- #1

AnniB

- 13

- 0

## Homework Statement

Let us consider a perfectly inelastic collision between two protons: an incident proton with mass m, kinetic energy K, and momentum magnitude p joins with an originally stationary target proton to form a single product particle of mass M. Due to conservation of momentum, there must be some kinetic energy after the collision. Show that the energy available to create a product particle is

Mc

^{2}= 2mc

^{2}(sqrt(1 + K/2mc

^{2}))

## Homework Equations

E

_{0}= E

_{f}

This is the only equation I can think of. Applying conservation of momentum in just seems like it would add a miscellaneous variable. The others are just figured out from the problem and a couple hints the teacher provided.

E

_{1}= K + mc

^{2}

E

_{2}= mc

^{2}

E

_{f}= Mc

^{2}+ K

## The Attempt at a Solution

E

_{1}+ E

_{2}= E

_{f}

(E

_{1}+ E

_{2})

^{2}= E

_{f}

^{2}

(Mc

^{2}+ K)

^{2}= (K + mc

^{2})

^{2}+ 2(K + mc

^{2})(mc

^{2}) + (mc

^{2})

^{2}

Solving this gave me:

(Mc

^{2})

^{2}+ 2KMc

^{2}= 4Kmc

^{2}+ 4(mc

^{2})

^{2}

I can't get this to look like the equation I need. I feel like I either have one term too many, or I'm missing one.