Question

A rocket traveling 1950 m/s away from the Earth at an altitude of 6400 km fires its rockets, which eject gas at a speed of 1300 m/s (relative to the rocket).

If the mass of the rocket at this moment is 2.65×10⁴ kg and an acceleration of 1.6 m/s2 is desired, at what rate must the gases be ejected?

The answer is not 132.19 it is coming up wrong.

Answer 1

The gravitational acceleration at that altitude is

g' = g(Re)^2/(Ro)^2

where the RHS takes advantage of the inverse-square-of-distance relationship for gravity,

where Ro = radius of orbit (or distance of object in this case)

and Re = radius at which "gravity" is known (e.g., radius of Earth, 6.371e6m)

and g = the gravitational value corresponding to Re (for Earth, 9.8 m/s^2)

g' = 9.8m/s^2 * 6.371^2 / (6.1 + 6.371)^2 = 2.56 m/s^2

Meaning that until the rocket fires its rockets, although it is traveling away from the earth, it is accelerating towards the earth. (Without firing its rockets, the rocket will return to earth.) In other words, the total acceleration required is 2.56m/s^2 + 1.6 m/s^2 = 4.16 m/s^2.

F = ma = 26500 kg * 4.16m/s^2 = 110240 N

F = 110240 N = (dm/dt) * v = (dm/dt) * 1300m/s

dm/dt = 84.8 kg/s

the rate must the gases be ejected is 84.8 kg/s