How much speed do I need to leave a celestial body (the moon, the sun...) ?
For "leave" I mean "go from the surface to a distance where the gravitational force of the celestial body is negligible" and for speed I mean the initial velocity that an object without a propulsion system must have
I suppose that the celestial body has no atmosphere, because the friction change completely the calculations
Let's talk about potential energy
The potential energy that a object has is m x g x h where m is the mass of the object, g is the acceleration, h is the height from the ground. But this is an approximation valid only near the surface. If i'm so far away from the celestial body, its attraction is less than near the surface, so g change.
To solve this problem we can use the definition of potential energy in figure
This is the energy you need to bring an object(of mass m) from a distance R from the center of celestial body (of mass M) to an infinite distance
Now, if our object has enough kinetic energy (greater or equal of potential energy), it can leave the body
where m is the mass of our object, M is the mass of celestial body, and v is the speed we were looking for :)
Ciao!