91Ó°ÊÓ

Given data:

The speed of a rocket is$v=\frac{3}{5}c$ .

The elapsed time is t=1hr .

a)

As the rocket clock runs slow, write the expression for the reading of the earth clock.

$t^{\text{'}}=\mathrm{γ}t$

${\mathit{t}}^{\mathbf{\text{'}}}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}$ …… (1)

Substitute the value of in the above expression.

$$\begin{gathered} t^{,}=\frac{1}{\sqrt{1-{\displaystyle \frac{{\left({\displaystyle \frac{3}{5}}c\right)}^2}{c^2}}}}\left(1hr\right) \\ {t}^{,}=\frac{1}{\sqrt{1-{\displaystyle \frac{9}{25}}}} \\ {t}^{,}=\frac{1}{\sqrt{{\displaystyle \frac{16}{25}}}} \end{gathered}$$

On further solving,

$$\begin{gathered} t^{,}=\sqrt{\frac{25}{16}} \\ {t}^{,}=\frac{5}{4}hr \\ {t}^{,}=1\mathrm{hr}15\min \end{gathered}$$

Therefore, the time of sending a signal after take-off is 1hr 15 min.

(b)

Write the relation between distance, speed, and time.

$d=v{t}^{\text{'}}$

Substitute the value of vand$t^{\text{'}}$in the above expression.

$$\begin{gathered} d=\left(\frac{3}{5}c\right)×\left(\frac{5}{4}hr\right) \\ d=\frac{3}{4}\mathrm{light}\mathrm{hours} \end{gathered}$$

As the rocket already travels 1 hr 15 min after departure, the light signal reaches the earth 2hrs after take-off.

Therefore, the time (according to earth clocks) taken by the light signal to reach the earth is 2hrs .

(c)

Substitute the value of the time (rocket) taken by a light signal reaches the earth, i.e.,t=2hrs in equation (1).

$$\begin{gathered} t^{\text{'}}=\frac{1}{\sqrt{1-{\displaystyle \frac{\left({\displaystyle \frac{3}{5}}c\right)}{c^2}}}}\left(2hrs\right) \\ {t}^{\text{'}}=\frac{1}{\sqrt{{\displaystyle \frac{25-9}{25}}}}\left(2hrs\right) \end{gathered}$$

On further solving,

$$\begin{gathered} t^{\text{'}}=\sqrt{\frac{25}{16}}×\left(2\mathrm{hrs}\right) \\ {t}^{\text{'}}=\frac{5}{4}×\left(2\mathrm{hrs}\right) \\ {t}^{\text{'}}=\frac{5}{2}\mathrm{hrs} \\ {\mathrm{t}}^{\text{'}}=2\mathrm{hrs}30\min \end{gathered}$$

Therefore, the time (according to the rocket observer) taken by the light signal to reach the earth is 2hrs 30 min .