Question

Wind is blowing in the north direction at a speed of 2 m/sec which causes the rain to fall at some angle with the vertical. With what velocity should a cyclist drive so that the rain appears vertical to him?
A. \[2\,{\text{m}}\,{{\text{s}}^{ - 1}}\] south
B. \[2\,{\text{m}}\,{{\text{s}}^{ - 1}}\] north
C. \[4\,{\text{m}}\,{{\text{s}}^{ - 1}}\] west
D. \[4\,{\text{m}}\,{{\text{s}}^{ - 1}}\] south

Answer 1

Hint:First of all, we will use the concept of relative velocity. We know, an object moving at a certain velocity appears to be still if an observer also continues its motion along the same direction with the same velocity. However, the same object appears to move fast, if the observer continues its motion in the opposite direction with the same velocity.

Complete step by step solution:
In the given problem, we are supplied the following data:
The direction in which the wind is blowing is along the north direction. The speed of the wind is \[2\,{\text{m}}\,{{\text{s}}^{ - 1}}\] .The rain is falling at a certain angle with the vertical. We are asked to find the velocity of a cyclist so that according to him the rain appears to be vertical.

To begin with, in this problem, we will introduce the concept of relative velocity. Relative velocity is not the actual velocity. It is the velocity that appears to an observer with respect to some other object or himself. The problem allows the rider to travel such that the relative velocity in the horizontal plane can be zero, such that vertical drops of rain continue to fall. Therefore, he has to travel at precisely the same speed as the rain drops in the horizontal plane, so that there is zero relative velocity in the horizontal plane. Now with the wind, the rain drops are borne away and have the same speed as the wind in the horizontal direction that is \[2\,{\text{m}}\,{{\text{s}}^{ - 1}}\] in the north direction, so he can travel in the north direction with \[2\,{\text{m}}\,{{\text{s}}^{ - 1}}\] .

If the horizontal and vertical components are equal, so the tangent of the angle equals to \[1\] . This however is that horizontal and vertical travel at the same speed. Thus, while the rain will land on the earth at an angle to an observer, the cyclist will have the same reference frame as the rain and it will seem to fall straight down.

Hence, the velocity of a cyclist so that according to him the rain appears to be vertical is \[2\,{\text{m}}\,{{\text{s}}^{ - 1}}\] north.The correct option is B.

Note: While answering this question remember that an individual standing still on the ground can see that the rain in the direction of the wind is falling at an angle. But if a person is riding a bicycle at a speed of 2 metres per second, also due north, then the person is going in the same direction and speed as the rain blowing wind. Direction of the cyclist can turn the tables around. If we do not consider the following facts, we will get a wrong answer.