Question

# Zero, a hypothetical planet, has a mass of 4.9 x 1023 kg, a radius of 2.8...

Zero, a hypothetical planet, has a mass of 4.9 x 1023 kg, a radius of 2.8 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?

We'll have to consider potential energy relative to the middle of the planet, since using Ep = mgh with these

distances would fail to consider the dimishing effects of gravity over distance.

Gravitational potential energy is the Weight times the height.

Weight = Gravitational force F = GmM/r^2

GmM = 6.674 x 10^-11 Nm^2/kg^2 x 10kg x 4.9 x 10^23kg

= 3.27 x 10^14 kgm^3/s^2

At the surface (r = 2.8 x 10^6m), then, F = 42.69 N

and Ep = F x r = 1.20 x 10^8 J

At r1= 4 x 10^6m, F = 20.44 N

and Ep = 8.175 x 10^7 J

from the ground, ∆Ep = 3.825 x 10^7 J case 1

At r2 = 8 x 10^6m, F = 5.11 N

and Ep = 4.1 x 10^7 J

from the ground, ∆Ep = 7.9 x 10^7 J case 2

case 1: ∆Ep = ∆Ek = 3.825 x 10^7 J = 5 x 10^7 J - Ek

Ek = 1.175 x 10^7 J ...................(a)

case 2: Ep = Ek = 7.9 x 10^7 J ................. (b)

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