(a) Explanation Given information: The acceleration due to gravity ( g ) is 9.81 m / s 2 . Take the radius of the earth ( R ) is 6370 km . Calculation: Convert the unit of radius of earth from kilometer to meter using the relation: R = R k m × ( 1000 m 1 km ) Here, R k m is the radius of the earth in kilometer. Substitute 6370 km for R k m . R = 6 , 370 × ( 1000 m 1 km ) = 6 .37 × 10 6 m Write the expression for calculating the geocentric force acting on the missile on the surface of the earth as follows: F = G M m R 2 Here, G is the universal gravitational constant, M is mass of the earth, m is the mass of the missile, R is the radius of the earth and F is the force acting on the missile on the surface of the earth. Write the expression for force acting on the missile on the surface as follows: F = m g Substitute G M m R 2 for F . m g = G M m R 2 G M = g R 2 Substitute 9.81 m / s 2 for g , and 6 .37 × 10 6 m for R . G M = ( 9.81 m / s 2 ) ( 6.37 × 10 6 m ) 2 = ( 9.81 ) ( 40.58 × 10 12 ) = 398.06 × 10 12 m 3 / s 2 Write the expression for the kinetic energy of the missile at position 1 ( T 1 ) when the missile is on the surface of earth as follows: T 1 = 1 2 m v 1 2 Here, v 1 is the initial velocity of the missile. Write the expression for the gravitational potential energy of the missile at position 1 ( V 1 ) as follows: V 1 = − G M m R Write the expression the kinetic energy of the missile at position 2 ( T 2 ) when the missile is at infinite distance from the surface of the earth as follows: T 2 = 1 2 m v 2 2 Here, v 2 is the final velocity of the missile. Write the expression the gravitational potential energy of the missile at position 2 ( V 2 ) as follows; V 2 = − G M m R ∞ Here, R ∞ is the infinite radius (b) To determine Find the initial velocity or escape velocity ( v e s c ) of the missile.

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 13.85P

(a)

To determine

Find the kinetic energy (T1m) per unit mass.

(b)

To determine

Find the initial velocity or escape velocity (vesc) of the missile.

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Chapter 13.2, Problem 13.84P

Chapter 13.2, Problem 13.86P

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Find the kinetic energy ( T 1 m ) per unit mass.

Solution Summary: The author explains how to calculate the geocentric force acting on a missile on the surface of the earth.