Once Newton had developed the mathematics for the trajectory of a cannon ball he realized that the Moon was something like a huge cannon ball, that is the same force that pulls the cannon ball to towards the Earth also pulls the Moon towards the Earth. You can test this out by varying the speed of the Satellite in this simulation.
The simulation on this page also includes a 'rocket', that starts its flight from near the planet's surface. Both the satellite and the rocket are simulated as having a very low mass and so they do not have any gravity interaction with each other, although they both are in the gravity field of the Earth. If you time the launch just right the rocket can rendevouz with the sattelite, although their speed will not match, that will be an excersize for a later simulation.