A passenger in an automobile observes that raindrops make an...

A passenger in an automobile observes that raindrops make an angle of $30^{\circ}$ with the horizontal as the auto travels forward with a speed of $60 \mathrm{~km} / \mathrm{h}$. Compute the terminal (constant) velocity $\mathbf{v}_{r}$ of the rain if it is assumed to fall vertically.

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Key Concepts

Relative Motion

Relative motion involves analyzing the movement of objects with respect to different frames of reference. In this context, the observer in the car sees the raindrops moving at an angle due to the combination of the rain’s vertical fall and the car’s horizontal motion. This concept explains how velocities add vectorially, influencing the observed direction and magnitude of the moving objects.

Reference Frames

Reference frames are coordinate systems from which an observer measures positions, velocities, and accelerations. In problems like this, one frame is attached to the earth (where the rain falls vertically) and another is attached to the moving car (where the rain appears to fall at an angle). Transforming between these frames using relative velocity principles allows us to understand how motion is perceived differently by observers in different frames.

Terminal Velocity

Terminal velocity is the constant speed achieved by an object falling through a fluid when the force of gravity is balanced by drag forces. In the context of this problem, the rain is assumed to reach its terminal velocity, meaning it falls with a steady speed vertically. This is critical when analyzing the raindrops’ motion because it allows us to treat the vertical component of rain’s velocity as constant.

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