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Question: A man can swim in still water with a speed of \(3\dfrac{m}{s}\), x and y axis are drawn along and no...

A man can swim in still water with a speed of 3ms3\dfrac{m}{s}, x and y axis are drawn along and normal to the bank of river flowing to right with a speed of 1ms1\dfrac{m}{s}. The man starts swimming from origin O at t=0st = 0s. Assume the size of man to be negligible. Find the equation of locus of all the possible points where man can reach at t=1st = 1s.

Explanation

Solution

A man can swim in still water with a speed of three metres per second, so we can say that the man is swimming a distance of three metres in one second. Similarly, one metre per second can be said as one metre distance travelled in one second. With the help of this concept, we will find the coordinates of the given point and hence, we will get the equation of the locus.

Complete step by step answer:
Considering that the river is still, then the man would have travelled a distance of 3m3m from origin OO after a time period of 1sec⁡1\sec . The locus of any point at which the man can reach at t=0st = 0s is a semicircle which is of a radius 3m3m and the centre is present at OO.
We know that the river flows towards the right at a speed of 1ms1\dfrac{m}{s}.
Therefore, there should be an additional shift in position by 1ms1\dfrac{m}{s}, that is 1sec⁡=1m1\sec = 1m towards the right. So, the locus of every point giving possible position after a time period of one second will be a semicircle shifted to right by 1m1m.
Thus, the locus of every point where the man can be at t=1st = 1s is also a semicircle which is of a radius t=1st = 1s and centres at O′(1m,0m){O'}(1m,0m).

Hence, the equation of locus of all the points comes out to be,
(x−1)2+(y−0)2=32{(x - 1)^2} + {(y - 0)^2} = {3^2}
On simplifying the above equation, we get,
(x−1)2+y2=9{(x - 1)^2} + {y^2} = 9
The equation of locus of all the possible points where man can reach at t=1st = 1s is (x−1)2+y2=9{(x - 1)^2} + {y^2} = 9 which is a circle.

Note:
The above equation (x−1)2+y2=9{(x - 1)^2} + {y^2} = 9 is the equation of a circle. By equating it with the standard equation of the circle, we will get the radius of this circle as 3m3m and the centre is present at the point (1,0)(1,0).