Question

Rain is falling vertically downwards with a velocity \[3\;{\text{kmh}}{{\text{r}}^{ - 1}}\]. A man walks in the rain with a velocity of \[4\;{\text{kmh}}{{\text{r}}^{ - 1}}\]. The raindrop will fall on the man with a velocity of
(A) \[5\;{\text{kmh}}{{\text{r}}^{ - 1}}\]
(B) \[4\;{\text{kmh}}{{\text{r}}^{ - 1}}\]
(C) \[1\;{\text{kmh}}{{\text{r}}^{ - 1}}\]
(D) \[3\;{\text{kmh}}{{\text{r}}^{ - 1}}\]

Answer 1

Hint: In this question, the concept of the vector addition will be used that is, in the vector addition the magnitude and direction of the quantities are considered while calculating the resultant of the quantities. Here, the vector addition will be used for the rain velocity and the velocity of the man to obtain the velocity of the raindrop falling on the man.

Complete step by step answer:
In this question, the velocity of rain with respect to the ground is given as, \[3\;{\text{kmh}}{{\text{r}}^{ - 1}}\] and the velocity of man with respect to the ground is given as \[4\;{\text{kmh}}{{\text{r}}^{ - 1}}\].
Let us assume that the relative velocity is ${v_r}$.
As we know that the relative velocity can be expressed as,
$ \Rightarrow v_r^2 = v_1^2 + v_2^2 + 2{v_1}{v_2}\cos \theta $
Here, ${v_1}$ is the velocity of rain with respect to ground, ${v_2}$ is the velocity of man with respect to the ground and the $\theta $ is angle between the ground and the rain falling on the ground.
In this question the rain is falling vertically so the value of $\theta $ will be $90^\circ $.
Now we substitute the given values in the above equation.
$ \Rightarrow v_r^2 = {\left( 3 \right)^2} + {\left( 4 \right)^2} + 2\left( 3 \right)\left( 4 \right)\cos 90^\circ $
As we know that the cos of $90^\circ $ is zero, so the expression become,
$ \Rightarrow v_r^2 = {\left( 3 \right)^2} + {\left( 4 \right)^2} + 2\left( 3 \right)\left( 4 \right)\left( 0 \right)$
Now, we simplify the above expression as,
\[ \Rightarrow {v_r} = \sqrt {25} \]
As we know that the square root of $25$ is $5$, so the velocity will be,
\[\therefore {v_r} = 5\;{\text{kmh}}{{\text{r}}^{ - 1}}\]

Therefore, the correct option is A.

Note: As we know that the velocity is the vector quantity and the resultant of the velocities will be calculated by using the vector addition that is it depends on the direction of the velocities.