(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:
