Answer
We need a launch speed of 2.38 km/s
Work Step by Step
We need to find the escape speed from the moon's surface. We can assume that kinetic energy and gravitational potential energy after a long time are both equal to zero. We can use conservation of energy to find the escape speed:
$KE+U_g = 0$
$\frac{1}{2}mv^2-\frac{G~M_{moon}~m}{R_{moon}} = 0$
$\frac{1}{2}mv^2 = \frac{G~M_{moon}~m}{R_{moon}}$
$\frac{1}{2}v^2 = \frac{G~M_{moon}}{R_{moon}}$
$v = \sqrt{\frac{2G~M_{moon}}{R_{moon}}}$
$v = \sqrt{\frac{(2)(6.67\times 10^{-11}~m^3/kg~s^2)~(7.35\times 10^{22})}{1.737\times 10^6~m}}$
$v = 2380~m/s = 2.38~km/s$
We need a launch speed of 2.38 km/s.
