Question

A motor ship covers the distance of $300km$ between two localities on a river in $10hrs$ downstream and in $12hrs$ upstream. Find the flow velocity of the river assuming that these velocities are constant.
$A)2.0km{{h}^{-1}}$
$B)2.5km{{h}^{-1}}$
$C)3km{{h}^{-1}}$
$D)3.5km{{h}^{-1}}$

Answer 1

Hint: During downstream, relative velocity of a ship with respect to the river is equal to the sum of the velocity of the ship and the velocity of flow of water in the river. During upstream, relative velocity of a ship with respect to the river is equal to the difference in velocities of the ship and the flow of the river.

Formula used:
$1){{v}_{downstream}}={{v}_{s}}+{{v}_{r}}$
$2){{v}_{upstream}}={{v}_{s}}-{{v}_{r}}$
$3){{t}_{downstream}}=\dfrac{D}{{{v}_{downstream}}}$
$4){{t}_{upstream}}=\dfrac{D}{{{v}_{upstream}}}$

Complete step-by-step answer:
Let us consider a motor ship moving between two localities, in a river. It is given that the ship covers a distance of $300km$ between these two localities. We are also provided that the time taken by the ship to cover this distance downstream is $10hrs$ and the time taken by the ship to cover this distance upstream is $12hrs$. We are required to find the velocity of flow of the river.
Let us assume that the velocity of flow of the river is ${{v}_{r}}$ and velocity of ship is ${{v}_{s}}$. We know that during downstream, relative velocity of ship with respect to river is given by
${{v}_{downstream}}={{v}_{s}}+{{v}_{r}}$
where
${{v}_{downstream}}$ is the relative velocity of ship with respect to river, downstream
${{v}_{r}}$ is the velocity of river
${{v}_{s}}$ is the velocity of ship
Let this be equation 1.
We also know that during upstream, relative velocity of ship with respect to river is given by
${{v}_{upstream}}={{v}_{s}}-{{v}_{r}}$
where
${{v}_{upstream}}$ is the relative velocity of ship with respect to river, upstream
${{v}_{r}}$ is the velocity of river
${{v}_{s}}$ is the velocity of ship
Let this be equation 2.

Moving on, we know that distance covered by the ship is equal to the product of time taken by the ship to cover the distance and the relative velocity of the ship with respect to river, downstream or upstream. If we denote the distance covered as $D$, the time taken downstream as ${{t}_{downstream}}$ and the time taken upstream as ${{t}_{upstream}}$, we have
${{t}_{downstream}}=\dfrac{D}{{{v}_{downstream}}}\Rightarrow 10hrs=\dfrac{300km}{({{v}_{s}}+{{v}_{r}})}\Rightarrow {{v}_{s}}+{{v}_{r}}=30km{{h}^{-1}}$
where
${{t}_{downstream}}=10hrs$, as provided in the question
$D=300km$, as provided
${{v}_{downstream}}={{v}_{s}}+{{v}_{r}}$, from equation 1
Let this be equation 3.
Similarly,
${{t}_{upstream}}=\dfrac{D}{{{v}_{upstream}}}\Rightarrow 12hrs=\dfrac{300km}{({{v}_{s}}-{{v}_{r}})}\Rightarrow {{v}_{s}}-{{v}_{r}}=25km{{h}^{-1}}$
where
${{t}_{upstream}}=12hrs$, as provided in the question
$D=300km$, as provided
${{v}_{upstream}}={{v}_{s}}-{{v}_{r}}$, from equation 2
Let this be equation 4.
Subtracting equation 4 from equation 3, we have
${{v}_{s}}+{{v}_{r}}-{{v}_{s}}+{{v}_{r}}=30km{{h}^{-1}}-25km{{h}^{-1}}\Rightarrow 2{{v}_{r}}=5km{{h}^{-1}}\Rightarrow {{v}_{r}}=2.5km{{h}^{-1}}$
Therefore, velocity of the ship is equal to $2.5km{{h}^{-1}}$. The correct option to be marked is $B$.

Note: The concepts of downstream and upstream can be easily understood by considering ourselves rowing a boat, in a flowing river. When we row along the direction of flow of the river, our speed increases and we are said to be moving downstream. At the same time, when we row in the opposite direction of flow of the river, our speed decreases and we are said to be moving upstream.