Zero, a hypothetical planet, has a mass of 5.3 x 1023 kg, a radius of 3.3 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?

(a)

The total mechanical energy of the space probe must be constant, so we can write:

(1)

where

is the kinetic energy at the surface, when the probe is launched

is the gravitational potential energy at the surface

is the final kinetic energy of the probe

is the final gravitational potential energy

Here we have

at the surface, (radius of the planet), (mass of the planet) and m=10 kg (mass of the probe), so the initial gravitational potential energy is

At the final point, the distance of the probe from the centre of Zero is

so the final potential energy is

So now we can use eq.(1) to find the final kinetic energy:

(b)

The probe reaches a maximum distance of

which means that at that point, the kinetic energy is zero: (the probe speed has become zero):

At that point, the gravitational potential energy is

So now we can use eq.(1) to find the initial kinetic energy: