Chapter 25, Problem 85QAP

To determine

(a)

What is the speed of the light signal as measured by the observer (i) at rest in the rocket and (ii) at rest on Earth if a rocket 642 m long is traveling parallel to Earth's surface at 0.5c from left to right, at time t = 0 a light flashes for an instant at the center of the rocket, detectors at opposite ends of the rocket record the arrival of the light signal, call event A the light striking the left detector and event B the light striking the right detector. Observers at rest in the rocket and on Earth record the events?

Explanation of Solution

Calculation:

A rocket that is 642 m long is traveling parallel to Earth's surface at a speed V = 0.5c from left to right. At t = 0, a light flash for an instant at the center of the rocket. Detectors at opposite ends of the rocket detect when the light arrives-event A is when the light strikes the left side of the rocket, and event B is when the light strikes the right side of the rocket. The events are recorded by observers at rest in the rocket and observers at rest on Earth. Since the speed of light must be constant in all inertial reference frames, the speed of the light measured by both sets of observers must be the same and equal to c. In the observations made on the rocket, the light travels an equal distance to the detectors on the left and right ends of the ship. Events A and B will take place at a time equal to half of the length of the rocket divided by the speed of light. Both length contraction of the rocket and the relative motion of the rocket and the light need to be considered in the measurements made on Earth. The light for event A must travel half of the contracted length of the rocket minus the distance the rocket travels during that time. The light for event B must travel half of the contracted length of the rocket plus the distance the rocket travels during that time. We can also perform this calculation using time dilation-the proper time is the time measured on the rocket and the observed time interval is the time difference between events A and B as measured in Earth's frame.
The speed of light is equal to c in all inertial reference frames, so it is the same for both the observer at rest in the rocket and the observer at rest on Earth.

Conclusion:

The speed of light is equal to c in all inertial reference frames, so it is the same for both the observer at rest in the rocket and the observer at rest on Earth.

To determine

(b)

At what time after the flash do events A and B occur as measured by (r) the observer in the rocket and (it) the observer at rest on Earth? Which event occurs first in each case??

Explanation of Solution

Calculation:

1. Observer in the rocket
   $t_A=t_B=\frac{x}{c}=\frac{321}{3\times {10}^8}=1.07\times {10}^{-6}s$
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2. Observer in the Earth
   Event A:
   $\begin{array}{l}c{t}_A=x_A=x\sqrt{1-\frac{v^2}{c^2}}-V{t}_A\\ t_A=\frac{x\sqrt{1-\frac{v^2}{c^2}}}{c+V}=\frac{x\sqrt{1-\frac{{(0.5c)}^2}{c^2}}}{c+0.5c}=\frac{321\sqrt{1-\frac{{(0.5c)}^2}{c^2}}}{c+0.5c}=6.178\times {10}^{-7}s\end{array}$

  Event B:
    $\begin{array}{l}c{t}_B=x_B=x\sqrt{1-\frac{v^2}{c^2}}+V{t}_B\\ t_B=\frac{x\sqrt{1-\frac{v^2}{c^2}}}{c-V}=\frac{x\sqrt{1-\frac{{(0.5c)}^2}{c^2}}}{c-0.5c}=\frac{321\sqrt{1-\frac{{(0.5c)}^2}{c^2}}}{0.5c}=1.853\times {10}^{-6}s\end{array}$

To determine

(c)

That the results in part (b) are consistent with time dilation?

Explanation of Solution

Calculation:

  $\begin{array}{l}\Delta t=\frac{\Delta t_{proper}}{\sqrt{1-\frac{v^2}{c^2}}}\\ 1.853\times {10}^{-6}-6.178\times {10}^{-7}=\frac{1.07\times {10}^{-6}}{\sqrt{1-\frac{{(0.5c)}^2}{c^2}}}\\ 1.235\times {10}^{-6}s=1.235\times {10}^{-6}s\end{array}$

Conclusion:

This is consistent with time dilation.