Problem 2

  In 1992, Jenny Thompson set a world's record for the 100-m freestyle. Suppose that this race had been monitored from a spaceship travelling at a speed of 0.8200c relative to the earth and that the space travellers measured the time interval of the race to be 95.18 s. What was the time recorded on earth?

SOLUTION:
We are measuring two different events; namely, the start and finish of the race. We on earth hold the stopwatch in our hand, so the ticks corresponding to start and finish both occur at the same point in space. So our measurement of the time interval is the proper time interval. (You might quibble that we aren't in the rest frame of the runner-and you'd be right, of course-but the runner is presumably running at a velocity with respect to us that is much less than the speed of light, whereas the space travellers are moving at an appreciable fraction (0.82) of the speed of light with respect to us, so, essentially, we are in the same inertial reference frame as the runner.) Given the time interval $\Delta t$ observed by the space travellers, we can plug into the time dilation formula to get the time interval on our stopwatch.

$$\Delta t_o = \Delta t \sqrt{1 - \frac{v^2}{c^2}}$$

$$\Delta t_o = 54.48 s$$