A "free falling" object is an object that is only submitted to the gravity force.
Strictly speaking, no object on the earth is "free falling", because of the air resistance force.
But it is said to be so if the resistance of the air is neglegtible compared to the gravity force.
Examples of "free falling" objects are a tennis ball, a socker ball and a gun bullet.
All these objects follow trajectories that are parabolic curves.
The proof of it relies on the solution of integral equations:...