Question

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 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 Me 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 sofar. (a) The work (energy) W needed to free yourself from Earth when its radius is R metres is W=\int_{R}^{\infty} G \frac{M_{E} \cdot m}{h^{2}} d h Show that this improper integral is equal to G \frac{M_{E} \cdot m}{R} 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 that W=\frac{1}{2} m v^{2} where v is the velocity at which you're escaping Earth (escape velocity). Using thisequation, show that v=\sqrt{\frac{2 G M_{E}}{R}} (c) Our result from part (b) tells us that, as Earth shrinks, we need a larger and larger velocity to escape. Find the radius of (shrunken) Earth, Rß, for which even light cannot escape (this is known as the Schwarzschild radius of Earth). Hint: Compute Rê by using the 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, MẸ, and c. (You may find that Råis about 9mm.)

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