P231. Time Dilation

Write your answers as requested.

Text reference:  The derivation in section 29.2 should be helpful as you do these problems.

By doing these problems, you'll see that the time dilation formula is simply a matter of using the very first equation you learned in physics (v = d/t) together with a counterintuitive but easy to apply postulate and some algebra.  

One of the postulates on which The Special Theory of Relativity is based is the following: The speed of light in a vacuum always has the same value, no matter how fast the source of light and the observer are moving relative to each other. As strange as this may seem, try to suspend judgment and simply use the postulate in these problems.

You're doing some EVA (extra-vehicular activity) on the top of the saucer section of the starship Interprize (see diagram to the right). The Interprize is moving at speed, v, relative to the space station, Depe Spaise Nein. You shine a beam of laser light perpendicular to the path of the ship toward a mirror mounted on the ship. The light travels a distance, D, to the mirror, reflects and returns to you for a total distance of 2D. You measure the total transit time of the light beam to leave and return to you to be Δty (subscript y = you). The speed of the light beam according to you is therefore c = 2D/Δty. Woldorf, an observer on DSN, is watching you as you do this experiment. He uses his built-in sensors to measure the transit time of the light beam to be Δtw (subscript w = Woldorf), a different value than what you measured.

Here is an applet that also illustrates the situation.

  1. Draw the path of the light beam as seen by someone looking down on the Interprize and moving with the same velocity as the Interprize.

  2. According to Woldorf, the light beam takes a diagonal path from you to the mirror. Sketch this path from you to the mirror and back to you.

  3. According to Woldorf, what distance, L, do you (and the Interprize) move during the transit time of the light?  Express your answer in terms of v and Δtw.  Tell why you should use Δtw rather than Δty.

  4. According to Woldorf, what distance, Z, does the light beam travel while you're moving distance L?  Express your answer in terms of D, v, and Δtw.

  5. According to Woldorf, who knows the speed of light postulate of Special Relativity, the total distance traveled by the light (from #4) divided by the total transit time, Δtw, must equal c, the same value as you would measure for the speed of light. Write this as a formula.

  6. Now you have two equations for the speed of light.  You have the one according to Woldorf from step 5 and you have the one that you calculate using c = 2D/Δty. Now it's time for some algebra. Combine the two equations, and eliminate D and L from the formula. Solve for Δtw in terms of Δty, v, and c only. The result should be the time dilation formula of special relativity.

  7. Which of the times Δtw and Δty is the proper time? Explain.


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