Program Explanation Given information: The differential equation for the projectile launched from the surface of the earth, attracted by earth and moon is given below. d 2 r d t 2 = − G M e r 2 + G M m ( S − r ) 2 ........... (1) Here r ( t ) is the distance from the center of the earth. Also M e and M m denote the masses of the earth and the moon, respectively. The radius of the earth is denoted by R and the distance between the centers of the earth and the moon is S = 384400 km . The initial conditions for the differential equation are given below. r ( 0 ) = R r ′ ( 0 ) = v 0 Explanation: Solve the differential equation (1), as follows: d 2 r d t 2 = − G M e r 2 + G M m ( S − r ) 2 d v d t = − G M e r 2 + G M m ( S − r ) 2 d v d r ⋅ d r d t = − G M e r 2 + G M m ( S − r ) 2 v d v d r = − G M e r 2 + G M m ( S − r ) 2 Further, solve the differential equation as, ∫ v d v = ∫ ( − G M e r 2 + G M m ( S − r ) 2 ) d r v 2 2 = − G M e ∫ 1 r 2 d r + G M m ∫ 1 ( S − r ) 2 d r v 2 2 = − G M e ( − 1 r ) + G M m ( 1 S − r ) + C v 2 2 = G M e r + G M m S − r + C Further solve the above differential equation as, v 2 2 = G [ M e ( S − r ) + M m r ] r ( S − r ) + C

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Recurrence relation

A recurrence relation or recursive relation is an equation that represents a function in terms of the values of its smaller inputs. Every recurrence relation T(n) is a recursive function of integer n and consists of a base case and a recursive case.

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Chapter 2.3, Problem 30P

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Program Description: Purpose of the problem is to obtain the minimal launch velocity v0 that suffices for the projectile to make it from the earth to the moon.

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Chapter 2.3, Problem 29P

Chapter 2.4, Problem 1P

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Chapter 2 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Program Description: Purpose of the problem is to obtain the minimal launch velocity v 0 that suffices for the projectile to make it from the earth to the moon.

Solution Summary: The author explains the differential equation for the projectile launched from the surface of the earth, attracted by earth and moon.

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Program Description: Purpose of the problem is to obtain the minimal launch velocity v0 that suffices for the projectile to make it from the earth to the moon.

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Chapter 2 Solutions