Question

Math 20 - Calculus I $4.2 - Extra Credit Applied Project: The Calculus of Rainbows Show your work! Good luck! (40pts) Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows. 1. The figure shows a ray of sunlight entering a spherical raindrop at A. Some of the light is reflected, but the line AB shows the path of the part that enters the drop. Notice that the light is refracted toward the normal line AO and in fact Snell's Law says that sing = ksing where is the angle of incidence. Bis the angle of refraction, and is the index of refraction for water. At B some of the light passes through the drop and is refracted into the air, but the line BC shows the part that is reflected. The angle of incidence equals the angle of reflection.) When the ray reaches C. part of it is reflected, but for the time being we are more interested in the part that leaves the raindrop at C. (Notice that it is refracted away from the normal line.) The angle of deviation D(a)is the amount of clockwise rotation that the ray has undergone during this three-stage process. Thus D(a) = (a - b) + (1 - 2B) + (@-B) = 1 + 2a - 48 Formation of the primary rainbow Show that the minimum value of the deviation is D(a) - 138 and occurs when a 59.4'. The significance of the minimum deviation is that when a = 59.4' we have D'(a) -0.50 0. This means that many rays with a 59.4 become deviated by approximately the same amount. It is the concentration of rays coming from near the direction of minimum deviation that creates the brightness of the primary rainbow.

Answer 1

1
Equation D(a) + q [having quantities a, b ] D(a) = (a - b) + (1 - 2B) + (a - b) D(a) = 180 + 20 - 4B sin a Sink +e (used equation | B = sin" (sina) D(a) = 180 + 20 - 4 sin" (sin a)

D'(a) + f (used function | D(a) = 180 + 2a-4 Lot 06) = ca [ 18020 – asin "sima) + [used result derivative of sin^-1(xl 4cosa D(a) = 2- Vo - Csip 9) 4k cosa 1 D(a) = 2-- Vk - sinak 4 cosa Da = 2-T - sinéa

+ e [used equation | D(a) = 0 4 cosa 2. VR-sina 50 4(k? - sin?a) = 16 cosa k? - sin’a = 4(1 - sinʼa) 3 sin’a = 4 - k + a [ with specific amount k = 4/3] am = 59.4

+ sina [used equation - =k sin P= sin" (sina) B = sin -11 +e (with specific amount a_m = 59.4 B-sin" (sin 59,4) B =40.21° D(a) + q [having quantities a, B] Da = 180 + 20 - 43 + a[ with specific amounta_m = 59.4, B = 40.21 ] D(a) = 180 + 2(59.4) – 4(40.21) D(a) = 138°