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Problem: Insufferable Enlightenment era scientists are a lot like insufferable frat boys/sorority girls - they just cannot get enough of themselves. One of the most well-known "big brain" contests in their history was a Europe-wide challenge presented by a certain Swiss scientist called Johann Bernoulli (who already had his own solution so that he could pompously show off once the submissions came in). He asked the "leading minds of the world" the following: What is the curve along which a particle, under constant gravitational acceleration g, would descend from one arbitrary point to another within the fastest duration of time? Fancily called the brachistochrone problem (derived from the Greek word for just "shortest time"), Bernoulli's initial solution used the law of refraction as the result of applying the principle of least time to a light particle. Let's give this solution a shot ourselves. (a) Consider an initial point of (xo, 0), both parameterised by time t. Our goal is to find the curve y(x) such that a particle along it would descend to a different point (x1, y₁) in the fastest time it can under constant gravity. In general, ₁ x0. Let us take the negative y-direction to be positive in y. What is then the speed v of a particle at any position (x, y) along such a curve? (b) Since the principle for light worked under the assumption that light is just some particle, it should also work effectively the same when we want any other particle to minimise its time. Bernoulli's insight was, just like how Fermat divided a system into two different media with different propagation speeds, to slice up a particle's descent path into infinitesimal "medium" zones, each zone being a place where the particle travels at some speed v. Let us consider two such slices with speed v₁ and v2 respectively, what is the equivalent of the law of refraction in terms of the angles 0₁ and 02 the trajectory in each slice makes with the normal?

Problem: Insufferable Enlightenment era scientists are a lot like insufferable frat boys/sorority girls - they just cannot get enough of themselves. One of the most well-known "big brain" contests in their history was a Europe-wide challenge presented by a certain Swiss scientist called Johann Bernoulli (who already had his own solution so that he could pompously show off once the submissions came in). He asked the "leading minds of the world" the following: What is the curve along which a particle, under constant gravitational acceleration g, would descend from one arbitrary point to another within the fastest duration of time? Fancily called the brachistochrone problem (derived from the Greek word for just "shortest time"), Bernoulli's initial solution used the law of refraction as the result of applying the principle of least time to a light particle. Let's give this solution a shot ourselves. (a) Consider an initial point of (xo, 0), both parameterised by time t. Our goal is to find the curve y(x) such that a particle along it would descend to a different point (x1, y₁) in the fastest time it can under constant gravity. In general, ₁ x0. Let us take the negative y-direction to be positive in y. What is then the speed v of a particle at any position (x, y) along such a curve? (b) Since the principle for light worked under the assumption that light is just some particle, it should also work effectively the same when we want any other particle to minimise its time. Bernoulli's insight was, just like how Fermat divided a system into two different media with different propagation speeds, to slice up a particle's descent path into infinitesimal "medium" zones, each zone being a place where the particle travels at some speed v. Let us consider two such slices with speed v₁ and v2 respectively, what is the equivalent of the law of refraction in terms of the angles 0₁ and 02 the trajectory in each slice makes with the normal?

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Problem: Insufferable Enlightenment era scientists are a lot like insufferable frat boys/sorority girls - they just cannot get enough of themselves. One of the most well-known "big brain" contests in their history was a Europe-wide challenge presented by a certain Swiss scientist called Johann Bernoulli (who already had his own solution so that he could pompously show off once the submissions came in). He asked the "leading minds of the world" the following: What is the curve along which a particle, under constant gravitational acceleration g, would descend from one arbitrary point to another within the fastest duration of time? Fancily called the brachistochrone problem (derived from the Greek word for just "shortest time"), Bernoulli's initial solution used the law of refraction as the result of applying the principle of least time to a light particle. Let's give this solution a shot ourselves. (a) Consider an initial point of (xo, 0), both parameterised by time t. Our goal is to find the curve y(x) such that a particle along it would descend to a different point (x₁, y₁) in the fastest time it can under constant gravity. In general, x₁ #x0. Let us take the negative y-direction to be positive in y. What is then the speed v of a particle at any position (x, y) along such a curve? (b) Since the principle for light worked under the assumption that light is just some particle, it should also work effectively the same when we want any other particle to minimise its time. Bernoulli's insight was, just like how Fermat divided a system into two different media with different propagation speeds, to slice up a particle's descent path into infinitesimal "medium" zones, each zone being a place where the particle travels at some speed v. Let us consider two such slices with speed v₁ and v2 respectively, what is the equivalent of the law of refraction in terms of the angles 0₁ and 02 the trajectory in each slice makes with the normal?
Transcribed Image Text:Problem: Insufferable Enlightenment era scientists are a lot like insufferable frat boys/sorority girls - they just cannot get enough of themselves. One of the most well-known "big brain" contests in their history was a Europe-wide challenge presented by a certain Swiss scientist called Johann Bernoulli (who already had his own solution so that he could pompously show off once the submissions came in). He asked the "leading minds of the world" the following: What is the curve along which a particle, under constant gravitational acceleration g, would descend from one arbitrary point to another within the fastest duration of time? Fancily called the brachistochrone problem (derived from the Greek word for just "shortest time"), Bernoulli's initial solution used the law of refraction as the result of applying the principle of least time to a light particle. Let's give this solution a shot ourselves. (a) Consider an initial point of (xo, 0), both parameterised by time t. Our goal is to find the curve y(x) such that a particle along it would descend to a different point (x₁, y₁) in the fastest time it can under constant gravity. In general, x₁ #x0. Let us take the negative y-direction to be positive in y. What is then the speed v of a particle at any position (x, y) along such a curve? (b) Since the principle for light worked under the assumption that light is just some particle, it should also work effectively the same when we want any other particle to minimise its time. Bernoulli's insight was, just like how Fermat divided a system into two different media with different propagation speeds, to slice up a particle's descent path into infinitesimal "medium" zones, each zone being a place where the particle travels at some speed v. Let us consider two such slices with speed v₁ and v2 respectively, what is the equivalent of the law of refraction in terms of the angles 0₁ and 02 the trajectory in each slice makes with the normal?
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