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## 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}= m

_{0}c

^{2}= 1,000 MeV

E

_{RestFinal}= 10,000 MeV

E

_{Total}= 100,000 MeV

## Homework Equations

γ

_{x}= 1 / √[1 - (v

_{x}

^{2}/ c

^{2})]

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

_{Rest}γ

Relativistic Momentum for the particles:

P

_{Initial}= m

_{0}/ c

^{2}* [v

_{1}γ

_{1}+ v

_{2}γ

_{2}]

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}[γ

_{1}+ γ

_{2}] = 100,000 MeV

100 = γ

_{1}+ γ

_{2}

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 = m

_{0}/ c

^{2}* [v

_{1}γ

_{1}+ v

_{2}γ

_{2}]

1 / c * 99,498 MeV / [1,000 MeV / c

^{2}] = [v

_{1}γ

_{1}+ v

_{2}γ

_{2}]

99.498c = [v

_{1}γ

_{1}+ v

_{2}γ

_{2}]

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 v

_{2}= 0:

γ

_{2}= 1

100 = γ

_{1}+ 1

99 = γ

_{1}

√[1-(v

_{1}

^{2}/ c

^{2})] = 1 / 99

v

_{1}

^{2}/ c

^{2}= 1 - (1 / 99)

^{2}

v

_{1}= √[1 - (1 / 99)

^{2}] c

v

_{1}= .99995c

**Is this answer right? v**

_{1}= .99995c and v_{2}= 0cCan someone please either verify my answer or give me a step in the right direction? Thank you.

Last edited: