Question

A man can swim in still water at a speed 2 times that of the river. Then at what angle downstream he must swim to cross the river along the shortest possible path?

Answer 1

Hint: In this question, we will use the relation between the velocity and the angle, by observing the diagram. Further, by substituting the values and solving for the angle, we will get the required answer. Also, we will discuss the basic difference between speed and velocity.
Formula used:
$2v\cos \theta = v$

Complete step by step solution:
Here, for the shortest path man velocity will be perpendicular with respect to the river.
Now, let us suppose that the velocity of the river is v and the velocity of the man is 2v.

From the given figure, we can write:
$2v\cos \theta = v$
$ \Rightarrow \cos \theta = \dfrac{1}{2}$
Now, putting the value, we get:
$\theta = {60^ \circ }$
Now, to find the angle with downstream the man must swim to cross the river with shortest distance will be given by:
${\theta _0} = {180^ \circ } - \theta $
By substituting the value of $\theta $ in above equation and solving, we get:
$\eqalign{ & {\theta _0} = {180^ \circ } - {60^ \circ } \cr
  & \therefore {\theta _0} = {120^ \circ } \cr} $
Therefore, we get the required answer, which tells the angle downstream the man must swim to cross the river with the shortest possible path.

Additional information:
We already know the basic difference of speed and velocity i.e., speed is the measure of how fast an object can travel, whereas velocity tells us the direction of this speed. Speed is a scalar quantity that means it has only magnitude, whereas velocity is a vector quantity that means it has both magnitude and direction. The S.I unit of velocity is meter per second (m/sec).
Also, as we know a speedometer also called a speed meter is a gauge that measures and displays the instantaneous speed of a vehicle. It is mostly mounted on bikes, cars, scooters. We also know that the speedometer used to measure the speed of a vehicle and the entire vehicle indicate speed in km/h.

Note:
We should know that a straight line makes an angle of 180 degrees. The angle more than 360 degrees is known as reflex angle. Also, angle is measured in two units that are: degree and radian. One should also remember that the velocity is a vector quantity i.e., it has both magnitude and direction.