Planet Zero has a mass of 4.0 x 1023 kg and a radius of 2.0 x 106 m. A 10-kg space probe is launched vertically from the surface of Zero with an initial kinetic energy of 8.0 x 107 J. What maximum distance from the center of Zero is achieved by the probe

r2 = 1.25 × 10^(6) m

Explanation:

We are given;

Initial kinetic energy; K_1= 8.0 x 10^(7) J

Mass of planet zero; M = 4.0 x 10^(23) kg

Mass of space probe; m = 10 kg

Radius of planet zero; r1 = 2 × 10^(6) m

From conservation of energy, we can say that;

Initial potential energy + Initial kinetic energy = Final potential energy + Final kinetic energy

This is;

U1 + K1 = U2 + K2

Using formula for potential energy: U = GMm/r, we can write it as;

GMm/r1 + K1 = GMm/r2 + K2

Where Maximum distance = r2

At maximum distance from the center of Zero, K2 = 0.

Thus;

GMm/r1 + K1 = GMm/r2

Making r2 the subject, we have;

r2 = GMm/((GMm/r1)+ K1)

Where G is gravitational constant = 6.67 × 10^(-11) N/m²/kg2

Thus;

r2 = (6.67 × 10^(-11) × 4.0 x 10^(23) × 10)/((6.67 × 10^(-11) × 4.0 x 10^(23) × 10)/(2 × 10^(6)) + (8.0 × 10^(7)))

r2 = 1.25 × 10^(6) m