Question

A man is crossing a river flowing with velocity of \[5m/s\]. He reaches a point directly across at a distance of 60m in 5sec. His velocity in still water should be

\[\begin{align}
  & \text{A}\text{. 12}m/s \\
 & \text{B}\text{. 13}m/s \\
 & \text{C}\text{. 5}m/s \\
 & \text{D}\text{. 10}m/s \\
\end{align}\]

Answer 1

Hint: Velocity of the river, width of the river and the time taken by man to cross the river is given and we have to find velocity of man in still water. Velocity is a vector quantity and it also gives direction. By using the formula of velocity and addition of vectors we can find the velocity of man in still water.

Formula used:
\[\begin{align}
  & V=\dfrac{s}{t} \\
 & V_{m}^{2}=V_{r}^{2}+V_{R}^{2} \\
\end{align}\]

b>Complete answer:
A man was at point A of the river and crossed the river to reach at point B. The width of the river or the distance between A and B is 60m and the time taken to cross is 5 second.
As the river is flowing with velocity \[5m/s\], the man should swim in the direction C as when he dives in the river he will experience the force of it and due to which there will be a shift in his position.

The velocity of river is \[{{V}_{r}}\] and the velocity of the man is \[{{V}_{m}}\] then the resultant velocity will be \[{{V}_{R}}\], the resultant velocity is the velocity which a man will have after taking the velocity of river in consideration.
As velocity is a vector quantity because it gives us magnitude and direction, the above diagram can be given as

This is a vector diagram and from the addition of vectors we can write
\[V_{m}^{2}=V_{r}^{2}+V_{R}^{2}\text{ }..................\text{(i)}\]
Now the value of velocity of the river is given \[{{V}_{r}}=5m/s\]and the value of resultant velocity can be given by the standard formula of velocity I.e.
\[V=\dfrac{s}{t}\]
Where S displacement and t is time. Here the width of river is the displacement i.e. 60m and time taken to cover it is 5 seconds, hence the resultant velocity is given as
\[\begin{align}
  & {{V}_{R}}=\dfrac{60}{5} \\
 & {{V}_{R}}=12m/s \\
\end{align}\]
Substituting the values of \[{{V}_{R}}\text{ and }{{V}_{r}}\]in equation (i) we get
\[\begin{align}
  & V_{m}^{2}={{5}^{2}}+{{12}^{2}} \\
 & V_{m}^{2}=25+144 \\
 & V_{m}^{2}=169 \\
 & {{V}_{m}}=\sqrt{169} \\
 & {{V}_{m}}=13m/s \\
\end{align}\]
Hence, the velocity of man is \[13m/s\] and this will be the velocity in still water too.

So, the correct answer is “Option B”.

Note:
Distance and displacement has a slight difference, the distance just gives as magnitude but displacement gives us magnitude and direction as in the given question we have given the displacement when we consider the motion of the man as it tells us the direction in which he is moving which is from A to B.