Special Relativity derivations ....

[VoteSaxon]

Homework Statement
Using the special relativity formulae
p = ^mv / _{[1 - (v/c)²]}
E² = p²c² + m²c⁴
derive linear relations between:
(i) momentum and mass;
(ii) energy and mass;
(iii) energy and momentum,
which involve only c, c², β = ^v/_c, and γ (= ¹/_{sqrt(1 - β²}))

The attempt at a solution
I am pretty sure the answer to (i) is p = γmv = γmβc, although I am unsure if this counts as a linear relation.
I suppose for (ii) I should be aiming for E = mc², and for (iii) maybe I should be trying to get to E = pc (although I think this only applies to massless particles), but I haven't had much luck thus far.
I know these aren't really that hard, but for some reason my brain is just drawing blanks with these.

Many thanks for help and patience.

 

[robphy]

I think that here "linear" means that the two variables being related are each of first-power (no squares or square roots).
So, (i) is fine.
It seems that you have three quantities: p,m,E (mutually related by the above)... and you wish to find the relations between pairs chosen from those three.
So, what would you get for (ii)?

 

[VoteSaxon]

I think that here "linear" means that the two variables being related are each of first-power (no squares or square roots).
So, (i) is fine.
It seems that you have three quantities: p,m,E (mutually related by the above)... and you wish to find the relations between pairs chosen from those three.
So, what would you get for (ii)?

Sorry, a bit confused. I am pretty sure for (ii) I am meant to get E = mc². My main problem is getting there from the formulae the question provided. Does that make sense? ...

 

[robphy]

Sorry, a bit confused. I am pretty sure for (ii) I am meant to get E = mc². My main problem is getting there from the formulae the question provided. Does that make sense? ...

From your given relations,
E is the relativistic energy, p is the relativistic momentum, and m is the [invariant] rest mass.
With these symbols, $E\neq mc^2$ in general.
You can see this immediately by plugging in "what you think your E should be" into "E² = p²c² + m²c⁴".
When (that is, Under what conditions) will E="what you think your E should be" be true?