Free fall

Since all objects fall at the same rate in the absence of other forces, objects and people will experience weightlessness in these situations.

This is the "textbook" case of the vertical motion of an object falling a small distance close to the surface of a planet. It is a good approximation in air as long as the force of gravity on the object is much greater than the force of air resistance, or equivalently the object's velocity is always much less than the terminal velocity (see below).

Acceleration of a small meteoroid when entering the Earth's atmosphere at different initial velocities

Assuming an object falling from rest and no change in air density with altitude, the solution is:

The object's speed versus time can be integrated over time to find the vertical position as a function of time:

The separation as a function of time is given by the inverse of the equation. The inverse is represented exactly by the analytic power series: