Question

The current velocity of river grows in proportion to the distance from its bank and reaches the maximum value $v_0$ in the middle. Near the banks the velocity is zero. A boat is moving along the river in such a manner that the boatman rows his boat always perpendicular to the current. The speed of the boat in still water is $u$.(width of the river is $c$). If the trajectory of the boat is given as $y^2=\frac{aucx}{v_0}$. Find $a$.

Answer 1

Let the current velocity be given by $v_x=ky$

Since velocity is $v_0$ at $y=\frac{c}{2}$,

$v_0=k\frac{c}{2}$

$⟹k=\frac{2v_0}{c}$

$⟹v_x=\frac{2v_0}{c}y$

At some time $t$,

$y=ut$

But $v_x$ changes as boat travels in y direction

Hence, $x={\int }_0^t{v}_{x}dt={\int }_0^t\frac{2v_0}{c}ydt$

$={\int }_0^t\frac{2v_0}{c}utdt=\frac{2v_{0}u}{c}\frac{t^2}{2}=\frac{v_{0}u{t}^2}{c}$

$⟹x=\frac{v_0{y}^2}{cu}$

$⟹y^2=\frac{ucx}{v_0}$\

$⟹a=1$