A spacecraft of mass 1,000 kg is in an elliptical orbit about the Earth, as show

Question

A spacecraft of mass 1,000 kg is in an elliptical orbit about the Earth, as shown on the right. At point A the spacecraft is at a distance r_A = 1.2 times 10^7 m from the center of the Earth and its velocity, of magnitude v_A = 7.1 times 10^3 m/s, is perpendicular to the line connecting the center of the Earth to the spacecraft. The mass and radius of the Earth are M_E = 6.0 times 10^24 kg and r_E = 6.4 times 10^6 meters, respectively. Determine each of the following for the spacecraft when it is at point A. (a) The total mechanical energy of the spacecraft, assuming that the gravitational potential energy is zero at an infinite distance from the Earth. (b) The magnitude of the angular momentum of the spacecraft about the center of the Earth. Later the spacecraft is at point B on the exact opposite side of the orbit at a distance r_B = 3.6 times 10^7 m from the center of the Earth. (c) Determine the speed v_B of the spacecraft at point B. Suppose that a different spacecraft is at point A, a distance r_A = 1.2 times 10^7 mfrom the center of the Earth. Determine each of the following. (d) The speed of the spacecraft if it is in a circular orbit around the Earth (e) The minimum speed of the spacecraft at point A if it is to escape completely from the Earth

Explanation / Answer

4a)

Total mechanical energy = - GMm/r + (0.5) mVa2 = - (6.67 x 10-11) (6 x 1024) (1000)/(1.2 x 107) + (0.5) (1000) (7100)2

Total mechanical energy = - 8.145 x 109 J

b)

L = mva ra = 1000 (7100) (1.2 x 107) = 8.52 x 1013

c)

using conservation of momentum

mva ra = mvb rb

7100 (1.2 x 107) = vb (3.6 x 107)

vb = 2.4 x 103 m/s

d)

V = sqrt(GMe/ra) = sqrt((6.67 x 10-11) (6 x 1024) /(1.2 x 107))

V = 5.77 x 103 m/s

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.