Question

Zero, a hypothetical planet, has a mass of 5.2 x 1023 kg, a radius of 3.2 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

(a) Here, we should consider potential energy relative to the middle of the planet.

The expression for potential energy, 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/r²

Now, GmM = 6.674 x 10^?11 N·m²/kg² · 10kg · 5.2 x 10^23 kg = 3.47 x 10^14 kg·m³/s²

At the surface (r= 3.2 x 10^6 m), then, F = (3.47 x 10^14) / (3.2 x 10^6)^2 = 33.9 N

and Ep = F·r = 10.84 x 10^7 J

At r?= 4.0 x 10^6 m, F = (3.47 x 10^14) / (4.0 x 10^6)^2 = 21.7 N

and Ep = 8.67 x 10^7 J

from the ground, ?Ep = 2.17 x 10^7 J ? First case

At r?= 8.0 x 10^6 m, F = (3.47 x 10^14) / (8.0 x 10^6)^2 = 5.42 N
and Ep = 4.34 x 10^7 J
from the ground, ?Ep = 6.50 x 10^7 J ? Second case.

(a) Required kinetic energy, ?Ep = ?Ek = 2.17 x 10^7 J = 4.0 x 10^7 - Ek

=> Ek = 1.83 x 10^7 J

(b) In this case, kinetic energy required, Ep = Ek = 6.50 x 10^7 J