Problem 1. Let R be the radius of earth, M be the mass of earth, and G be the gravitational constant of earth. Chris wishes to use a jetpack to fly infinitely far away from earth. Due to fuel constraints, he can start at a velocity vo >0 and then can accelerate no more. On the other hand, earth accelerates Chris according to the law of gravity. Letting r(t) be Chris's distance from the center of the earth at time t, we thus have MG "(t)= r(t)² Here r' (t) is the velocity at time t and r"(t) is the acceleration at time t. (i) What are the initial values r(0) and r'(0) in terms of the constants in the paragraph above? (ii) Multiply both sides of (+) by -r' (t), Then integrate both sides of it between t= 0 and t = tf. Use u-substitution on each side to deduce (iii) Justify that (ty) ½v6 = ½ 7′(1,)² + [267) MG dy. MG 6 ≥ 2 2 fra 1 (+) 2 dy. (iv) The goal (Chris flying infinitely far away from earth) means what about r(t) as tfoo? Use this to right the right hand side of (**) in terms of an improper integral, and compute this integral. (v) Use the result of (iv) to give a condition² on to so that Chris will make it infinitely far away from earth.
Problem 1. Let R be the radius of earth, M be the mass of earth, and G be the gravitational constant of earth. Chris wishes to use a jetpack to fly infinitely far away from earth. Due to fuel constraints, he can start at a velocity vo >0 and then can accelerate no more. On the other hand, earth accelerates Chris according to the law of gravity. Letting r(t) be Chris's distance from the center of the earth at time t, we thus have MG "(t)= r(t)² Here r' (t) is the velocity at time t and r"(t) is the acceleration at time t. (i) What are the initial values r(0) and r'(0) in terms of the constants in the paragraph above? (ii) Multiply both sides of (+) by -r' (t), Then integrate both sides of it between t= 0 and t = tf. Use u-substitution on each side to deduce (iii) Justify that (ty) ½v6 = ½ 7′(1,)² + [267) MG dy. MG 6 ≥ 2 2 fra 1 (+) 2 dy. (iv) The goal (Chris flying infinitely far away from earth) means what about r(t) as tfoo? Use this to right the right hand side of (**) in terms of an improper integral, and compute this integral. (v) Use the result of (iv) to give a condition² on to so that Chris will make it infinitely far away from earth.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 1. Let R be the radius of earth, M be the mass of earth, and G be the gravitational
constant of earth. Chris wishes to use a jetpack to fly infinitely far away from earth. Due to
fuel constraints, he can start at a velocity vo >0 and then can accelerate no more. On the
other hand, earth accelerates Chris according to the law of gravity. Letting r(t) be Chris's
distance from the center of the earth at time t, we thus have
MG
r(t)²
Here r' (t) is the velocity at time t and r"(t) is the acceleration at time t.
(i) What are the initial values r(0) and r'(0) in terms of the constants in the paragraph
above?
"(t).
(iii) Justify that
==
(ii) Multiply both sides of (+) by -r' (t), Then integrate both sides of it between t= 0 and
t = tf. Use u-substitution on each side to deduce
r(ty)
½v6 = 17′(1,)² + (x6¹7) MG dy.
(+)
(1) MG
1 ≥ 2
(iv) The goal (Chris flying infinitely far away from earth) means what about r(t) as
tfoo? Use this to right the right hand side of (**) in terms of an improper
integral, and compute this integral.
(v) Use the result of (iv) to give a condition² on vo so that Chris will make it infinitely
far away from earth.
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