(a) Explanation Given information: The velocity of the spacecraft ( v A ) is 10.4 km / s . The maximum altitude of the spacecraft to reach the earth ( h A ) is 990 km . The radius of the earth ( R ) is 6370 km . The altitude of the point B from earth ( h B ) is 8350 km . The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: 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 spacecraft 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 spacecraft 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 , 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 Calculate the altitude of the point A from the center of the earth ( r A ) using the relation: r A = R + h A Substitute 6.370 × 10 6 m for R and 990 km for h A . r A = 6 .370 × 10 6 m + 990 × 10 3 m = 7.36 × 10 6 m Calculate the altitude of the point B from the center of the earth ( r B ) using the relation: r B = R + h B Substitute 6.370 × 10 6 m for R and 8350 km for h B . r B = 6 .370 × 10 6 m + 8350 × 10 3 m = 14.72 × 10 6 m Convert the unit of velocity of the spacecraft at point A ( v A ) from km / s to m / s : v A = ( v A ) km / s × ( 1 , 000 m 1 km ) Here, ( v A ) km / s is the velocity of the spacecraft at point A in km / s . Substitute 10.4 km / s for ( v A ) km / s . v A = 10.4 × ( 1000 m 1 km ) = 1 (b) To determine Find the angle ( ϕ B ) at point B .

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Chapter 13.2, Problem 13.106P

(a)

To determine

Find the magnitude of the velocity at point B (vB).

(b)

To determine

Find the angle (ϕB) at point B.

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Chapter 13.2, Problem 13.105P

Chapter 13.2, Problem 13.107P

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Find the magnitude of the velocity at point B ( v B ) .

Solution Summary: The author calculates the radius of the earth (R) from kilometer to meter. The expression for the geocentric force acting on the spacecraft is as follows: