Question

consider a spacecraft in an elliptical orbit around the earth. At the low point, or perigee, of its orbit, it is 300 km above the earth's surface; at the high point or apogee, it is 2500 km above the earth's surface.

Part A: find ratio of the spacecraft's speed at perigee to its speed at apogee?

(V_perigee / V_{apogee) = .....}

Part B: find the speed at the apogee?

V apogee = ........ m/s

Part C: find speed at perigee?

V perigee = ..... m/s

(I only have 1 try left in mastering please help me thanks)

Answer 1

Part A : The ratio of spacecraft's speed at perigee to its speed at apogee which will be given by -

(V_perigee / V_apogee) = (R_a + R_E) / (R_p + R_E)

(V_perigee / V_apogee) = [(2500 km) + (6380 km)] / [(300 km) + (6380 km)]

(V_perigee / V_apogee) = [(8880 km) / (6680 km)]

(V_perigee / V_apogee) = 1.32

Part B : The speed at an apogee which will be given by -

V_apogee = [2 G m_E (R_a + R_E) / (R_a + R_p + 2 R_E)] / (R_p + R_E)

V_apogee = {[(2) (6.67 x 10^-11 N.m²/kg²) (5.98 x 10²⁴ kg) (8880 x 10³ m)] / [(2500 x 10³ m) + (300 x 10³ m) + (12760 x 10³ m)]} / (6680 x 10³ m)

V_apogee = [(4.552 x 10¹⁴ N.m²/kg) / (6680 x 10³ m)]

V_apogee = 6.814 x 10⁷ m²/s²

V_apogee = 8255 m/s

Part C : The speed at an perigee which will be given by -

V_perigee = [2 G m_E (R_p + R_E) / (R_a + R_p + 2 R_E)] / (R_a + R_E)

V_perigee = {[(2) (6.67 x 10^-11 N.m²/kg²) (5.98 x 10²⁴ kg) (6680 x 10³ m)] / [(2500 x 10³ m) + (300 x 10³ m) + (12760 x 10³ m)]} / (8880 x 10³ m)

V_perigee = [(3.424 x 10¹⁴ N.m²/kg) / (8880 x 10³ m)]

V_perigee = 3.855 x 10⁷ m²/s²

V_perigee = 6209.5 m/s