Question

A man is trying to cross a river to cross a river flowing at a speed of $5m{s^{ - 1}}$ to least possible displacements by swimming at an angle of $143^\circ$ to the stream. The drift he suffers when he is crossing the same river in least possible time is (Width of river = 1km).

1. 160m
2. 800m
3. 400m
4. 200m

Answer 1

This is a classic question of kinematics. Here we need to resolve the angle into sin and cos relative to the river and equate it with the velocity of the man swimming. Then find out the least time and put it in the speed-distance formula and find out the drift.

Complete step by step solution:
The man is crossing a river which is flowing at a speed of 5m/s to least possible displacements by swimming at an angle of $143^\circ$ to the stream.
So, the deflection would be:
$\sin (143 - 90) = \sin 53^\circ$;
Now, the velocity is 5m/s:
${v_m}\sin 53^\circ = 5$;
$\Rightarrow {v_m} \times \left( {\dfrac{4}{5}} \right) = 5$;
Do the needed calculation:
$\Rightarrow {v_m} \times \left( {\dfrac{4}{5}} \right) = 5$;
$\Rightarrow {v_m} = \dfrac{{25}}{4}$;
Now, the drift he suffers when he is crossing the same river in least possible time is:
$v = \dfrac{d}{t}$ ;
Write the above formula in terms of time t:
$\Rightarrow t = \dfrac{d}{v}$;
Put in the given value in the above equation:
$\Rightarrow t = \dfrac{{1000}}{{25}} \times 4$;
$\Rightarrow t = 160s$;
Now, for the drift:
$D = v \times t$;
Put the given value in the above relation:
$D = 5 \times 160$
$\Rightarrow D = 800m$;

Final Answer: Option “2” is correct. Therefore, the drift he suffers when he is crossing the same river in the least possible time is 800m.

Note: Here we have been given the angle of swimming so, relative to the river it would be 90 minus the angle of swimming. The given velocity would be equal to the horizontal component of the velocity times the relative angle. To find out the drift we need to find the least time, apply the formula for time equals distance upon velocity.