Chapter 25, Problem 29E

(a)

To determine

The time taken by boat upstream.

Answer to Problem 29E

The time taken by boat upstream is $5\text{hours}$.

Explanation of Solution

Write the expression for time during upstream.

  $t_u=\frac{d}{v_b-v_w}$        (I)

Here, $d$ is the distance between two islands, $v_b$ is velocity of boat, $v_w$ is velocity of water and $t_u$ is time required during upstream.

Conclusion:

Substitute $5\text{km}\cdot {\text{h}}^{-1}$ for $v_b$, $3\text{km}\cdot {\text{h}}^{-1}$ for $v_w$ and $10\text{km}$ for $d$ in equation (I).

  $\begin{array}{c}{t}_u=\frac{10\text{km}}{5\text{km}\cdot {\text{h}}^{-1}-3\text{km}\cdot {\text{h}}^{-1}}\\ =5\text{h}\end{array}$

Thus, the time taken by boat upstream is $5\text{hours}$.

(b)

To determine

The time taken by boat downstream.

Answer to Problem 29E

The time taken by boat downstream is $1.25\text{hours}$.

Explanation of Solution

Write the expression for time during downstream.

  $t_d=\frac{d}{v_b+v_w}$        (II)

Here, $t_d$ is time required during downstream.

Conclusion:

Substitute $5\text{km}\cdot {\text{h}}^{-1}$ for $v_b$, $3\text{km}\cdot {\text{h}}^{-1}$ for $v_w$ and $10\text{km}$ for $d$ in equation (II).

  $\begin{array}{c}{t}_d=\frac{10\text{km}}{5\text{km}\cdot {\text{h}}^{-1}+3\text{km}\cdot {\text{h}}^{-1}}\\ =1.25\text{h}\end{array}$

Thus, the time taken by boat upstream is $1.25\text{hours}$.

(c)

To determine

The time taken in total round trip.

Answer to Problem 29E

The time taken in total round trip is $6.25\text{hours}$.

Explanation of Solution

Write the expression for total time.

  $t=t_d+t_u$        (III)

Here, $t$ is time taken in total round trip.

Conclusion:

Substitute $1.25\text{hours}$ for $t_d$ and $5\text{hours}$ for $t_u$ in equation (III).

  $\begin{array}{c}t=1.25\text{hours}+5\text{hours}\\ =6.25\text{hours}\end{array}$

Thus, the time taken in total round trip is $6.25\text{hours}$.

(d)

To determine

The time taken by boat in still water for round trip.

Answer to Problem 29E

The time taken by boat in still water for round trip is $4\text{hours}$.

Explanation of Solution

Write the expression for time by boat in still water.

  $t_s=\frac{2d}{v_b}$        (IV)

Here, $t_s$ is time taken by boat in still water.

Conclusion:

Substitute $5\text{km}\cdot {\text{h}}^{-1}$ for $v_b$ and $10\text{km}$ for $d$ in equation (IV).

  $\begin{array}{c}t=\frac{2\left(10\text{km}\right)}{5\text{km}\cdot {\text{h}}^{-1}}\\ =4\text{h}\end{array}$

Thus, the time taken by boat in still water for round trip is $4\text{hours}$.

(e)

To determine

The analogy between results and equation 25.1.

Answer to Problem 29E

The time taken by object decreases when its speed increases when observed by observer.

Explanation of Solution

The above results also show that time taken by boat decreases when the boat moves downstream. Equation 25.1 gives the relation between actual time taken and time dilation.

Write the expression for time dilation.

  $t^{\prime }=\frac{t}{\sqrt{1-{\left(u/c\right)}^2}}$

Here, $t$ is the time taken by light, $t^{\prime }$ is the time taken by required by object, $u$ is the speed of object and $c$ is the speed of light in vacuum.

The time required by object moving at speed $u$ will be more observed by person on the Earth than its actual time. However, $t^{\prime }$ and $t$ will be equal when the object moves at a speed of light.

Conclusion:

Thus, the time taken by object decreases when its speed increases when observed by observer.