## Tuesday, 15 April 2008

### Relativistic Velocity Transformations

If a body is moving with speed v in an inertial frame S, what is the speed v' measured in an inertial frame S' moving at speed u relative to S?

In frame S:
v = \frac{x}{t}

In S':
v^\prime = \frac{x^\prime}{t^\prime}

= \frac{\gamma(x-ut)}{\gamma(t-\frac{ux}{c^2})}

= \frac{vt-ut}{t-\frac{uvt}{c^2}}

v^\prime = \frac{v-u}{1-\frac{uv}{c^2}}

It is possible, from here, to prove that the speed of light is constant under velocity transformations (and thus that c is a limiting speed). Setting v = c:
v^\prime = \frac{c - u}{1-\frac{u}{c}} = c\frac{c-u}{c-u} = c

Cool, huh?