Relativistic version of Newton's second law with parallel force

[MasterNewbie]

Homework Statement

"Given F=dp/dt. If the force is parallel to velocity show that F=γ³ma."

Homework Equations

F=dp/dt and p=γmv

The Attempt at a Solution

Since the force is the first derivative of the momentum with respect to time, and γ and v both vary with time since there is a net force causing both quantities to change, take the derivative of p using multiplication rule: dp/dt = m[γ'v+γa] γ'=γ³*v/c² so this would change dp/dt to dp/dt = m[γ³(v/c)²+γa] After this I can't think of anywhere to go that could simplify this to the desired equation F=γ³ma.

 

[PullMeOut]

are you sure about $$\gamma$$ and its derivative? there must be something wrong about it

 

[MasterNewbie]

Yeah, I did the derivative wrong, I forgot to do chain rule for the v² so the derivative of γ would be γ³*va/c²

So then dp/dt = γma[γ²(v/c)²+1]. Assuming the math I did was right the only way I could get what I'm looking for is if (v/c)² is equal to 1-1/γ² so when you distribute you'd get γ²-1+1, then it would give dp/dt = F = γ³ma what I'm looking for.

I'll have to mess around with this.

[edit]
Coincidentally (v/c)² does equal what it needs to equal to simplify to what the equation needs to be.

Thank you a lot Pull :)
[/edit]