Chapter 25, Problem 48E

(a)

To determine

Velocity of rocket $P$ relative to observer in rocket $Q$.

Answer to Problem 48E

The velocity of rocket $P$ relative to observer in rocket $Q$ is $0.75\times {10}^8\text{m}\cdot {\text{s}}^{-1}$.

Explanation of Solution

Rocket $P$ moves away from earth at speed $2\times {10}^8\text{m}/\text{s}$ relative to earth frame and the second rocket $Q$ that moves away from earth at speed $1.5\times {10}^8\text{m}/\text{s}$ is overtaken by $P$.

Write the expression for speed ${u^{\prime }}_{\text{P}}$ of rocket $P$ relative to rocket $Q$

  ${u^{\prime }}_{\text{P}}=\frac{u_{\text{P}}-u_{\text{Q}}}{1-\frac{u_{\text{P}}{u}_{\text{Q}}}{c^2}}$        (I)

Here, $u_{\text{P}}$ is velocity of rocket $P$ relative to earth and $u_{\text{Q}}$ is velocity of rocket $Q$ relative to earth.

Conclusion:

Substitute $2\times {10}^8\text{m}/\text{s}$ for $u_{\text{P}}$, $3\times {10}^8\text{m}/\text{s}$ for $c$ and $1.5\times {10}^8\text{m}/\text{s}$ for $u_{\text{Q}}$ in expression (I)

  $\begin{array}{c}{u^{\prime }}_{\text{P}}=\frac{\left(2\times {10}^8\text{m}/\text{s}\right)-\left(1.5\times {10}^8\text{m}/\text{s}\right)}{1-\frac{\left(2\times {10}^8\text{m}/\text{s}\right)\left(1.5\times {10}^8\text{m}/\text{s}\right)}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^2}}\\ =0.75\times {10}^8\text{m}/\text{s}\end{array}$

Thus, the velocity of rocket $P$ is $0.75\times {10}^8\text{m}/\text{s}$ relative to observer in rocket $Q$.

(b)

To determine

Velocity of rocket $Q$ relative to observer in rocket $P$.

Answer to Problem 48E

Thus, the velocity of rocket $Q$ is $-0.75\times {10}^8\text{m}\cdot {\text{s}}^{-1}$  relative to observer in rocket $P$.

Explanation of Solution

Rocket $P$ moves away from earth at speed $2\times {10}^8\text{m}/\text{s}$ relative to earth frame. Second rocket $Q$ that moves away from earth at speed $1.5\times {10}^8\text{m}/\text{s}$ is overtaken by $P$.

Write the expression for speed ${u^{\prime }}_{\text{Q}}$ of rocket $Q$ relative to rocket $P$

  ${u^{\prime }}_{\text{Q}}=\frac{u_{\text{Q}}-u_{\text{P}}}{1-\frac{u_{\text{Q}}{u}_{\text{P}}}{c^2}}$        (II)

Conclusion:

Substitute $2\times {10}^8\text{m}/\text{s}$ for $u_{\text{P}}$, $3\times {10}^8\text{m}/\text{s}$ for $c$ and $1.5\times {10}^8\text{m}/\text{s}$ for $u_{\text{Q}}$ in expression (II).

  $\begin{array}{c}{u^{\prime }}_{\text{Q}}=\frac{\left(1.5\times {10}^8\text{m}/\text{s}\right)-\left(2\times {10}^8\text{m}/\text{s}\right)}{1-\frac{\left(1.5\times {10}^8\text{m}/\text{s}\right)\left(2\times {10}^8\text{m}/\text{s}\right)}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^2}}\\ =-0.75\times {10}^8\text{m}/\text{s}\end{array}$

Thus, the velocity of rocket $Q$ is $-0.75\times {10}^8\text{m}\cdot {\text{s}}^{-1}$ relative to observer in rocket $P$.