Question

Wind is blowing in the north direction at a speed of $2m/s$ which causes the rain to fall at some angle with the vertical. With what velocity should a cyclist drive so that the horizontal component of rain appears $2m/s$ south to him?
(A) $2m/s$ south
(B) $2m/s$ north
(C) $4m/s$ north
(D) $4m/s$ south

Answer 1

Hint: We know that the relative velocity is the net velocity of one body with respect to another body. When the first body and second body both are in motion, then the relative rate is the difference between the velocities of the first and second.

Complete step-by-step solution:
Let’s take the north direction as $ + \hat i$.
Since the wind is blowing in the north direction which causes the rain to fall at some angle with the vertical.
So, the horizontal component of the velocity of rain can be represented as,
${V_R} = + 2\hat im/s$
According to the question, the horizontal component of rain appears in the south direction to the cyclist.
Therefore the velocity of rain with respect to the cyclist can be represented as,
${V_{R/C}} = - 2\hat i$
And also, the relative of rain with respect to the cyclist can be determined as,
${V_{R/C}} = {V_R} - {V_C}$
where, ${V_C}$ is the velocity of the cyclist.
We will now substitute ${V_{R/C}} = - 2\hat im/s$ , ${V_R} = + 2\hat im/s$ in the above equation.
${V_{R/C}} = {V_R} - {V_C}$
$ \Rightarrow - 2\hat i = + 2\hat i - {V_C}$
On solving the above equation for ${V_C}$ we get,
${V_C} = 2\hat i + 2\hat i = 4\hat i$
Thus, the magnitude of the velocity of the cyclist is $4m/s$ in $ + \hat i$ direction i.e. North direction.

Hence, the correct answer is option (C) $4m/s$ north.

Additional information:
We know that after the resultant velocity is calculated, the direction of the velocity is also of utmost importance. The line of the resultant of two vectors is close to the vector with large magnitude, and the direction of the resultant velocities and vectors are generally calculated with the positive direction of the x-axis in radians or degree. If the angle is taken in an anti-clockwise order from the positive x-axis, then the value of the angle is positive and vice-versa.

Note: We must follow a proper sign convention for determining the relative velocities and taking the directions appropriately. The velocity in the positive x-direction is taken as positive, and the velocity in the positive y-direction is taken as positive.