Question

Zero, a hypothetical planet, has a mass of 5.0 x 10²³ kg, a radius of 3.1 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 can use the conservation of mechanical energy. Initial total energy is equal to final total energy.
At the surface o f the planet, the initial kinetic energy of the probe was Ke and the potential energy is U
E_i = KE + U = 5.0 x 10⁷ J + G M m / R
Where M is the mass of the planet, m is the mass of the probe and R is the radius of the radius of the planet
The final mechanical energy of the planet is
E_f = KE - G Mm / r
Where r is the distance of the planet from the probe.
r = 4.0 x 10⁶ m
E_i = E_f
5.0 x 10⁷ J - G M m / R = KE - G Mm / r
KE = 5.0 x 10⁷ J - G M m (1/ R - 1/r)
KE =  5.0 x 10⁷ J - 6.67 x 10^-11 x 5.0 x 10²³ x 10 x (1 / 3.1 x 10⁶ - 1/4.0 x 10⁶)
KE = 2.58 x 10⁷ J
b) For maximum distance achieved, the final kinetic energy of the probe is zero.
Again using the conservation of energy equation
KE_f = KE_i - G M m (1/ R - 1/r)
KE_i = KE_f + G M m (1/ R - 1/r)
KE_i = 0 + 6.67 x 10^-11 x 5.0 x 10²³ x 10 x (1 / 3.1 x 10⁶ - 1/ 8.0x 10⁶)
KE_i = 6.589 x 10⁷ J