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: the acceleration, that is the second derivate of the position, is the ratio of the weight and the mass, that is the gravity acceleretion g, vertically and in a descending direction.
We may notice here that the acceleration does not depend on the mass, so that the trajectories of free falling objects with the same initial speed are independant of the mass!
If we decompose the equation on the horizontal and vertical direction, we have a horizontal speed that is constant, so that the horizontal distance to the origin point is proportional to the horizontal intial speed and to the time.
And we have a vertical acceleration that is constant negative if the vertical axis is oriented upwards. So that the height of the object is a quadratic function of the time, with a quadratic coefficent being negative -g/2.
But the integral equations may be solved with any starting point on the trajectory, even with negative times. And a quadratic function of the real variable with a negative quadratic coefficent passes through a maximum, where its derivative is 0.
So, there is a point on the two sides trajectory where the height is maximum, with a vertical speed 0. If we shift the time to the "instant" (positive or negative) of that point, then the horizontal distance to that new origin is proportional to the time difference, and the vertical coordinate is proportional to the square of the time difference.
Thus the time difference is proportional to the horizontal distance and so are the squares of them. Consequently, the vertical coordinate is propotional to the square of the horizontal distance.
That is that the two-sides trajectory is a parabol, so that the effective trajectory from initial time to final time is a parabolic arch.
So, we have proved that the trajectories of free falling objects are parabolic…