Relative Velocity Of Rain And Observer

The umbrella must be aligned with the velocity of the rain relative to the cyclist, not with the rain's velocity relative to the ground, because in the cyclist's moving frame the rain appears to approach from a slanted direction. The relative velocity is found by subtracting the cyclist's velocity from the rain's velocity. The rain has velocity $(0,-10)\text{ m/s}$ (downward) and the cyclist has velocity $(10,0)\text{ m/s}$ (horizontal). The rain relative to the cyclist is $(0-10,-10-0)=(-10,-10)\text{ m/s}$, meaning it appears to come down and toward the cyclist from the front. The tilt angle from the vertical satisfies $\tan \theta =\frac{v_{horizontal}}{v_{vertical}}=\frac{10}{10}=1$, so $\theta =45^{\circ }$, and the umbrella must lean forward into the direction of motion. The option '45 degrees away' has the correct magnitude but the wrong direction of tilt. The options 30 and 60 degrees use incorrect velocity ratios. This is the NCERT rain-man relative-velocity result. The deeper idea is that what protects the cyclist is the apparent direction of the rain in the cyclist's own frame, which always tilts toward the direction of travel because the observer's motion adds a forward-pointing component to the rain's apparent velocity. As a check, equal horizontal and vertical relative components must give a symmetric 45-degree slant, confirming both the magnitude and the forward lean of the answer.