physics ...plzzzz help...

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

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  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.

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  2. thank you ...so much..

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

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