Question

A person can swim in still water at $5m{{s}^{-1}}$. He moves in a river of velocity $3m{{s}^{-1}}$, first down the stream next same distance up the stream. Then the ratio of the two times is
A. $1:1$
B. $1:2$
C. $1:4$
D. $4:1$

Answer 1

Hint: While going down the stream, a person swims along the flow of water and due to his speed increases. Whereas, while going up the stream his speed decreases because he swims against the flow. Calculate the time taken in both the cases and divide them.

Formula used:
${{v}_{up}}={{v}_{s}}+{{v}_{r}}$
${{v}_{down}}={{v}_{s}}-{{v}_{r}}$
$v=\dfrac{d}{t}$

Complete step by step answer:
It is given that a person is able to swim at a speed of $5m{{s}^{-1}}$ in still water, i.e. when the water is not moving. Before proceeding further, let us understand what we mean when we say that the person is swimming up the stream or down the stream.When the person is swimming down the stream, he is travelling along the flow of the water. This means that he is swimming in the direction of the flow of the water.

When the person is swimming up the stream, he is travelling against the flow of the water. This means that he is swimming in the direction that is opposite to the direction of the flow of the water.While going down the stream, a person swims along the flow of water and due to his speed increases. Whereas, while going up the stream his speed decreases because he swims against the flow.

If the speed of the river is ${{v}_{r}}$ and the speed of the person in still water is ${{v}_{s}}$, then his speed in downstream his speed is ${{v}_{up}}={{v}_{s}}+{{v}_{r}}$.
And in upstream his speed is ${{v}_{down}}={{v}_{s}}-{{v}_{r}}$.
In the given case ${{v}_{r}}=3m{{s}^{-1}}$ and ${{v}_{s}}=5m{{s}^{-1}}$.
This means that ${{v}_{down}}=5+3=8m{{s}^{-1}}$
And ${{v}_{up}}=5-3=2m{{s}^{-1}}$.

It is said that the person moves equal distances down the stream and up the stream. Let that distance be d.Now, let us sue the formula $v=\dfrac{d}{t}$,
where d is the distance travelled by the person, v is his speed and t is the time taken to cover the distance.
$\Rightarrow {{v}_{down}}=\dfrac{d}{{{t}_{down}}}$ and ${{v}_{up}}=\dfrac{d}{{{t}_{up}}}$
Substitute the values of ${{v}_{down}}$ and ${{v}_{up}}$.
$\Rightarrow 8=\dfrac{d}{{{t}_{down}}}$ …. (i)
And
$2=\dfrac{d}{{{t}_{up}}}$ …. (ii).
Now, divide (ii) by (i).
$\Rightarrow \dfrac{2}{8}=\dfrac{\dfrac{d}{{{t}_{up}}}}{\dfrac{d}{{{t}_{down}}}}$
$\therefore \dfrac{{{t}_{down}}}{{{t}_{up}}}=\dfrac{1}{4}$.
Therefore, the ratio of the two times is $1 : 4$.

Hence, the correct option is C.

Note: Sometimes students may get confused between upstream and downstream and may interchange their meaning. Always remember that to move down the stream is to swim along the flow and to move up the stream is to swim against the flow.