91Ó°ÊÓ

In this problem, consider the x-direction as the direction of the train or the direction the passenger is facing, and the y-direction to be upward. Use the relation of velocities in determining the required raindrops’ speed.

Given data:

The angle made by the raindrops is $\mathrm{θ}$.

The speed of the train is $v_T$.

The following is the diagram used to find the relation of the speed of the raindrops.

The relation of velocities can be written as:

${\stackrel{→}{v}}_0={\stackrel{→}{v}}_1+{\stackrel{→}{v}}_2$

Here, ${\stackrel{→}{v}}_0$is the velocity of the raindrops relative to the ground, ${\stackrel{→}{v}}_1$is the velocity of the raindrops relative to the train, and ${\stackrel{→}{v}}_2$is the velocity of the train relative to the ground.

The relation of the angle from the diagram is given by:

$\begin{array}{c}\tan \mathrm{θ}=\frac{{\stackrel{→}{v}}_2}{{\stackrel{→}{v}}_0}\\ \tan \mathrm{θ}=\frac{v_T}{{\stackrel{→}{v}}_0}\\ {\stackrel{→}{v}}_0=\frac{v_T}{\tan \mathrm{θ}}\end{array}$

Thus, the required speed is ${\stackrel{→}{v}}_0=\frac{v_T}{\tan \mathrm{θ}}$.