Vertically falling rain makes slanted streaks on the side...

Vertically falling rain makes slanted streaks on the side windows of a moving automobile. If the streaks make an angle of $45^{\circ}$ , how does the speed of the automobile compare with the speed of the falling rain?

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Key Concepts

Reference Frames

Understanding reference frames is essential because the appearance of the rain’s path (i.e., the angle of the streaks) depends on the observer's perspective. The rain falls vertically in the stationary ground frame, but appears slanted in the moving car’s frame due to the relative motion between the car and the rain.

Trigonometric Relationships

Trigonometric relationships are used to relate the magnitudes of the horizontal and vertical velocity components via angles. A 45° angle indicates that these two components are equal in magnitude, which is key to comparing the speed of the automobile with the speed of the falling rain.

Relative Velocity

Relative velocity refers to the velocity of one object as observed from the reference frame of another object. In this context, it involves understanding how the motion of the rain appears different when viewed from the moving automobile compared to an observer on the ground.

Vector Components

Analyzing vector components involves breaking down a velocity into its horizontal and vertical elements. For this problem, the horizontal component is due to the automobile's motion and the vertical component is due to the falling rain, and considering these separately is crucial for interpreting the streak angle on the window.

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