Physics and Mathematics
Relative Velocity of Rain w.r.t. a Moving Man
1. Introduction
Relative velocity describes the velocity of one object as observed from another moving object.
When it rains and a man walks, the rain appears to fall at an angle due to the relative motion between the rain and the man.
To the man, the rain’s velocity combines its vertical fall and his horizontal motion.
2. Understanding the Scenario
- Let the rain be falling vertically downwards with velocity [v_r] (relative to the ground).
- The man walks horizontally with velocity [v_m] (relative to the ground).
- To the man, the rain seems to come at an angle because he perceives both its vertical and horizontal components.
3. SI Unit and Dimensional Formula
- SI Unit: [m/s]
- Dimensional Formula: [M^0 L^1 T^{-1}]
4. Velocity Components
- Vertical component of rain’s velocity: [v_r] (remains unchanged).
- Horizontal component: due to the man’s motion, equals [v_m].
The rain’s relative velocity with respect to the man:
[v_{rm} = \sqrt{v_r^2 + v_m^2}]
Angle at which rain appears to fall (with vertical):
[\tan \theta = \dfrac{v_m}{v_r}]
Where:
- [\theta] = angle of apparent direction of rain with the vertical.
5. Important Formulas to Remember
| No. | Formula | Description |
|---|---|---|
| 1 | [v_{rm} = \sqrt{v_r^2 + v_m^2}] | Relative velocity of rain with respect to a moving man. |
| 2 | [\tan \theta = \dfrac{v_m}{v_r}] | Angle of apparent direction of rain with the vertical. |
| 3 | [v_r] = vertical velocity of rain | Component perpendicular to ground. |
| 4 | [v_m] = horizontal velocity of man | Component parallel to ground. |
6. Graphical Understanding
Imagine arrows representing velocities:
- A vertical arrow for [v_r] (rain’s downward velocity).
- A horizontal arrow for [v_m] (man’s velocity).
- The diagonal resultant arrow represents [v_{rm}], the apparent velocity of the rain as seen by the man.
7. Practice Questions
- A man walks at 4 m/s and rain falls vertically at 3 m/s. Find the angle at which rain appears to fall.
Solution:
[\tan \theta = \dfrac{v_m}{v_r} = \dfrac{4}{3} \implies \theta = \tan^{-1}\left(\dfrac{4}{3}\right)]
⇒ [\theta \approx 53.1^\circ] with the vertical. - Rain is falling vertically at 6 m/s. A man walks at 8 m/s. Find the relative velocity of the rain with respect to the man.
Solution:
[v_{rm} = \sqrt{v_r^2 + v_m^2}] = [\sqrt{6^2 + 8^2} = \sqrt{36 + 64}] = [\sqrt{100} = 10 , \text{m/s}] - A man walks at 5 m/s towards the east while the rain falls at 12 m/s vertically downwards. Find the angle at which he should hold his umbrella.
Solution:
[\tan \theta = \dfrac{v_m}{v_r} = \dfrac{5}{12} \implies \theta = \tan^{-1}\left(\dfrac{5}{12}\right)]
⇒ [\theta \approx 22.6^\circ] with the vertical, tilted towards the opposite direction of motion.
8. Conceptual Questions
1. Why does rain appear slanted to a moving person but vertical to a stationary person?
Because of relative motion: the horizontal component is due to the person’s movement.
2. Does the rain’s vertical velocity change when the man starts walking?
No. The vertical velocity remains unchanged; only the horizontal component appears due to the man’s motion.
3. Why should a man tilt his umbrella forward when walking in rain?
To block the apparent direction of rain caused by the combined effect of rain’s vertical velocity and his own horizontal motion.
4. If the man’s speed increases, what happens to the angle of apparent rain?
The angle with the vertical increases because the horizontal component increases.
5. Is relative velocity always greater than the actual velocity of rain?
Yes, because it combines both vertical and horizontal components using vector addition.
9. FAQs / Common Misconceptions
1. Does rain actually fall at an angle when the man walks?
No. Rain continues to fall vertically relative to the ground; the slant is only apparent to the moving man.
2. Does holding the umbrella vertically protect a walking man from rain?
No. The umbrella must be tilted forward to counteract the apparent slant of rain.
3. If the man walks opposite to the rain’s slant, will it fall vertically to him?
If the man moves in the opposite direction with a suitable speed equal to the horizontal component, the rain appears vertical to him.
4. Is the relative velocity vector always perpendicular to the ground?
No. It is diagonal, forming an angle with the vertical.
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