Question

Using the concept of relative motion, answer the following question. A swimmer swims (with respect to water) perpendicular to the current with acceleration a = 2t (where t is time) starting from rest form the origin O at t = 0. Velocity of the river with respect to ground is given by $v_0$ . Width of the river is d.

The time of crossing the river is

• A. $(d{)}^{\frac{1}{3}}$
• B. $(2d{)}^{\frac{1}{3}}$
• C. $(3d{)}^{\frac{1}{3}}$
• D. Information is insufficient

Answer 1

The correct option is C $(3d{)}^{\frac{1}{3}}$

We have from relative motion
${\overrightarrow{V}}_{SO}={\overrightarrow{V}}_{SW}+{\overrightarrow{V}}_{WO}$
$=v\hat{j}+v_0\hat{i}$
Where
$v=\int adt$
$=\int 2tdt$
$=t^2$ therefore,
${\overrightarrow{V}}_{SO}=v_0\hat{i}+t^2\hat{j}$
$\Rightarrow x=v_0$ t(since $v_0$ is constant)
If the time of travel is T, then $v=t^2$
$\Rightarrow \frac{dy}{dt}=t^2$
${\int }_0^{d}dy={}_0^T{t}^{2}dt$
$d=\frac{T^3}{3}\Rightarrow T=(3t{)}^{\frac{1}{3}}$