Program Explanation Given information: The differential equation for the projectile launched straight upward from the surface of the earth is given below. d 2 y d t 2 = − G M ( y + R ) 2 ........... (1) Here, y ( t ) is the height above the surface. The initial velocity v 0 is defined by the below inequality. v 0 < 2 G M R The initial conditions for the differential equation are given below. y ( 0 ) = 0 y ′ ( 0 ) = v 0 Explanation: Solve the differential equation (1), as follows: d 2 y d t 2 = − G M ( y + R ) 2 d v d t = − G M ( y + R ) 2 d v d y ⋅ d y d t = − G M ( y + R ) 2 v d v d y = − G M ( y + R ) 2 Further, solve the differential equation as, v d v d y = − G M ( y + R ) 2 ∫ v d v = ∫ ( − G M ( y + R ) 2 ) d y v 2 2 = − G M ∫ 1 ( y + R ) 2 d y Further solve the above differential equation as, v 2 2 = − G M y + R + C ........... (2) Impose the initial condition on the equation (2) and obtain the value of C . C = v 0 2 2 + G M R Therefore, equation (2) can be written as shown below. v 2 2 = − G M y + R + v 0 2 2 + G M R v 2 2 = v 0 2 2 + G M R − G M y + R v 2 2 = v 0 2 2 + G M ( 1 R − 1 y + R ) v 2 = v 0 2 − 2 G M y R ( y + R ) Therefore, it is proved that v 2 = v 0 2 − 2 G M y R ( y + R ) for the differential equation d 2 y d t 2 = − G M ( y + R ) 2 with initial condition y ( 0 ) = 0 and y ′ ( 0 ) = v 0

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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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 2.3, Problem 29P

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Program Description: Purpose of the problem is to obtain the maximum altitude that projectile achieved, if its initial velocity is 1 km/s .

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Chapter 2.3, Problem 28P

Chapter 2.3, Problem 30P

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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 maximum altitude that projectile achieved, if its initial velocity is 1 km / s .

Solution Summary: The author explains how to solve the differential equation for the projectile launched straight upward from the surface of the earth.

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Program Description: Purpose of the problem is to obtain the maximum altitude that projectile achieved, if its initial velocity is 1 km/s .

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Chapter 2 Solutions