually figured it out, no need to reply

Wait what did you do? I'm stuck.

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.