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:

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

and<br /> <br /> \frac{1}{v} \approx \frac{1}{c} \left[1+\frac{1}{2}\left(\frac{mc^2}{E}\right) ^2 \right]<br /> <br />

...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:

<br /> E^2 = p^2 c^2 + m^2 c^4 <br />

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

<br /> \frac{E^2}{(mc^2)^2}-1 = \frac{p^2 c^2}{(mc^2)^2}<br />

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

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

<br /> \frac{E^2}{(mc^2)^2}-1 = \frac{E^2}{(mc^2)^2}. \frac{v^2}{c^2}<br />

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

<br /> \frac{v^2}{c^2} = 1 - \frac{(mc^2)^2}{E^2}<br />

so

<br /> \frac{v}{c} = 1 - \frac{(mc^2)^2}{E^2}<br />

i.e

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

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:

<br /> \frac{v^2}{c^2} = 1 - \frac{(mc^2)^2}{E^2}<br />

so

<br /> \frac{v}{c} = 1 - \frac{(mc^2)^2}{E^2}<br />

...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!