Chapter 27, Problem 1CQ

Suppose the speed of light were 30 m/s instead of its actual value. Describe various ways in which the behavior of your everyday world would be different than it is now.

To determine

Explain the consequences of a light speed of 30ms^-1 on our everyday life.

Answer to Problem 1CQ

Slowing down the speed of light to 30 ms^-1 makes apparent changes in length, time and color observed by an observer.

Explanation of Solution

Write the expression for length contraction,

$L=\frac{L^{\prime }}{\gamma }$

Where,

$\begin{array}{l}\gamma ={\left(\sqrt{1-{\beta }^2}\right)}^{-1}\\ \beta =\frac{v}{c}\end{array}$

Here, $L$ is the length of an object observed by an observer in S frame,$L^{\prime }$ is the length observed by an observer in S´ frame where the object is at rest relative to the observer,$v$ is the velocity with which S´ is moving with respect to S frame and $c$ is the velocity of light.

In this case, $L^{\prime }$ is the proper length.

As $c$ is reduced to 30 ms^-1, $\beta$ becomes considerable and can’t be neglected.

If a person moving with a velocity $v$ with respect to earth sees an object on earth, he observes it to be reduced in length.(Note: In this case the observer on the ground at rest is the one who measures proper length since Earth is at rest relative to him.)

Write the expression for time dilation,

$\Delta t=\gamma \Delta t^{\prime }$

Here, $\Delta t$ is the time observed in S frame and $\Delta t^{\prime }$ is the time observed in the S´ frame.

Since the object is considered to be at rest in S´ frame, time observed in this frame is considered proper time.

As $\beta$ is not negligible, time observed by the observer in S´ is prolonged whereas for the observer in S, it is shortened.

Thus a person moving with a velocity $v$ on Earth, observes the clock on Earth ticking slower than his own clock.

Write the expression for Doppler effect when the source is approaching

$f=\frac{f^{\prime }\sqrt{1+\beta }}{\sqrt{1-\beta }}$

Where,

$\beta =\frac{v}{c}$

Write the expression for Doppler effect when the source is receding,

$f=\frac{f^{\prime }\sqrt{1-\beta }}{\sqrt{1+\beta }}$

Here, $f$ is the observed frequency and $f^{\prime }$ is the source frequency.

As $\beta$ is not negligible $(\beta >>0)$ when the speed of light is 30 ms^-1 the change in observed frequency becomes prominent. Thus the color of an object seems different while moving closer, moving away and at rest.