(a) Explanation Given information: The radius of the earth ( R ) is 3 , 960 mi . The altitude of point A ( h A ) from the surface of the earth is 40.3. The altitude of point B ( h B ) from the surface of the earth is 336 mi. Calculation: Find the product of the constant of gravitation G and the mass M of the earth ( G M ) using the equation: G M = g R 2 Here, G is the gravitational constant. Substitute 32.2 ft/s 2 for g and 3 , 960 mi for R . G M = 32.2 × ( 3 , 960 mi × 5 , 280 ft 1 mi ) 2 = 14.077 × 10 15 ft 3 /s 2 Find the radius of the point A ( r A ) using the equation: r A = R + h A Substitute 3 , 960 mi for R and 40.3 mi for h A . r A = 3 , 960 + 40.3 = 4 , 000.3 mi × 5 , 280 ft 1 mi = 21.1216 × 10 6 ft Find the radius of the point A ( r B ) using the equation: r B = R + h B Substitute 3 , 960 mi for R and 336 mi for h A . r B = 3 , 960 + 336 = 4 , 296 mi × 5 , 280 ft 1 mi = 22.683 × 10 6 ft Consider the minimum radius Write the equation of motion satellite and the planet at A of elliptical trajectory. 1 r A = G M h 2 + C cos θ A (3) Here, G is the gravitational constant, M is the planet, h is the angular momentum per unit mass of the satellite, C is the constant, and θ A is the trajectory angle at A . Consider the maximum radius. Write the equation of motion satellite and the planet at B of elliptical trajectory. 1 r B = G M h 2 + C cos θ B (4) Here, θ B is the trajectory angle at B . Write the equation of angle at B . θ B = θ A + 180 ° Apply cosine on both sides. cos θ B = cos ( θ A + 180 ° ) cos θ B = − cos θ A Find the angular momentum per unit mass of the satellite ( h ) (b) To determine Find the increase in speed required at B to insert the shuttle into the circular orbit.

12th Edition

ISBN: 9781260663778

Author: BEER

Publisher: MCG

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Chapter 12, Problem 12.131RP

(a)

To determine

Find the speed v0 of the shuttle at A.

(b)

To determine

Find the increase in speed required at B to insert the shuttle into the circular orbit.

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Chapter 12, Problem 12.130RP

Chapter 12, Problem 12.132RP

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A spacecraft approaching the planet Saturn reaches point A with a velocity vA of magnitude 68.8 × 103 ft/s. It is to be placed in an elliptic orbit about Saturn so that it will be able to periodically examine Tethys, one of Saturn’s moons. Tethys is in a circular orbit of radius 183 × 103 mi about the center of Saturn, traveling at a speed of 37.2 × 103 ft/s. Determine (a) the decrease in speed required by the spacecraft at A to achieve the desired orbit, (b) the speed of the spacecraft when it reaches the orbit of Tethys at B.

A satellite describes an elliptic orbit of minimum altitude 606 km above the surface of the earth. The semimajor and semiminor axes are 17 440 km and 13 950 km, respectively. Knowing that the speed of the satellite at point C is 4.78 km/s, determine (a) the speed at point A, the perigee, (b) the speed at point B,the apogee.

A satellite describes an elliptic orbit of minimum altitude 606 km above the surface of the earth. The semimajor and semiminor axes are 17,440 km and 13,950 km, respectively. Knowing that the speed of the satellite at Point C is 4.78 km/s, determine (a) the speed at Point A, the perigee, (b) the speed at Point B, the apogee. 606 km R = 6370 km 13 950 km B, A 17 440 km 17 440 km

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Find the speed v 0 of the shuttle at A .

Solution Summary: The author calculates the constant of gravitation G and the mass M of the earth using the equation: GM=gR2.