(a) Explanation Given information: The initial velocity is v 0 . The angle forming in vertical is ϕ 0 . The maximum altitude is α R . The radius of the earth is R . The escape velocity is v e s c . Calculation: Show the figure with the velocities acting as in Figure (1). The expression for the minimum distance between the center of the earth to the point A ( r A ) using the relation: Since the point A is acting at the surface of the earth. r A = R The expression for the maximum distance between the center of the earth to the point B ( r B ) using the relation: r B = R + α R = R ( 1 + α ) The expression or the principle of conservation of angular momentum at point A to the point B as follows; R m v 0 sin ϕ 0 = r B m v B R v 0 sin ϕ 0 = r B v B v B = R v 0 sin ϕ 0 r B Substitute R ( 1 + α ) for r B . v B = R v 0 sin ϕ 0 R ( 1 + α ) = v 0 sin ϕ 0 ( 1 + α ) The expression for the kinetic energy at point A ( T A ) as follows: T A = 1 2 m ( v 0 ) 2 Here, m is the mass of the satellite. Calculate the gravitational potential energy at point A ( V A ) using the formula: V A = − G M m r A Substitute R for r A . V A = − G M m R The expression for the kinetic energy of the orbit at point B ( T B ) as follows: T B = 1 2 m v B 2 Calculate the gravitational potential energy at point B ( V B ) using the formula: V B = − G M m r B Substitute R ( 1 + α ) for r B . V B = − G M m R ( 1 + α ) The expression for the principle of conservation of energy at the point A to point B as follows: T A + V A = T B + V B Substitute 1 2 m ( v 0 ) 2 for T A , − G M m R for V A , 1 2 m v B 2 for T B , and − G M m R ( 1 + α ) for V B . 1 2 m ( v 0 ) 2 − G M m R = 1 2 m v B 2 − G M m R ( 1 + α ) v 0 2 2 − G M R = v B 2 2 − G M R ( 1 + α ) v 0 2 2 − v B 2 2 = G M R − G M R ( 1 + α ) v 0 2 − v (b) To determine Find the range of allowable values of the initial velocity ( v 0 ) .

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 13.2, Problem 13.115P

(a)

To determine

Show that required angle (ϕ0) is defined by the relation, sinϕ0=(1+α)1−α1+α(vescv0)2.

(b)

To determine

Find the range of allowable values of the initial velocity (v0).

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Chapter 13.2, Problem 13.114P

Chapter 13.2, Problem 13.116P

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Vector Mechanics for Engineers: Statics and Dynamics

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Show that required angle ( ϕ 0 ) is defined by the relation, sin ϕ 0 = ( 1 + α ) 1 − α 1 + α ( v e s c v 0 ) 2 .

Solution Summary: The author illustrates how the velocities are defined by the relation, underset_mathrmsin