Question

While driving north at $25\mathrm{m}/\mathrm{s}$ during a rainstorm you notice that the rain makes an angle of $38°$ with the vertical. While driving back home moments later at the same speed but in the opposite direction, you see that the rain is falling straight down. From these observations, determine the speed and angle of the raindrops relative to the ground.

Short answer

The speed and angle of the raindrops relative to the ground is $68.7\mathrm{m}/\mathrm{s}$and$21.3°$ respectively.

Step-by-step solution

Given Information

While driving north at $25\mathrm{m}/\mathrm{s}$ during a rainstorm you notice that the rain makes an angle of $38°$ with the vertical.

Calculation

The velocity of rain with respect to the car with vertical direction make angle of $38°$.

role="math" localid="1647939501776" $$\begin{gathered} \tan 38°=\frac{{\mathrm{v}}_1+{\mathrm{v}}_2}{{\mathrm{u}}_2} \\ {\mathrm{u}}_2=\frac{25+25}{\tan 38°} \\ {\mathrm{u}}_2=64\mathrm{m}/\mathrm{s} \end{gathered}$$

The speed and angle of the raindrops relative to the ground is:

role="math" localid="1647939626538" $$\begin{gathered} \mathrm{v}=\sqrt{{{\mathrm{u}}_2}^2+{{\mathrm{v}}_2}^2} \\ \mathrm{v}=\sqrt{{64}^2+{25}^2} \\ \mathrm{v}=68.7\mathrm{m}/\mathrm{s} \\ \mathrm{\theta }={\tan }^{-1}\left(\frac{{\mathrm{v}}_2}{{\mathrm{u}}_2}\right) \\ \mathrm{\theta }={\tan }^{-1}\left(\frac{25}{64}\right) \\ \mathrm{\theta }=21.3° \end{gathered}$$