Special Relativity: Collision of a Particle and Antiparticle

[szimmy]

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.

E_Rest0 = m₀c² = 1,000 MeV
E_RestFinal = 10,000 MeV
E_Total = 100,000 MeV

Homework Equations

γ_x = 1 / √[1 - (v_x² / c²)]

Conservation of Energy:
E_Initial = E_Final
Conservation of Momentum:
P_Initial = P_Final

Total energy:
E_Total = √[(pc)² + (E_Rest)²]
E_Total = E_Restγ

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

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

The Attempt at a Solution

First I laid out conservation of energy.
E_Initial = E_Final
E_Rest0[γ₁ + 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.
P_Final = 1 / c * √[(100,000)² + (10,000)²]
P_Final = 1 / c * 99,498 MeV

The I setup conservation of momentum.
P_Final = P_Initial
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.

 

[szimmy]

I actually figured it out, no need to reply

 

[JaneWolf]

Wait what did you do? I'm stuck.

 

[vela]

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