How mathematical do you want to get?
The gravitational force of attraction between two objects is given by:
F = (G .m.M) / r^2
where G is the Gravity Constant (6.674 x 10^-11 N m^2 kg^-2)
m is the mass of the smaller body
M is the mass of the larger body
r is the distance between the centres of mass of the two bodies
The centrepetal force acting on a body in circular motion is given by:
F = (m.V^2)/r
where m is the mass of the body
V is the instantaneous velocity of the body
r is the radius of the circle
Putting these two equations together we end up with:
V^2 = G.M/r
G.M is a constant, mass of the earth times the gravity constant
Putting this into words, the lower the satellite orbit (r) the faster the satellite must move (V)
So for a geosynchronous orbit, the velocity must be such that the satellite completes the distance of its orbit in 24 hours.
So from all this, when you know the mass of the earth and the gravity constant, you can calculate how fast and at what height the satellite must move to stay in a fixed position in the sky.
As for a small object half way between the earth and the moon, the gravity force on it are very small, so it will not go far very quickly.
But, over time it will move, either it will drift off into space, or it will eventually fall to earth.
In this case, yes it is rocket science!
JimB