First slide

Relative motion in 2 dimension

Question

Velocity of the river with respect to ground is given by $V_{0}$ and width of the river is ‘d’. A swimmer swims (with respective to water) perpendicular to the current with acceleration a=2t (where t is time ) starting from rest from the origin ‘O’ at t=0. The equation of trajectory of the path following by the swimmer

Difficult

A

$y = \frac{x^{3}}{3 V_{o}^{3}}$
B

$y = \frac{x^{2}}{2 V_{o}^{3}}$
C

$y = \frac{x^{2}}{V_{o}}$
D

$y = \sqrt{\frac{x}{V_{o}}}$

Solution

Displacement along the x-axis is only due to water velocity
$x = v_{o} \times t$
$\Rightarrow t = \frac{x}{V_{o}}$
Along the $Y -$ axis
$\begin{array}{l} a = 2 t (∵ a = \frac{d V}{d t}) \\ \Rightarrow \frac{d V}{d t} = 2 t \\ \Rightarrow d V = 2 t d t \end{array}$
Integrating on both sides
$\Rightarrow V = \frac{2 t^{2}}{2}$
$\Rightarrow \frac{d y}{d t} = t^{2} (∵ V = \frac{d y}{d t})$
$\Rightarrow d y = t^{2} d t$
Integrating on both sides
$\Rightarrow y = \frac{t^{3}}{3}$
$∴ y = \frac{x^{3}}{3 V_{o}^{3}}$

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