Question

Problem 3: You and your friend are both starship captains. Your friend is orbiting a planet in a circular orbit at an altitude of 2 x 106m above the surface. Your ship is also in a circular orbit but is at an altitude of 7 x 106 m above the surface. (Mplanet = 2 x 1023 kg and RPlanet = 4 x 106 m) a) [4 points) What are the orbital speeds of the two ships in their circular orbits? b) [6 points] By what amount would you Diagram for parts B&C need to alter your velocity to drop your orbit down so that it touches your Your friend's orbit as shown in the diagram с Friend to the right? Would this be an increase or decrease in velocity?

Answer 1

a) Let the speed of your's to be v₁ and that of your friend's to be v₂.

Now balancing the forces along radial direction;

GM_planetm₁/(R_planet + 7 x 10⁶m)² = m₁v₁²/(R_planet + 7 x 10⁶m)

=> v₁ = [(GM_planet)/(R_planet + 7 x 10⁶ m)]^1/2

=> v₁ = 1101.6 m/s

similarly;

GM_planetm₂/(R_planet + 2 x 10⁶m)² = m₂v₂²/(R_planet + 2 x 10⁶m)

=> v₂ = [(GM_planet)/(R_planet + 2 x 10⁶ m)]^1/2

=> v₂ = 1491.563 m/s

(b) Let the altered velocity of your spacecraft to be v; and v_f to be the velocity when you reach your friend's orbit

Now if you have to enter your friend's orbit then you have to conserve the energy and the angular momentum to adjust the velocity; thus

m₁v(R_planet + 7 x 10⁶m) = m₁v_f/(R_planet + 2 x 10⁶m)

=> 11v = 6v_f ..........(i)

By energy conservation;

m₁v²/2 - GM_planetm₁/(R_planet + 7 x 10⁶m) = m₁v_f²/2 - GM_planetm₁/(R_planet + 2 x 10⁶m)

=> v² = 2GM_planet/(R_planet + 7 x 10⁶m) + v_f² - 2GM_planet/(R_planet + 2 x 10⁶m)

From equation (i);

=> v² - (11v/6)² = 2GM_planet/(R_planet + 7 x 10⁶m) - 2GM_planet/(R_planet + 2 x 10⁶m)

=> v² = 856592.0856

=> v = 925.5226 m/s

You have to decrease your velocity.

c) From equation (i)

v_f = 11v/6 = 1696.8 m/s will be the velocity of you at C.

d) The total energy of your spaceship at this particular moment has to be equal to the kinetic energy when you have escaped;

Assuming escape velocity to be v_e;

m₁v_e²/2 - Gm₁M_planet/(R_planet + 2 x 10⁶m) - Gm₁M_moon/(40 x 10⁶m) = 0

=> v_e² = 4649748.4

=> v_e = 2156.33 m/s