- #1

martinhiggs

- 24

- 0

## Homework Statement

Using:

particle velocity, beta

particle momentum, p

total energy, E

Lorentz factor, gamma

kinetic energy, KE

Derive an equation for momentum as a function of kinetic energy. The functions have to depend either on the variable in the bracket, p(KE), or on a constant.

## The Attempt at a Solution

This is what I've done so far, and I am now stuck, and unsure if the way I am doing it is correct or if there is a different approach.

[tex]E^{2} = p^{2}c^{2} + m^{2}c^{4}[/tex]

[tex]KE = E - m_{0}c^{2}[/tex]

[tex]KE = \sqrt{p^{2}c^{2} + m^{2}c^{4}} - m_{0}c^{2}[/tex]

[tex]p^{2} = \frac{KE^{2}}{c^{2}} - m^{2}c^{2} - m_{0}^{2}c^{4}[/tex]

The only thing I could think of doing next is:

[tex]KE = \frac{p^{2}}{2m_{0}} , m_{0} = \frac{p^{2}}{2KE}[/tex]

[tex] p^{2} = \frac{KE}{c^{2}} - m^{2}c^{2} - \frac{p^{4}}{4KE^{2}}c^{2}[/tex]

[tex]p^{2} + \frac{p^{4}}{4KE^{2}}c^{2} = \frac{KE}{c^{2}} - m^{2}c^{2}[/tex]

[tex]p^{2}(1 + \frac{p^{2}}{4KE^{2}}c^{2}) = \frac{KE}{c^{2}} - m^{2}c^{2}[/tex]

I'm not sure if this is the best or easiest way to do this, as it seems to be pretty messy, and I also have one more m in the equation that I need to get rid of but am not sure of the best way of doing so.

Any help will be greatly appreciated :)