# Proving one of Einsteins formulas (Kinetic energy of an object)

1. Jan 11, 2004

### Gunni

My physics teacher is having me prove two of Einsteins formulas as a homework assignment. One is his formula for the relationship between the energy of an object and it's momentum, E = sqrt(p^2c^ + m(original)^2c^4), that one was no problem, insert m = m(original) * gamma into E = mc^2 and so on. The other one, his formula for the kinetic energy of an object, is a trickier thing. My book says that it's found by defining K as such:

$$K = \int_{0}^{s} Fds = \int_{0}^{s} \frac{d(mv)}{dt}ds$$

Then it says it's possible to calculate this integer by inserting for the mass from equation 1.3 (Below) and do it by parts.

$$m = \frac{m_o}{\sqrt(1 - v^2/c^2)} (1.3)$$

The end result being:

$$K = mc^2 - m_oc^2$$

I'm completely stuck. I'd really appreciate any piece of advice you might have on integrating this by parts, or other means. I've done a little algebraic manipulation to try to simplify it but with no succes. What I'd really like to know if you can treat gamma as a constant when integrating (or diffrentiating) for time or distance? It'd simplify things a lot, but I suspect it can't be done that way since gamma contains velocity which is a function of distance and time, and would then be affected.

Any thoughts or hints?