Answer to Question #162990 in Astronomy | Astrophysics for Mac Roy

Question

An astronaut working on the Moon tries to determine the gravitational constant G by throwing a Moon rock of mass m with a velocity of v vertically into the sky. The astronaut knows that the Moon has a density ρ of 3340 kg/m3 and a radius R of 1740 km.

(a) Show with F = G · mM/ R²  that the potential energy of the rock at height h above the surface is given by: E = − (4πG /3) mρ · R³ / (R + h) (2)

(b) Next, show that the gravitational constant can be determined by: G = 3v² /(8π x ρR²) (1 − R /(R + h)^-1 (3)

(c) What is the resulting G if the rock is thrown with 30 km/h and reaches 21.5 m?

Answer 1

a)

$$U=\frac{dF}{dr}=-\frac{GmM}{r}=-\frac{4\pi Gm\rho R^3}{3r}\\ U=-\frac{4\pi Gm\rho R^3}{3(R+h)}$$

b)

$$g=\frac{4\pi G\rho R}{3}$$

c)

$$G=\frac{3(30)^2}{8\pi(3.6)^2(3340)(1740000)^2}\cdot \frac{1}{1-\frac{1740000}{1740000+21.5}}\\G=6.63\cdot10^{-11}\frac{m^3}{kgs^2}$$