Chapter 7.2, Problem 78E

To determine

To calculate : the speed of the current and the speed of the motorboat with relative to the current.

Answer to Problem 78E

The speed of the current is $5\text{ miles per hour}$ and the speed of the motorboat with relative to the current in upstreamis $20\text{ miles per hour}$ and in downstream $30\text{ miles per hour}$ .

Explanation of Solution

Given information :

Total distance covered by the motorboat going upstream or downstream $=20\text{ miles}$

Time take by the motorboat to travel downstream

  $\begin{array}{l}=\text{40 minutes}\\ =\left(\frac{40}{60}\right)\text{ hours}\\ =\frac{2}{3}\text{ hours}\end{array}$

Time take by the motorboat to travel upstream

  $\begin{array}{l}=60\text{ minutes}\\ \text{= }\frac{60}{60}\text{ hour}\\ =1\text{ hour}\end{array}$

Calculation :

Let us suppose the speed of the current with relative to the ground be $x$ miles per hour and the speed of the motorboat with relative to the ground be $y$ miles per hour.

Since, the motorboat is going downstream , the net speed of motorboat with relative to the current is calculated as $\left(x+y\right)\text{ miles per hour}$ .

Now, according to the question:

  $\begin{array}{l}\text{Speed = }\frac{\text{Total distance covered}}{\text{Total time taken}}\\ \left(x+y\right)=\frac{20}{\left(\frac{2}{3}\right)}\\ x+y=\frac{20\times 3}{2}\\ x+y=10\times 3\end{array}$

  $x+y=30$  ...(1)

Similarly, the motorboat is going upstream, the net speed of the motorboat with respect to the current is calculated as $\left(x-y\right)\text{ miles per hour}$ .

  $\begin{array}{l}\text{Speed = }\frac{\text{Total distance covered}}{\text{Total time taken}}\\ \left(x-y\right)=\frac{20}{1}\end{array}$

  $x-y=20$  ...(2)

Solve equation (1) and (2):

  $\underset{\_}{\begin{array}{l}x-\cancel{y}=20\\ \text{+ }x+\cancel{y}=30\end{array}}$

  $\begin{array}{l}2x=50\\ x=\frac{50}{2}\\ x=25\end{array}$

Substitute value of $x$ in equation (1) to calculate $y$ .

  $\begin{array}{l}x-y=20\\ y=x-20\\ y=25-20\\ y=5\end{array}$

Calculate the speed of the motorboat with respect to the current:

For upstream:

  $\begin{array}{l}=\text{Speed of motorboat - speed of current}\\ =\left(25-5\right)\text{miles per hour}\\ =20\text{ miles per hour}\end{array}$

For downstream:

  $\begin{array}{l}=\text{Speed of motorboat + speed of current}\\ =\left(25+5\right)\text{miles per hour}\\ =30\text{ miles per hour}\end{array}$

Therefore, the speed of the current is $5\text{ miles per hour}$ and the speed of the motorboat with relative to the current in upstream is $20\text{ miles per hour}$ and in downstream $30\text{ miles per hour}$ .