Question

1 ) If the mass of the Moon is 7.4 x 1022 kg and its radius is 1.74 x 106 m, compute the speed with which an object would have to be fired in order to sail away from it, completely overcoming the Moon’s  gravity  pull.

2 ) Show that if a satellite orbits very near the surface of a planet with period T, the density (mass
per unit volume) of the planet is p(rho) = 3pi/(GT)^2

Answer 1

(1)
mv²/2 = GmM/R,
v =sqrt(2•G•M/R)
This is escape velocity (it is the speed needed to "break free" from a gravitational field without further propulsion).
The gravitational constant
G =6.67•10^-11 N•m²/kg².
(2)

The volume of the sphere
V=(4•π•R^3)/3
The planet mass
M=ρ•V
Its density
ρ =M/V=3•M/4•π•R³ …..(1)
The period of satellite
T =2•π•r/v
r is the radius of the satellite orbit (given that r≈R)
v is the orbital speed of a body ( satellite) at which it orbits around a more massive body
From m•v²/R =G•m•M/R²
this speed is v =sqrt(g•M•/R)
Then period is
T=2πR /sqrt(G•M/R)
Square this equation
T² =4•π²•R³/G•M,
R³ =G•M•T²/, …. (2)
Subsitute (2) in (1)
ρ = 3•M/4•π•R³ =
=3•M•4•π²/4•π•G•M• T² =
= 3• π /G•T².
It seems to me that it is mistake in your expression 3pi/(GT)^2.

Answer 2

thank you ...so much..

Answer 3

ya i m sorry ...i wrote the wrong expression by mistake