A car travels due east with a speed of 50.0 km/h. Rain is falling vertically with respect to Earth. The traces of the rain on the side windows of the car make an angle of 60.0° with the vertical. Find the velocity of the rain with respect to the following:
a. the car
b. Earth
I'm confused on how you are supposed to find the velocity in this question. There is no height given, so I don't think you can use the free fall equation, and I don't think it means after the rain hits the car either.
Let the positive $x$ direction be East, and let the positive $y$ direction be upward, then,
The velocity of the car is $\mathbf{v}_1 = (50, 0) $.
The unknown velocity of the rain is $\mathbf{v}_2 = (0, -a)$.
The minus sign is introduced because the rain is falling downward.
Relative velocity of rain with respect to the car is
$\mathbf{v}_{21} = b (- \sin 60^\circ, -\cos 60^\circ ) $
where the $a \gt 0$ is to be determined.
$\mathbf{v}_{21} = \mathbf{v_2} - \mathbf{v_1} = (0, -a) - (50, 0) = (-50, -a)$
Therefore,
$50 = b sin 60^\circ$
From which $b = \dfrac{100}{\sqrt{3}} $ and this is the speed of rain relative to the car.
Now $ a = b \cos 60^\circ = \dfrac{50}{\sqrt{3}} $ and this is the speed of rain relative to Earth.