3. Let's imagine that the earth is shrinking and we want to escape before it is too late. Let'sset up some notation:R: the radius of EarthMẸ : the mass of Earthm: your massG : the universal gravitational constantc: the speed of lightNote that, since Earth is shrinking, R is not constant, but MẸ is constant (the values of ME,G and c are available on Wikipedia). In this question, we will compute the velocity neededto escape Earth and the radius of (shrunken) Earth for which even light cannot escape (whenEarth becomes a black hole). In fact, all of the related formulae are well known and thepurpose of this question is to justify our work using what we have learned in this course sofar.(a) The work (energy) W needed to free yourself from Earth when its radius is R metres isw = GME m dh.h2RShow that this improper integral is equal toGME mR Your justification should show all necessary steps of the computation including the defi-nition of the improper integral.(b) By the Work-Energy Principle, we know that1w = zmv,where v is the velocity at which you're escaping Earth (escape velocity). Using thisequation, show that2GMEv =(c) Our result from part (b) tells us that, as Earth shrinks, we need a larger and largervelocity to escape. Find the radius of (shrunken) Earth, RB, for which even light cannotescape (this is known as the Schwarzschild radius of Earth). Hint: Compute Rg by usingthe fact that it occurs exactly when the escape velocity is equal to the speed of light c.Your answer should be an expression in terms of G, ME, and c. (You may find that RBis about 9mm.)

Answered: Image /qna-images/answer/f5d49fea-c841-4fb4-b9d0-940c80fd8796.jpg

3. Let's imagine that the earth is shrinking and we want to escape before it is too late. Let's set up some notation: R: the radius of Earth ME : the mass of Earth m : your mass G : the universal gravitational constant c: the speed of light Note that, since Earth is shrinking, R is not constant, but Mg is constant (the values of ME, G and c are available on Wikipedia). In this question, we will compute the velocity needed to escape Earth and the radius of (shrunken) Earth for which even light cannot escape (when Earth becomes a black hole). In fact, all of the related formulae are well known and the purpose of this question is to justify our work using what we have learned in this course so far. (a) The work (energy) W needed to free yourself from Earth when its radius is R metres is W = GME m dh. Show that this improper integral is equal to ‚ME m R

3. Let's imagine that the earth is shrinking and we want to escape before it is too late. Let's set up some notation: R: the radius of Earth ME : the mass of Earth m : your mass G : the universal gravitational constant c: the speed of light Note that, since Earth is shrinking, R is not constant, but Mg is constant (the values of ME, G and c are available on Wikipedia). In this question, we will compute the velocity needed to escape Earth and the radius of (shrunken) Earth for which even light cannot escape (when Earth becomes a black hole). In fact, all of the related formulae are well known and the purpose of this question is to justify our work using what we have learned in this course so far. (a) The work (energy) W needed to free yourself from Earth when its radius is R metres is W = GME m dh. Show that this improper integral is equal to ‚ME m R

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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