Probably Fairly Simple Special Relativity Calculation

[Astrofiend]

Homework Statement

I am trying to show that the velocity of an ultra-relativistic particle can be approximated by the following expressions:

[tex]

v \approx c \left[1-\frac{1}{2}\left(\frac{mc^2}{E}\right) ^2 \right]

[/tex]

and[tex]

\frac{1}{v} \approx \frac{1}{c} \left[1+\frac{1}{2}\left(\frac{mc^2}{E}\right) ^2 \right]

[/tex]

...but I'm struggling. I know it can't be that hard, but I just can't quite get there. It seems I can get close, but I miss the factor of 1/2 out the front of the mass/energy term. I'm definitely doing something wrong, so any help would be much appreciated. Here's how I went about trying to obtain the first expression:

The Attempt at a Solution

So we have our usual relativistic expression:

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

Then, I divided through by m^2 c^4 to get:

[tex]
\frac{E^2}{(mc^2)^2}-1 = \frac{p^2 c^2}{(mc^2)^2}
[/tex]

Now, using a relation that I pulled from somewhere - [tex] (pc)^2 = E^2 \frac{v^2}{c^2}[/tex]

I simply sub that in for (pc)^2, getting

[tex]
\frac{E^2}{(mc^2)^2}-1 = \frac{E^2}{(mc^2)^2}. \frac{v^2}{c^2}
[/tex]

and dividing through by [tex]\frac{E^2}{(mc^2)^2}[/tex], get

[tex]
\frac{v^2}{c^2} = 1 - \frac{(mc^2)^2}{E^2}
[/tex]

so

[tex]
\frac{v}{c} = 1 - \frac{(mc^2)^2}{E^2}
[/tex]

i.e

[tex]
v = c\left[1 - \frac{(mc^2)^2}{E^2}\right]
[/tex]

Hmmmm - the problem is that a) this is an exact equality not an approximation, b) I'm missing the damn factor of 1/2 out the front of the mass-energy fraction, and c) nowhere have I used the fact that the particle is highly relativistic. These facts are almost certainly all related, but I can't see how and I've been staring at it for some time now.

Any help or suggestions, or more ideas for more rigorous derivations would be exceedingly welcome!

 

[Doc Al]

I didn't look over the rest of your derivation, but this step:

[tex]
\frac{v^2}{c^2} = 1 - \frac{(mc^2)^2}{E^2}
[/tex]

so

[tex]
\frac{v}{c} = 1 - \frac{(mc^2)^2}{E^2}
[/tex]

...doesn't follow. (You forgot to take the square root of the RHS!)

Hint: Use a Taylor series expansion to approximate the square root of 1 - x, where x << 1.

 

[Astrofiend]

"(You forgot to take the square root of the RHS!)"

Der! Must be tired - or just plain dumb...

Thanks a lot Doc Al - I got it out easily after that little hint. I appreciate it a lot!