Question

River stream velocity grows in proportion to the distance from the bank and reaches its maximum velocity $2m{s}^{-1}$ in the middle. Near the bank velocity is zero. The velocity of a swimmer in still water is $5m{s}^{-1}$ and is directed perpendicular to river stream. The width of river is $100m$. The drifting in swimmer is $5n$ meter. The value of $n$ is _____.

Answer 1

Answer: 4

$v_x=ky$ and $v_y=5$

$\therefore y=5t,$ At middle point, $50=5t$
$\therefore t=10s$
$\because v_x=ky$
At middle, $v_x=2$
$\therefore k=\frac{2}{50}$
$\therefore v_x=\frac{2}{50}y=\frac{y}{25}=\frac{5t}{25}=\frac{t}{5}$
$\Rightarrow \frac{dx}{dt}=\frac{t}{5}$
$\therefore \underset{0}{\overset{x_0}{\int }}dx=\underset{0}{\overset{10}{\int }}\frac{t}{5}dt$
$\therefore x_0=10m$ up to middle
$\therefore$ Total drifting $=2\times 10=20m$
$\therefore n=4$