Equations for a falling body

A set of equations describing the trajectories of objects subject to a constant gravitational force under normal Earth-bound conditions.

Assuming constant acceleration g due to Earth's gravity, Newton's law of universal gravitation simplifies to F = mg, where F is the force exerted on a mass m by the Earth's gravitational field of strength g. Assuming constant g is reasonable for objects falling to Earth over the relatively short vertical distances of our everyday experience, but is not valid for greater distances involved in calculating more distant effects, such as spacecraft trajectories.

The effect of air resistance varies enormously depending on the size and geometry of the falling object—for example, the equations are hopelessly wrong for a feather, which has a low mass but offers a large resistance to the air.

(In the absence of an atmosphere all objects fall at the same rate, as astronaut David Scott demonstrated by dropping a hammer and a feather on the surface of the Moon.)

The equations also ignore the rotation of the Earth, failing to describe the Coriolis effect for example.

Nevertheless, they are usually accurate enough for dense and compact objects falling over heights not exceeding the tallest man-made structures.

In all cases, the body is assumed to start from rest, and air resistance is neglected.

Generally, in Earth's atmosphere, all results below will therefore be quite inaccurate after only 5 seconds of fall (at which time an object's velocity will be a little less than the vacuum value of 49 m/s (9.8 m/s2 × 5 s) due to air resistance).

and equation for universal gravitation (r+d= distance of object above the ground from the center of mass of planet): The first equation shows that, after one second, an object will have fallen a distance of 1/2 × 9.8 × 12 = 4.9 m. After two seconds it will have fallen 1/2 × 9.8 × 22 = 19.6 m; and so on.

On the other hand, the penultimate equation becomes grossly inaccurate at great distances.

Based on wind resistance, for example, the terminal velocity of a skydiver in a belly-to-earth (i.e., face down) free-fall position is about 195 km/h (122 mph or 54 m/s).

Higher speeds can be attained if the skydiver pulls in his or her limbs (see also freeflying).

[5] For astronomical bodies other than Earth, and for short distances of fall at other than "ground" level, g in the above equations may be replaced by

Removing the simplifying assumption of uniform gravitational acceleration provides more accurate results.

Centripetal force causes the acceleration measured on the rotating surface of the Earth to differ from the acceleration that is measured for a free-falling body: the apparent acceleration in the rotating frame of reference is the total gravity vector minus a small vector toward the north–south axis of the Earth, corresponding to staying stationary in that frame of reference.

An initially stationary object which is allowed to fall freely under gravity falls a distance proportional to the square of the elapsed time. This image, spanning half a second, was captured with a stroboscopic flash at 20 flashes per second. During the first 0.05 s the ball drops one unit of distance (about 12 mm), by 0.10 s it has dropped at total of 4 units, by 0.15 s 9 units, and so on.
Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of , where h is the height and g is the acceleration of gravity.