Question

Zero, a hypothetical planet, has a mass of 5.8 x 10²³ kg, a radius of 3.4 x 10⁶ m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 x 10⁷ J, what will be its kinetic energy when it is 4.0 x 10⁶ m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 10⁶ m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

Answer 1

Since the gravitational force is a conservative force, the mechanical energy of the probe-planet
system is conserved. The energy conservation equation yields
Ei = Ef ⇒ Ki – GMm/Ri = Kf – GMm/Rf

⇒ Kf = Ki – GMm(1/Ri – 1/Rf)

G = 6.67 x 10^-11 m3 kg-1 s-2
M = 5.8 x 10^23 kg
ri =3.4 x 10^6 m,
rf = 106
m = 10 kg
Ki =5.0 x 10^7 J

Kf = Ki – GMm(1/Ri – 1/Rf) = 5.0 x 10^7J - 6.67 x 10^-11*5.8 x 10^23*10*(1/3.4 x 10^6 - 1/106) =3.649*10^12J

b)
When the probe reaches the maximum height, its speed becomes zero. The energy conservation
equation leads to

Ei = Ef ⇒ Ki – GMm/Ri= -GMm/Rf
ri =3.4 x 10^6 m,
rf = 8*10^6m

⇒ Ki = GMm(1/Ri– 1/Rf) = 6.542*10^7J