Rain and Wind Vector Simulation
Determining the optimal umbrella angle when wind blows
Theory
When rain is falling vertically and wind starts blowing horizontally, the apparent direction of rain changes due to the vector addition of the rain's velocity and the wind's velocity.
Given in the problem:
Vertical rain velocity (\( v_r \)) = 35 m/s downward
Horizontal wind velocity (\( v_w \)) = 12 m/s (east to west)
Resultant velocity (R):
Magnitude: \( R = \sqrt{v_r^2 + v_w^2} = \sqrt{35^2 + 12^2} = 37 \, \text{m/s} \)
Direction: \( \theta = \tan^{-1}\left(\frac{v_w}{v_r}\right) = \tan^{-1}\left(\frac{12}{35}\right) \approx 19^\circ \) west of vertical
The boy should hold his umbrella at an angle of about 19° with the vertical towards the east (opposite the wind direction) to protect himself from the rain.
