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Question: A river is flowing from west to east with a speed of \(5m{\min ^{ - 1}}\). A man can swim in still w...

A river is flowing from west to east with a speed of 5mmin⁡−15m{\min ^{ - 1}}. A man can swim in still water with a velocity of 10mmin⁡−110m{\min ^{ - 1}}. In which direction should the man swim so as to take the shortest possible path to go to the south?
A. 30∘{30^ \circ } east of south
B. 30∘{30^ \circ } west of south
C. South
D. 30∘{30^ \circ } west of north

Explanation

Solution

Here we have to first find the resultant directions and then find the sine of the given angle to get the answer.
Here we have to apply the concepts of relative motion velocity.

Complete step by step answer:

Given,
Speed of river =5mmin⁡−1 = 5m{\min ^{ - 1}}
Speed of the man in still water =10mmin⁡−1 = 10m{\min ^{ - 1}}
According to the diagram we can see that the man has to travel the distance PQ to go in the south direction.
Let, m=m = man, g=g = ground and r=r = river.
Since, the river is going from west to east, so,
{v_{rg}} = {v_{mg}}\sin \theta \\\
\implies \sin \theta = \dfrac{{{v_{rg}}}}{{{v_{mg}}}} \\\
\implies \sin \theta = \dfrac{5}{{10}} \\\
∴θ=30∘ \therefore \theta = {30^ \circ } \\\
So, the man will have to swim 30∘{30^ \circ } west of south.

So, the correct answer is “Option B”.

Additional Information:
The relative velocity of motion refers to an object that might be stationary, travelling at the same velocity, or slowly, moving at a higher velocity, or moving in the opposite direction relative to any other object.
Relative velocity is the rate of time at which one object’s relative motion varies relative to another. When two objects travel in the same direction, the value of the relative of one object relative to another is proportional to the value difference of the two velocities.
Absolute motion is the transfer of a body’s location from one absolute place to another; relative motion is the transfer of a body’s location from one relative position to another.

Note:
Here we have to be careful about the value of sine because if we swap the numerator and denominator then we shall get a different answer.
For motion along a straight line, the broad notion of relative can be very conveniently generalised to include motion in a plane or in three dimensions.