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?

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