Derive lorentz transform for energy

[gboff21]

Homework Statement

Derive the relation:
$E' = \gamma (E + \beta p c)$

Homework Equations

$p' = \gamma p \\ E^{2} = p^{2} c^{2} + M^{2}c^{4}$

The Attempt at a Solution

start off with stationary frame S $E=mc^{2}$
then in moving frame S' $E'^{2} = p'^{2} c^{2} + E^{2}$:
lorent transform momentum:
$E'^{2} = \gamma^{2} m^{2} v^{2}c^{2} + E^{2}$
and that's as far as I get!

 

[Simon Bridge]

$E=\gamma mc^2$

That "p", in the equation you are supposed to prove, is unprimed.
Is that significant?

 

[gboff21]

Yes the p is supposed to be unprimed.

I've solved it now. Thanks anyway.
Here it is for anyone who's having the same problem:
You start off (or derive it as I had to, to understand it) with the lorentz transform for velocities in two frames $u = \frac{u'+v}{1+frac{uv}{c^{2}}}$
Know that $E'=\gamma m_{0}c^{2}$ because in the stationary frame S, only rest mass provides energy.
You expand out gamma with u' given above and recognise that $p'=\gamma p$
simplify and you get an answer!