# A man can swim in still water with a speed of 3 m/s. x and y axes are drawn along and normal to the bank of the river flowing to right with a speed of 1 m/s. The man starts swimming from origin O at t=0 second. Assume size ............?

## A man can swim in still water with a speed of 3 m/s. $x$ and $y$ axes are drawn along and normal to the bank of the river flowing to right with a speed of 1 m/s. The man starts swimming from origin $O$ at $t = 0$ second. Assume size of man to be negligible. Find the equation of locus of all possible points where man can reach at $t = 1 \sec$?

Dec 29, 2017

Suppose the swimmer swims with velocity $3$ m/s in the direction making an angle $\theta$ with the bank i.e positive direction of X-axis,OX

So the velocity components of the swimmer will be

${V}_{O X} = 3 \cos \theta$

and

${V}_{O Y} = 3 \sin \theta$

As the river is flowing along $O X$

Net velocity along $O X$ will be $3 \cos \theta + 1$

These two velocities are independent on each other as they are orthogonal. The swimmer starts at origin $O$ If the displacement of the swimmer after 1 sec along X-axis and Y- axis be $x \mathmr{and} y$ respectively then

$x = 3 \cos \theta + 1. \ldots . . \left[1\right]$
and
$y = 3 \sin \theta \ldots \ldots \ldots \ldots . \left[2\right]$

From [1] and [2] we get

${\left(x - 1\right)}^{2} + {y}^{2} = {3}^{2} {\cos}^{2} \theta + {3}^{2} {\sin}^{2} \theta = 9$

So equation of locus of all possible points where man can reach at t=1sec will be

$\textcolor{m a \ge n t a}{{\left(x - 1\right)}^{2} + {y}^{2} = {3}^{2}}$, the possible positions are on the blue semicircular line of radius 3m and center $C \left(1 , 0\right)$ as shown in figure above.