Question

A row-boat is observed travelling downstream on a flowing river at a speed of
$8\,m/s$ with respect to the shore. A motor-boat comes from the opposite direction at a speed of
$10\,m/s$ with respect to the water. 10 seconds after meeting each other they are at 160 meters from each other.
(a) At what speed does the river flow?
(b) Find the speed of the row-boat in still water.

Answer 1

Express the velocity of row-boat and motor-boat with respect to still water using the quantities given in the question. The combined velocity of the two boats is the velocity that we require to calculate the velocity of the river.

Formula used:
$s = vt$
Here, s is the displacement, v is the velocity and t is the time.

Complete step by step answer:
(a) Assume the velocity of river water is ${v_r}$, the velocity of the row-boat with respect to river is ${v_{rb}}$ and the velocity of the motor boat with respect to river is ${v_{mb}}$.
Since the speed of row-boat with respect to shore is $8\,m/s$ and the row-boat is moving along the downstream of the river, we can write,
${v_{rb}} + {v_r} = 8$
${v_{rb}} = 8 - {v_r}$ …… (1)
The motor-boat is moving along the opposite direction of the river flow, we can write the velocity of the motor-boat with respect to shore as follows,
${v_{mb}} - {v_r} = 10$
${v_{mb}} = {v_r} + 10$ …… (2)
Since the boats move in the opposite directions, the velocity of the boats is also in the opposite direction as shown in the figure below,

We can express the combined velocity of motor boat and row boat as follows,
$v = {v_{mb}} - {v_{rb}}$
The negative sign for velocity of a row-boat implies that the boat is travelling along the negative x-axis.
Using equation (1) and (2) we can rewrite the above equation as follows,
$v = \left( {{v_r} + 10} \right) - \left( {8 - {v_r}} \right)$
$\Rightarrow v = 2{v_r} + 2$
We use the relation between velocity, distance and time to determine the velocity of river as follows,
$s = vt$
Substitute 160 m for s, $2{v_r} + 2$ for v and 10 s for t in the above equation.
$160 = \left( {2{v_r} + 2} \right)\left( {10} \right)$
$\Rightarrow 14 = 2{v_r}$
$\Rightarrow {v_r} = 7\,m/s$
Therefore, the river is flowing with the speed 7m/s.

(b)Substitute 7 m/s for ${v_r}$ in equation (1).
${v_{rb}} + 7 = 8$
$\Rightarrow {v_{rb}} = 1\,m/s$
Therefore, the speed of a river-boat with respect to still water is 1 m/s.

Note:
After you get the answer of the velocity of the row-boat, check whether the speed of the motor-boat is greater or less than the row-boat. It should be greater than the speed of a row-boat because it is motor-driven. Define the direction of motor-boat and row-boat and take the difference in their speeds to determine the total velocity.