- #1

Ken Miller

- 26

- 1

## Homework Statement

I have been working through “Basic Concepts in Relativity and Early Quantum Theory” by Resnick and Halliday. I've read about and done most of the problems about time-dilation, length-contraction, and Doppler effect. But then I got to problem 2-76, and I’ve been swirling in confusion for several days about the essential difference between a clock and a series of radar pulses.

Problem statement: A radar transmitter T is fixed to a reference frame S’ that is moving to the right with speed v relative to reference frame S. A mechanical time (essentially a clock) in frame S’, having a period \tau_0 (measured in S’), causes transmitter T to emit radar pulses, which travel at the speed of light and are received by R, a receiver fixed in frame S.

(a) What would be the period τ of the timer relative to observer A, who is fixed in frame S?

(b) Show that the receiver R would observe the time interval between pulses arriving from T, not as ##\tau## or ##\tau_0##, but as $$ \tau_R = \tau_0 \sqrt{\frac {c+v} {c-v}}.$$

(c) Explain why the observer at R measure a different period for the transmitter than does observer A, who is in the same reference frame. [Hint: A clock and a radar pulse are not the same thing.]

## Homework Equations

Time-dilation: $$ \tau = \frac {\tau_0} {\sqrt {1-\frac{v^2}{c^2}} }.$$

Doppler effect (with ##angle = 180^o##): $$ f = f_0 \sqrt {\frac {1-\beta} {1+\beta} } $$

## The Attempt at a Solution

Before I worked on part (a) and before I saw the question to part (c), I used the Doppler-effect equation to solve part (b), and came up with the correct answer. Then I went to part (a), and thought that A should detect the same period as R does. Then when I read part (c), I realized I was wrong about (a), and that I should probably use the time-dilation formula to calculate the period measured by A.

So, I have two questions.

--Am I correct in my answer to part (a)? (i.e. use time-dilation equation)

--What is really puzzling me is why a clock and a radar pulse are not the same thing?! (By the way, in the hint, I suspect that the authors meant that “a clock and a series of radar pulses...”)