Crashing Shuttle?

Question

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 47.0 km. To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 17.0 m/s.

If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

I suspect you use v=sqrt(Gm/r) but I don't think I have enough variables....

Answer 1

h = height of spacecraft = 47 km

Rm = radius of the moon = 1738 km

r = radial distance to satellite = Rm + h = 1738 + 47 = 1785 km = 1.785x10^6 m

g = acceleration of gravity on the moon = 1.6 m/s^2

M = mass of the moon = 7.36 x10^22  kg

G = gravitational constant =  6.673 x 10-11 N•m^2/kg^2

The velocity at impact will have vertical and horizontal components. No air, no friction, assuming we can treat this as being a flat surface crash, etc. then the horizontal component will just be the horizontal speed of the spacecraft and the vertical speed will just depend on gravity.

Vertical speed at impact Vv = SQRT(2gh)

Vv = SQRT[ 2(1.6)(47000)] = 387.8 m/s

Horizontal speed at impact:

Vo = orbital speed at 47 km

Vo = SQRT(GM/r) = SQRT[(6.673x10-11)(7.36 x10^22)/(1.785x10^6)]

Vo = SQRT[27.51x10^5] = 1659 m/s

Horizontal speed = Vh = Vo - 17 m/s 1659 - 17 = 1642

Now combine these to get the total speed:

V = SQRT(Vh^2 + Vv^2)

V = SQRT[(1642)^2 + (378.8)^2]

V = SQRT[2696164 + 150400]

V = 1687 m/s