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PS4-4

# PS4-4 - Problem Set 4 Problem 4 Problem During a flyby of...

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Problem Set 4: Problem 4. Problem: During a flyby of Earth, the speed of a spacecraft is V as it reaches its minimum altitude of 1 6 R above the surface at Point O, where R is Earth’s radius. At Point B the spacecraft’s altitude is 4 3 R . (a) Verify that the spacecraft’s trajectory is parabolic and determine its velocity at Point O as a function of R , the universal constant of gravitation, G , and Earth’s mass, M . (b) Using the energy-conservation principle, compute V A . Express your answer as a function of V . Solution: (a) In general, the shape of the trajectory is 1 r = GM h 2 (1 + 6 cos θ ) At Point O, r = 7 6 R and θ =0 o ,sotha t 6 7 R = GM h 2 (1 + 6 · 1) = GM h 2 (1 + 6 ) At Point B, r = 7 3 R and θ =90 o , which gives 3 7 R = GM h 2 (1 + 6 · 0) = GM h 2 Combining these two equations yields 6 7 R = w 3 7 R W (1 + 6 )= 2=1+ 6 Therefore, we conclude that 6 =1 wherefore the orbit is indeed parabolic. Consequently, its velocity at Point O, which is the perigee, is the escape velocity, i.e., V = 5 2 GM r O = 5 12 7 GM R wherewehaveusedthefac

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PS4-4 - Problem Set 4 Problem 4 Problem During a flyby of...

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