Explanation Given information: The altitude of the circular orbit from center of earth ( A ) is 300 km . The magnitude of relative velocity ( v A ) R is 3.44 km / s . The radius of the earth ( R ) is 6370 km . The radius of the geosynchronous orbit ( r C ) is 42.14 × 10 6 m . Calculation: Calculate the radius of the circular orbit at point A ( r A ) using the relation: r A = R + A Substitute 6370 km for R and 300 km for A . r A = 6370 + 300 = 6670 km Convert the unit of radius of the circular orbit at point A ( r A ) from kilometer to meter: r A = ( r A ) km × 1000 m km Here, ( r A ) km is the radius of the circular orbit at point A in kilometer. Substitute 6670 km for ( r A ) km . r A = 6670 × 1000 m km = 6.670 × 10 6 m Convert the radius of the earth ( R ) from kilometer to meter: R = R E × 1000 m km Here, R E is the radius of the earth in kilometer. Substitute 6370 km for R E . R = 6370 × 1000 m km = 6.370 × 10 6 m The expression for the geocentric force acting on the satellite when it is on the surface of earth ( F 0 ) as follows: F 0 = G M m R 2 Here, G is the universal gravitational constant, M is the mass of the earth and m is the mass of the space vehicle. The expression for the force acting on the satellite on the surface of the earth due to gravity ( F 0 ) as follows: F 0 = m g Substitute m g for F . F = G M m R 2 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 Show the figure of the circular orbit with the forces as in Figure (1). The expression for the normal acceleration ( a n ) as follows: a n = v c i r c 2 r Calculate the velocity at the circular orbit ( v c i r c ) by considering the force equilibrium using the relation: ∑ F n = m a n Substitute v c i r c 2 r for a n and G M m r 2 for F n . G M m r 2 = m ( v c i r c 2 r ) G M r = v c i r c 2 v c i r c = G M r Substitute 398.06 × 10 12 m 3 / s 2 for GM and 6.670 × 10 6 m for r A . v c i r c = 398.06 × 10 12 m 3 / s 2 6.670 × 10 6 m = 7.726 × 10 3 m / s Calculate the velocity at the point A ( v A ) using the relation: v A = v c i r c + ( v A ) R Substitute 7.726 × 10 3 m / s for v c i r c and 3.44 km / s for ( v A ) R . v A = 7.726 × 10 3 m / s + 3.44 × 10 3 m / s = 11.166 × 10 3 m / s The expression for the kinetic energy of the satellite at point A ( T A ) as follows: T A = 1 2 m v A 2 Here, m is the mass of the satellite

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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 13.107P

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Find the angle (γ) at which the rocket will cross the intended orbit at point C.

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Chapter 13.2, Problem 13.106P

Chapter 13.2, Problem 13.108P

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Vector Mechanics for Engineers: Statics and Dynamics

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Find the angle ( γ ) at which the rocket will cross the intended orbit at point C .

Solution Summary: The author calculates the radius of the circular orbit at point A using the relation: r_A=R+A.

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Find the angle (γ) at which the rocket will cross the intended orbit at point C.

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