Why Do You Tilt an Umbrella Forward When Walking in Rain? The Physics of Relative Velocity
On a windless day the rain falls straight down. Yet when you walk, it hits you in the face and you tilt your umbrella forward. Nothing about the rain changed. What changed is the frame of reference: you are moving. This is relative velocity, and the lean angle is a one-line calculation.
Two points of view
Someone standing still on the pavement sees the rain falling vertically. You, walking forward, see the same raindrops coming at you from in front and above. Both are correct, because velocity depends on who is measuring it.
The rule for relative velocity
The velocity of the rain relative to you is the rain’s velocity minus your velocity: v(rain relative to you) = v(rain) − v(you). Taking away your forward velocity leaves the rain with a backward component. So from your point of view the rain moves down and backward, which means it comes from the front.
Worked example
Rain falls vertically at 8 m/s. A person walks at 2 m/s. In the walker’s frame the rain has components 8 m/s down and 2 m/s backward.
The angle from the vertical satisfies tan θ = 2 ÷ 8 = 0.25, so θ = 14.0°. The umbrella shaft should lean forward by 14.0°, pointing toward where the rain seems to come from. The speed at which the rain meets the walker is √(8² + 2²) = 8.25 m/s.
How the lean changes with speed
With the same 8 m/s rain, tan θ = walking speed ÷ 8, so the lean grows as you speed up:
- •Stroll at 1.5 m/s: lean 10.6° forward.
- •Walk at 2 m/s: lean 14.0°.
- •Brisk walk at 3 m/s: lean 20.6°.
- •Jog at 5 m/s: lean 32.0°.
- •Standing still (0 m/s): no lean at all.
Why the shaft points where the rain comes from
An umbrella blocks the most rain when the canopy faces the rain head-on, which means the shaft is lined up with the direction the rain seems to come from. Lean it less and your legs get wet. A small umbrella and a fast pace mean you may need to lean a little more than the formula says, because the canopy only covers a small area.
Try it: the walking-in-the-rain game
Our free DP Physics lesson on adding velocities has a walking-in-the-rain game. Change your speed, lean the umbrella, and watch the "body kept dry" meter. Switch between watching from the pavement (rain falls straight) and from the walker (rain slants) to see the same event from two frames.
Frequently asked questions
Because you are moving into it. Relative to you, the rain’s velocity is the rain’s velocity minus your velocity, which gives it a backward component, so it seems to come from in front.
Lean it forward by θ where tan θ = walking speed ÷ rain speed. For a 2 m/s walk in 8 m/s rain that is 14 degrees.
In still air it falls vertically relative to the ground. Relative to a moving person it falls at an angle. Both statements are true in their own frame of reference.
The method is the same: add or subtract velocity vectors with a triangle. The numbers and which side you know differ, so the trigonometry (tan instead of sin) differs.