DUE TOMORROW!!!)? D(\alpha ) = (\alpha - \beta ) + (\pi - 2\beta ) + (\alpha - \beta) = \pi + 2\alpha - 4\beta. \qquad \qquad (4) That results from the part of a ray that enters a raindrop and is refracted at A, reflected twice (at B and C), and refracted as it leaves the drop at D (see the figure at the left).
Calculus: Early Transcendentals 8th Edition answers to Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises - Page 302 40 including work step by step written by community members like you. \cos\alpha = \sqrt\frac{7}{27}, \Rightarrow \alpha = \arccos\sqrt\frac{7}{27} \approx 59.4^\circ Answers and Replies Related Other Physics Topics News on Phys.org. 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. Applied Project in Sec.4.1, Calculus by Stewart $$ Applied Project in Sec.4.1, Calculus by Stewart Chinese version: 彩虹的微積分學 Rainbows are created when raindrops scatter sunlight. \qquad \qquad (2) 2\cos\beta = 3\cos\alpha. Some of the light is reflected, but the line AB shows the path of the part that enters the drop. 274 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION APPLIED PROJECT The Calculus of Rainbows 1. $$ Scribd will begin operating the SlideShare business on December 1, 2020 Textbook Authors: Stewart, James , ISBN-10: 1285741552, ISBN-13: 978-1-28574-155-0, Publisher: Cengage Learning As of this date, Scribd will manage your SlideShare account and any content you may have on SlideShare, and Scribd's General Terms of Use and Privacy Policy will apply. View Notes - Chapter 4 12 from MTH 222 at Essex County College. \cos\alpha = \sqrt{\frac{k^2-1}{8}}, \Rightarrow \alpha = \arccos\sqrt{\frac{k^2-1}{8}}. Notice that the light is refracted toward the normal line AO and in fact Snell"s Law says that $\sin\alpha=k \sin\beta$, where $\alpha$ is the angle of incidence, $\beta$ is the angle of refraction, and $k\approx\frac{4}{3}$ is the index of refraction for water. (The angle of incidence equals the angle of reflection.) One can solve (1) and (2) to have So the rainbow really consists of seven individual bows corresponding to the seven colors. $$ For red light: $k\approx 1.3318\Rightarrow \alpha_1\approx 59.473^{\circ}$, the rainbow angle$= \pi - D(\alpha_1)\approx 42.3^{\circ}$. Since $D(\alpha ) = 2\alpha - 6\beta + 2\pi$, we look for the solution of $D’(\alpha)=0$ to obtain This is from a ray that enters a raindrop and is refracted at A, reflected twice at B&C and refracted as it leaves the drop at D THE CALCULUS OF RAINBOWS Part 1: Location Part 1: Looks like you’ve clipped this slide to already. $$ Join Yahoo Answers and get 100 points today. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. $$. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. It it the concentration of rays coming from near the direction of minimum deviation that creates the brightness of the primary rainbow. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. \qquad \qquad (1)
i dont have a clue on how to do this and it's due tomorrow for a test grade. If you continue browsing the site, you agree to the use of cookies on this website. We teat the angle $\alpha$ as a function of $\beta$ and take derivative with respect to $\alpha$ to Snell’s law to obtain $$ Apr 12, 2020, Sec.3.1 - Building a better roller coaster, Sec.4.7 - Planes and birds: Minimizing energy. This time the deviation angle $D(\alpha)$ is the total amount of counterclockwise rotation that the ray undergoes in this four-stage process. By repeating the calculation of Problem 1 for these values of $k$, show that the rainbow angle is about $42.3^{\circ}$ for red bow and $40.6^{\circ}$ for the violet bow.
$$ Calculus reveals all the important aspects of graphs of functions. When the ray reaches C, part of it is reflected, but for the time being we are more interested in the part that leaves the raindrop at C.(Notice that it is refracted away from the normal line.) (This angle is called the rainbow angle .). $$ Learn more. When the great mathematician Isaac Newton explained the colours of the rainbow with refraction the poet John Keats was horrified.
The angle of deviation $D(\alpha )$ is the amount of clockwise rotation that the ray has undergone during this three-stage process. $$ Therefore, the primary rainbow is lower than the secondary rainbow, and the colors in the secondary rainbow appear the opposite order from those in the primary rainbow. Show that $D(\alpha ) = 2\alpha - 6\beta + 2\pi$ and $D(\alpha)$ has a minimum value when $\cos\alpha = \sqrt\frac{k^2-1}{8}$. \qquad \qquad (5) 2\cos\alpha = k\cos\beta. AP Calculus Project 2010, describes the calculus behind the rainbow. $$ Ask Question + 100. Perhaps you have seen a fainter secondary rainbow above the primary bow. $$ One can solve (3) and (4) to have PLZZZ help me in anyway you can thank you! … For red light the refractive index $k \approx 1.3318$, whereas for violet light it is $k\approx 1.3435$. Trending Questions. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. \frac{d\beta}{d\alpha} = \frac1 2 = \frac{3\cos\alpha}{4\cos\beta}. For violet light: $k\approx 1.3435\Rightarrow\alpha_2\approx 58.786^{\circ}$, the rainbow angle $=\pi - D(\alpha_2)\approx 40.6^{\circ}$.
$$ Clipping is a handy way to collect important slides you want to go back to later. In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows. D(59.4^\circ)\approx138^\circ. For $k=\frac43\Rightarrow \alpha\approx 71.849^{\circ}\Rightarrow D(\alpha)\approx231^{\circ}$. \sin\alpha =k\sin\beta. Accessibility, "The Mathematics of the Rainbow, Part II. Similar to the approach of problem 1, we first write down the Snell’s law ----------, Supplement to the previous paper. CALCULUS OF RAINBOW project (PLZ! As Newton discovered in his prism experiments of 1666, the index of refraction is different for each color, (The effect is called dispersion.) One can solve (3) and (5) to have 彩虹的微積分學. Last updated on
So if we want to understand it, we look at what happens to all the rays hitting the rainbow and coming from a fixed direction. $$ The Calculus of Rainbows By Angela Han, Rabeea Abbas, Laura Boardman, and Cathy Ma Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. \qquad \qquad (3) Part 3: Secondary Rainbow Part 2: Colors There is a secondary rainbow above the primary rainbow.
Show that the colors in the secondary rainbow appear in the opposite order from those in the primary rainbow. So you do not need to waste the time on rewritings. $$ So we find the minimal value $$ The Calculus of Rainbows
By Angela Han, Rabeea Abbas, Laura Boardman, and Cathy Ma
. 3\cos\alpha = k\cos\beta.
Show that the minimum value of deviation is $D(\alpha)\approx 138^{\circ}$; and occurs when $\alpha\approx 59.4^{\circ}$.
But what I was trying to explain was why the colors are in a specific order.
$$ Thus But as we will see, the mathematical explanation, requiring just the basic geometry of … Based on (6) we find, for red light: $k\approx 1.3318\Rightarrow \alpha_1\approx 71.906^{\circ}$, the rainbow angle: $D(\alpha_1)-\pi\approx 50.6^{\circ}.$, For violet light: $k\approx 1.3435\Rightarrow\alpha_2\approx 71.505^{\circ}$, the rainbow angle: $D(\alpha_2)-\pi\approx 53.6^{\circ}.$. Chinese version: Problem 1 explains the location of the primary rainbow, but how do we explain the colors? The total deflection of the ray from its original direction is D = 180+2i-4r, where i is the angle of incidence when the ray first hits the drop and r the refraction angle at first contact.. For a rainbow, all rays come from the same direction, that of the sun. 8am to 4pm? Keats complained (through poetry of course) that a mathematical explanation robbed these marvels of nature of their magic, conquering "all mysteries by rule and line". See our User Agreement and Privacy Policy.
To find the minimum value of $D(\alpha)$, we look for the solution of $D’(\alpha)=0$:
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