Chapter 39, Problem 21P

(a)

To determine

Reading of clock in tail of rocket.

Explanation of Solution

Given:

Rocket has proper length $700\text{m}$ is moving to right at speed $0.900c$ .

Rocket has one clock in nose and one in tail that are synchronized in frame of rocket.

Clock in nose of rocket and clock on ground both read zero as they pass each other.

Formula used:

Write the expression of time measured by clock in tail in frame of rocket

  $t_1{}^{\prime }=\gamma \left(t_1-\frac{v{x}_1}{c^2}\right)$   ....... (1)

Here, $t_1$ is time measured by clock in tail in frame of ground, $\gamma$ is the gamma factor, $x_1$ is length of rocket in frame of ground and $v$ is speed of rocket with respect to ground.

Write the expression of length of rocket according to ground frame

  $x_1=\left(-\frac{L}{\gamma }\right)$

Here, $L$ is proper length of rocket.

Substitute $0$ for $t_1$ and $\left(-\frac{L}{\gamma }\right)$ for $x_1$ in expression (1)

  $t_1{}^{\prime }=\gamma \left(0-\frac{v}{c^2}\left(-\frac{L}{\gamma }\right)\right)$

Simplify the above expression

  $t_1{}^{\prime }=\frac{vL}{c^2}$   ....... (2)

Calculation:

Substitute $0.900c$ for $v$ , $700\text{m}$ for $L$ and $2.99\times {10}^8\text{m}/\text{s}$ for $c$ in expression (2)

  $\begin{array}{c}{t}_1{}^{\prime }=\frac{\left(0.900\right)\left(700\text{m}\right)}{\left(2.99\times {10}^8\frac{\text{m}}{\text{s}}\right)}\\ =2.10\mu \text{s}\end{array}$

Conclusion:

Thus, the clock in tail of rocket reads $2.10\mu \text{s}$ .

(b)

To determine

Reading of clock in tail of rocket with respect to ground.

Explanation of Solution

Given:

Rocket has proper length $700\text{m}$ is moving to right at speed $0.900c$ .

Rocket has one clock in nose and one in tail that are synchronized in frame of rocket.

Clock in nose of rocket and clock on ground both read zero as they pass each other.

Formula used:

Write the expression of time measured by clock in tail of rocket according to ground frame

  $t_1=\frac{L}{v}$   ....... (3)

Calculation:

Substitute $0.900c$ for $v$ , $700\text{m}$ for $L$ and $2.99\times {10}^8\text{m}/\text{s}$ for $c$ in expression (3)

  $\begin{array}{c}{t}_1=\frac{\left(700\text{m}\right)}{\left(0.900\left(2.99\times {10}^8\frac{\text{m}}{\text{s}}\right)\right)}\\ =2.59\mu \text{s}\end{array}$

Conclusion:

Thus, the clock in tail of rocket reads $2.59\mu \text{s}$ in frame of ground.

(c)

To determine

Reading of clock in nose of rocket according to ground.

Explanation of Solution

Given:

Rocket has proper length $700\text{m}$ is moving to right at speed $0.900c$ .

Rocket has one clock in nose and one in tail that are synchronized in frame of rocket.

Clock in nose of rocket and clock on ground both read zero as they pass each other.

Formula used:

Write the expression of time measured by clock in nose of rocket in ground frame

  $t_2=\left(t_1-t_1{}^{\prime }\right)$   ....... (4)

Calculation:

Substitute $2.59\mu \text{s}$ for $t_1$ and $2.10\mu \text{s}$ for $t_1{}^{\prime }$

  $\begin{array}{c}{t}_2=\left(2.59\mu \text{s}-2.10\mu \text{s}\right)\\ =0.49\mu \text{s}\end{array}$

Conclusion:

Thus, the clock in nose of rocket reads $0.49\mu \text{s}$ with respect to ground frame.

(d)

To determine

Reading of clock in nose with respect to frame of rocket.

Explanation of Solution

Given:

Rocket has proper length $700\text{m}$ is moving to right at speed $0.900c$ .

Rocket has one clock in nose and one in tail that are synchronized in frame of rocket.

Clock in nose of rocket and clock on ground both read zero as they pass each other.

Formula used:

Since the clock in nose of rocket is synchronized with clock in tail of rocket in frame of rocket therefore

  $t_2{}^{\prime }=t_1{}^{\prime }$   ....... (5)

Here, $t_2{}^{\prime }$ is reading of clock in nose of rocket in rocket frame.

Calculation:

Substitute $2.59\mu \text{s}$ for $t_1{}^{\prime }$ in expression (5)

  $t_2{}^{\prime }=2.59\mu \text{s}$

Conclusion:

Thus, the clock in nose of rocket reads $2.59\mu \text{s}$ in frame of rocket.

(e)

To determine

Reading of clock on ground when signal is received by observer on ground.

Explanation of Solution

Given:

Rocket has proper length $700\text{m}$ is moving to right at speed $0.900c$ .

Rocket has one clock in nose and one in tail that are synchronized in frame of rocket.

Clock in nose of rocket and clock on ground both read zero as they pass each other.

Signal is sent from nose of rocket to ground when clock in nose of rocket reads $1.00\text{h}$ .

Formula used:

Write the expression of time $T$ measured by clock on ground upon reception of signal

  $T=T_1+T_2$   ....... (6)

Here, $T_1$ is elapsed time when signal is sent and $T_2$ is time taken by signal to reach ground.

Write the expression of time when signal is sent

  $T_1=\frac{T_0}{{\left(1-\frac{v^2}{c^2}\right)}^{\frac{1}{2}}}$   ....... (7)

Here, $T_0$ is elapsed time in frame of rocket when signal is sent.

Write the expression of the traveling time $T_2$ of signal

  $T_2=\frac{v{T}_1}{c}$   ....... (8)

Calculation:

Substitute $1.00\text{h}$ for $T_0$ and $0.900c$ for $v$ in expression (7)

  $\begin{array}{c}{T}_1=\frac{\left(1.00\text{h}\right)}{{\left(1-\frac{{\left(0.900c\right)}^2}{c^2}\right)}^{\frac{1}{2}}}\\ =2.294\text{h}\end{array}$

Substitute $2.294\text{h}$ for $T_1$ and $0.900c$ for $v$ in expression (8)

  $\begin{array}{c}{T}_2=\frac{\left(0.900c\right)\left(2.294\text{h}\right)}{c}\\ =2.065\text{h}\end{array}$

Substitute $2.294\text{h}$ for $T_1$ and $2.065\text{h}$ for $T_2$ in expression (6)

  $\begin{array}{c}T=\left(2.294\text{h}\right)+\left(2.065\text{h}\right)\\ =4.36\text{h}\end{array}$

Conclusion:

Thus, the clock on ground reads $4.36\text{h}$ .

(f)

To determine

Reading of clock in nose of rocket when signal is received at nose of rocket.

Explanation of Solution

Given:

Rocket has proper length $700\text{m}$ is moving to right at speed $0.900c$ .

Rocket has one clock in nose and one in tail that are synchronized in frame of rocket.

Clock in nose of rocket and clock on ground both read zero as they pass each other.

Observer on ground sends return signal to nose of rocket upon reception of the signal.

Formula used:

Write the expression of time $T_3$ elapsed in ground frame when signal reaches rocket

  $T_3=\frac{vT}{0.100c}$

Substitute $0.900c$ for $v$ and $4.36\text{h}$ for $T$ in above expression

  $\begin{array}{c}{T}_3=\frac{\left(0.900c\right)\left(4.36\text{h}\right)}{0.100c}\\ =39.23\text{h}\end{array}$

Write the expression of total time $T_4$ in ground frame when signal travels to rocket

  $T_4=\left(1.1\right)T_3$

Write the expression of time dilatation equation

  $T_4{}^{\prime }={\left(1-\frac{v^2}{c^2}\right)}^{\frac{1}{2}}{T}_4$

Here, $T_4{}^{\prime }$ is time measured by clock in nose of rocket.

Substitute $\left(1.1\right)T_3$ for $T_4$ in above expression

  $T_4{}^{\prime }={\left(1-\frac{v^2}{c^2}\right)}^{\frac{1}{2}}\left(1.1T_3\right)$   ....... (9)

Calculation:

Substitute $39.23\text{h}$ for $T_3$ and $0.900c$ for $v$ in expression (9)

  $\begin{array}{c}{T}_4{}^{\prime }={\left(1-\frac{{\left(0.900c\right)}^2}{c^2}\right)}^{\frac{1}{2}}\left(1.1\left(39.23\text{h}\right)\right)\\ =18.8\text{h}\end{array}$

Conclusion:

Thus, the clock in nose of rocket reads $18.8\text{h}$ .