# The Dry Canvas Problem

Image: Cloe Liane Shasha

• ### Gravity's Engines

We’ve long understood black holes to be the points at which the universe as we know it comes to an end. Often billions of times more massive than the Sun, they...

Have you ever wondered whether you would get wetter if you ran through a rain storm rather than walked? At which angle should you incline your head to minimize the amount of water that hits you? This puzzle asks the same question, but concerns moving an object through the rain: a museum’s large, precious Delacroix canvas.

To start, assume your museum’s canvas is of height H and side-to-side width W. Raindrops fall vertically at velocity vr . It is raining steadily and heavily and you don't want the canvas to get wet.

Warm-Up:
If the canvas is waterproof on its very top surface (and the water that hits the top will flow to the sides of the canvas without dripping on it) but the canvas is equally sensitive to water in the front and back, how fast and at which angle should you move the canvas so the front and back receive as little rain as possible?

Solution to Warm-Up:
When the canvas moves with velocity vc no rain will fall on the front or back of the canvas if the canvas is tilted forward at an angle theta (from the vertical direction) where theta = arctan(vc / vr ) and vr is the velocity of the raindrops.

In case your trigonometry is rusty, recall that the tangent of a right triangle is the opposite length over the adjacent length. If a raindrop starts in front of the canvas it won't hit the canvas (the canvas will never catch up) and if it starts behind, it will never catch up to the canvas.

The curator finds the prospect of keeping a steady pace a little worrisome, so he manages to find an awning that can be attached to the top of the canvas.

Problems:
1. If there is an awning of length A attached at right angles to the top of the canvas of height H, and if both the canvas and the rain are vertical,then how fast should you move to ensure that no rain hits the front or back of the canvas and you arrive to the destination as quickly as possible?

This strategy doesn't work well if there is a head wind, especially a variable wind. In that case, you may want to tilt the canvas and swivel the awning in surprising ways.

2. Suppose that the awning can be swiveled to any position relative to the canvas. The head wind  vw varies from 0 to 1.5 meters per second. The vertical velocity of the raindrops vr is 5 meters per second, the height H of the canvas is 3 meters, and the awning A is 0.8 meters off the canvas. With which velocity vc must you move? At which angle should the canvas be to the ground? At which angle should the awning be to the canvas? Does it help to move faster?

By Dennis E. Shasha, Courant Institute of Mathematical Sciences, New York University.

Dennis's most recent puzzle book, Puzzles for Programmers and Pros, was published this past May by John Wiley and Sons/Wrox.

Dennis would also like to congratulate the two lone solvers of the contest puzzles in Puzzlers Elusion: a tale of fraud, pursuit, and the art of logic (Avalon/Perseus books). Those extremely hard puzzles were solved by a high school teacher Deborah Logan and a financial trading strategist Michael Birken.

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