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A man can row a boat with $4km/h$ in still water. If he is crossing a river where the current is $2km/h$ and the width of the river is $4km$. How long will it take him to row $2km$ up the stream and then back to his starting point?
(A) 2 hours
(B) 1 hour
(C) $\dfrac{4}{3}$hours
(D) None of the above.

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Last updated date: 13th Jun 2024
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Answer
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Hint: In the downstream motion, the boat and the water move in the same direction and thus the speed is more than what it will be when the two are moving opposite to each other. In other words, the upstream motion is slower.
For upstream motion of the boat, the velocity of the man is given by the difference of the relative velocity of the man to the river and the velocity of water. For downward motion of the boat, , the velocity of the man is given by the sum of the relative velocity of the man to the river and the velocity of water.

Complete Step by Step Solution: It has been given that a man can row a boat with $4km/h$ in still water. If he is crossing a river where the current is $2km/h$ and the width of the river is $4km$.
Thus, we can write that, velocity of man with respect to water ${v_{mw}} = 4km/h$.
Velocity of river, ${v_r} = 2km/h$.
Since, the width of the river is $4km$, it can be said that the distance covered in upstream or downward motion $d = 2km$.
The motion of the boat with respect to the river banks can be of two types.
One of the types is when the boat goes in the same direction as the direction of the river. This is what we know as the downstream motion. However, when the boat goes in the opposite direction as that of the stream or the river, it is what we call the upstream motion. In the downstream motion, the boat and the water move in the same direction and thus the speed is more than what it will be when the two are moving opposite to each other. In other words, the upstream motion is slower.
For upstream motion of the boat, the velocity of the man is given by the difference of the relative velocity of the man to the river and the velocity of water.
The velocity of the man ${v_m}$, is given by,
${v_m} = {v_{mw}} - {v_w} = \left( {4 - 2} \right)km/h$
$ \Rightarrow {v_m} = 2km/h$
The time taken to go upstream ${t_1} = \dfrac{d}{{{v_m}}} = \dfrac{2}{2} = 1hr$.
For downward motion of the boat, the velocity of the man is given by the sum of the relative velocity of the man to the river and the velocity of water.
The velocity of the man ${v_m}$, is given by,
${v_m} = {v_{mw}} + {v_w} = \left( {4 + 2} \right)km/h$
$ \Rightarrow {v_m} = 6km/h$
The time taken to go downstream ${t_2}$ is given by,
${t_2} = \dfrac{d}{{{v_m}}} = \dfrac{2}{6} = \dfrac{1}{3}hr$.
The total time taken to cross the river is given by the sum of the time taken to cross the river in upstream and downstream direction.
$\therefore $ Total time taken
$t = {t_1} + {t_2} = 1 + \dfrac{1}{3}$
$ \Rightarrow t = \dfrac{4}{3}hr$

Hence, the correct answer is Option C.

Note: A man sitting in the boat will feel that the boat is moving but the water is still. This is the velocity of the boat with respect to water or the speed of the boat in still water. This speed is constant irrespective of the motion of the boat i.e. for both upstream and downstream motions. We denote this speed by ‘$b$’.
The speed of the river with respect to the banks of the river is what we call the rate of the flow of the stream. This is what we denote with the letter ‘$r$’. Therefore the speed upstream is ($b - r$) and the speed downstream is ($b + r$).