Question

Velocity of the river with respect to ground is given by $v_0$. Width of the river is $d$. A swimmer swims (with respect 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 followed by the swimmer is

• A. $y=\frac{x^3}{3v_0^3}$
• B. $y=\frac{x^2}{2v_0^2}$
• C. $y=\frac{x}{v_0}$
• D. $y=\sqrt{\frac{x}{v_0}}$

Answer 1

The correct option is A $y=\frac{x^3}{3v_0^3}$

From the given data, we have,
$\frac{d{v}_y}{dt}=2t$

Integrating w.r.t $t$,
$v_y=t^2+c$
Since $v_y=0$ at $t=0$, $c=0$
$\therefore v_y=\frac{dy}{dt}=t^2$

Integrating again w.r.t $t$
$y=\frac{t^3}{3}+c^{\prime }$

Again, we can choose the coordinate axes such that $y=0$ at $t=0$
$\therefore y=\frac{t^3}{3}...\left(i\right)$
Since in the horizontal direction, $x=v_{0}t⟹t=x/v_0$

Substituting in eq. (i) we have,
$y=\frac{x^3}{3v_0^3}$