November 2007 Puzzle Solution

Join Our Community of Science Lovers!

1. With an awning of length A at right angles to the canvas and the canvas vertical, no rain drops will hit the canvas if v_c / v_r = A/H. This follows directly by looking at the figure.

2. The rain velocity v_r is the vertical component of the velocity of the rain, while the wind moves with velocity v_w. By the Pythagorean theorem therefore, the rain moves at a speed of sqrt(v_r² + v_w²)with respect to the ground. These are illustrated in the following figures where the canvas of height H moves from left to right with an awning of length A. Segment T hits the bottom horizontal axis at the leftmost point at which it could hit without causing rain to hit the bottom of the canvas. That point is the base of the vertical segment B.

First, the canvas will be tilted forward in the same way as would be required for zero wind velocity. The triangles HBD and the one formed by velocities v_c and v_r are similar; therefore,

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

D = H v_c /sqrt(v_c² + v_r² ) B = H v_r / sqrt(v_c² + v_r² )

Secondly, it is clear from the cartoon that the optimum position of the swiveled awning is such that the drop trajectory T at maximum wind velocity (just missing the bottom point p of the canvas) is a tangent to the circle with the radius A and with the center at the upper end of the canvas. The triangles TEF and BTA are similar because the angle between A and T is 90 degrees as is the angle between E and F. Also, the angle between F and T is 90 minus the angle between B and T and therefore it is equal to the angle between A and B. This implies that the angle between B and T is equal to the angle between T and E. Hence all three angles are the same in the two triangles so TEF and BTA are similar, when the wind pushes the rain as far left as possible while still missing the canvas. Further the triangle TEF and the one formed by velocities v_w and v_r is similar. Intuitively, the awning takes care of the wind velocity v_w whereas the incline of the painting takes care of the movement velocity v_c. This intuition is tempered only by the fact that the incline of the painting determines the length of B. The smaller B, the less the danger the canvas will be hit.

So, we have

A/B = F/T
F/T = v_w /sqrt(v_w² + v_r² )
A = B F/T = H v_r v_w /sqrt{(v_c² + v_r² ) (v_w² + v_r² )}.

(v_c² + v_r² ) (v_w² + v_r² ) = (H v_r v_w/ A)²

v_c²(v_w² + v_r² ) = (H v_r v_w/ A)²– v_r²(v_w² + v_r² )
v_c²= ((H v_r v_w/ A)²/(v_w² + v_r² ) ) – v_r²

From that it follows
v_w= (A/H) v_r sqrt{(v_c² + v_r² )/[ v_r² - (A/H)²(v_c² + v_r² )]}

Similarly, the canvas velocity is given as
v_c = v_r sqrt{( v_wH /A)² /(v_w² + v_r² ) – 1}

So, suppose the raindrops are falling 5 meters per second, the height is 3 meters, and the awning A is 0.8 meters off the canvas. How fast should you move if the headwind velocity may range from 0 up to 1.5 meters per second? What angle should the canvas be to the ground? What angle should the awning be to the canvas?

You have to move at least 2 meters per second (about twice a normal walking pace). If so, the canvas will be tilted about 22 degrees from the vertical and the awning will be about 95 degrees from the canvas (but still slanted downwards to the right). Moving faster will force a greater tilt which will allow you to tolerate a greater windspeed. Move fast, stay dry.

The solutions are due to Ivan Rezanka.

It’s Time to Stand Up for Science

If you enjoyed this article, I’d like to ask for your support. Scientific American has served as an advocate for science and industry for 180 years, and right now may be the most critical moment in that two-century history.

I’ve been a Scientific American subscriber since I was 12 years old, and it helped shape the way I look at the world. SciAm always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does that for you, too.

If you subscribe to Scientific American, you help ensure that our coverage is centered on meaningful research and discovery; that we have the resources to report on the decisions that threaten labs across the U.S.; and that we support both budding and working scientists at a time when the value of science itself too often goes unrecognized.

In return, you get essential news, captivating podcasts, brilliant infographics, can't-miss newsletters, must-watch videos, challenging games, and the science world's best writing and reporting. You can even gift someone a subscription.

There has never been a more important time for us to stand up and show why science matters. I hope you’ll support us in that mission.

Thank you,

David M. Ewalt, Editor in Chief, Scientific American