Question bank/JEE Main/Physics

A river is flowing with a speed of $1\ kmh^{- 1}$. A swimmer
wants to go to point c starting from A. He swims with a speed of
$5\ kmh^{- 1}$ at an angle $\theta$w.r.t. the river flow, at what angle
with the river bank should the swimmer swim?

A30ºB37ºC53ºD60º

Show answer and solution▾

Path of swimmer relative to ground is a straight line at 45º
with bank, therefore, x and y components of swimmer's resultant
velocity must be equal.
Let velocity of swimmer$|{\overset{\rightarrow}{v}}_{m}| = v$
$V_{x} = V_{r} + V_{m,r}cos\theta$
$V_{y} = V_{m,r}sin\theta$
[IMAGE] Condition for reaching the point C,
$tan45^{\circ} = \frac{v_{y}}{v_{x}}$
$V_{y} = V_{x}$.
$(V_{r} + V_{m,r}cos\theta) = V_{m,r}sin\theta$
$1 + 5cos\theta = 5sin\theta$
On squaring$1 + 25\cos^{2}\theta + 10cos\theta = 25 - 25\cos^{2}\theta$
$50\cos^{2}\theta + 10cos\theta - 24 = 0$
On solving, we get$\theta = 53^{\circ}$

Source: Grass-G / P-16 Physics paper 25 Nov First.docx