Question

A man can swim in a still water with a speed of 2m/s. If he wants to cross a river of water speed √3m/s along the shortest path then in which direction should he swim​

Answer 1

answer : at an angle 150° to the water current.

explanation : A man can swim in a still water with a speed of 2m/s.

i.e., speed of man , $v_m$ = 2m/s

now he wants to cross a river of water speed √3m/s along shortest path.

it is possible only when horizontal component of velocity of man with respect to river will be zero.

Let man starts to swim at an angle α with vertical line.

then, velocity of man, $V_m=-2sin\alpha\hat{i}+2cos\alpha\hat{j}$

so, velocity of man with respect to river , $V_{mr}=V_m-V_r$

=$-2sin\alpha\hat{i}+2cos\alpha\hat{j}-\sqrt{3}\hat{i}$

= $(\sqrt{3}-2sin\alpha)\hat{i}+cos\alpha\hat{j}$

so, horizontal component of velocity of man with respect to river = (√3 - 2sinα) = 0

or, √3 = 2sinα

or, sinα = √3/2 = sin60°

or, α = 60°

hence, he should swim, at an angle (60° + 90°) = 150° to the water current.