Question

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?

Answer 1

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)