Question

Raindrops make an angle$\theta$with the vertical when viewed through a moving train window (Fig. 3–52). If the speed of the train is$v_T$, what is the speed of the raindrops in the reference frame of the Earth in which they are assumed to fall vertically?

Short answer

The obtained value of the speed of the raindrops in the reference frame of the Earth in which they are assumed to fall vertically is ${\overrightarrow{v}}_0=\frac{v_T}{\tan \theta }$.

Step-by-step solution

Step 1. Vector analysis

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 $\theta$.

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:

${\overrightarrow{v}}_0={\overrightarrow{v}}_1+{\overrightarrow{v}}_2$

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

Step 2. Determine the speed of the raindrops

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

$\begin{array}{c}\tan \theta =\frac{{\overrightarrow{v}}_2}{{\overrightarrow{v}}_0}\\ \tan \theta =\frac{v_T}{{\overrightarrow{v}}_0}\\ {\overrightarrow{v}}_0=\frac{v_T}{\tan \theta }\end{array}$

Thus, the required speed is ${\overrightarrow{v}}_0=\frac{v_T}{\tan \theta }$.