Relativistic Momentum Help With Equation Reduction

[jimbobian]

Hi, so basically have been looking at http://en.wikibooks.org/wiki/Special_Relativity:_Dynamics#Momentum" and working my way through the maths for myself. However I have hit this point and can't get past it:

\begin{align}
u = \frac{v - u}{1-\frac{uv}{c^2}}
\end{align}
Which should be able to become:
\begin{align}
u = \frac{c^2}{v(1-\sqrt{1-\frac{v^2}{c^2}})}
\end{align}
I have tried and tried but can't seem to get it to work. Can anyone help me out on this one please?!

Thanks
James

 

[G01]

Hi, so basically have been looking at http://en.wikibooks.org/wiki/Special_Relativity:_Dynamics#Momentum" and working my way through the maths for myself. However I have hit this point and can't get past it:

\begin{align}
u = \frac{v - u}{1-\frac{uv}{c^2}}
\end{align}
Which should be able to become:
\begin{align}
u = \frac{c^2}{v(1-\sqrt{1-\frac{v^2}{c^2}})}
\end{align}
I have tried and tried but can't seem to get it to work. Can anyone help me out on this one please?!

Thanks
James

That wiki entry is not being clear with it's order of operations!

What you should be getting is:

$$u = \frac{c^2}{v}(1-\sqrt{1-\frac{v^2}{c^2}})$$

 

[jimbobian]

That wiki entry is not being clear with it's order of operations!

What you should be getting is:

$$u = \frac{c^2}{v}(1-\sqrt{1-\frac{v^2}{c^2}})$$

Hi G01, thanks for your reply. Have tried aiming for that equation instead and still can't get there. I have tried all sorts of different approaches, such as getting it in the form of a quadratic or dividing the top and bottom of the original fraction by c^2, but I just end up nowhere. Could you perhaps give me a hand in the right direction, maybe the first step or two - but don't make it too easy for me!

 

[jtbell]

Show us exactly what you've tried, and where you get stuck, and tell us why you're stuck.

 

[jimbobian]

Ok, have shut my computer down now and am replying on my phone. I will post my attempts tomorrow.
Thanks,
James

 

[PAllen]

Well, brute force use of the quadratic formula leads directly to the desired answer.

 

[jimbobian]

Haha, went back over my general quadratic attempt after PAllen's suggestion and realized that I had made a rather fundamental cross multiplication error! Fixed that and got to the desired equation. Lovely!

Last question, the equation that I get is:
\begin{align}
u = \frac{c^2}{v}(1\pm\sqrt{1-\frac{v^2}{c^2}})
\end{align}
I assume that we choose the negative version of the equation, because the positive version would yield a value for u which is greater than c?

Cheers