Overview
Near the surface of the Earth, use g = 9.8 m/s² (metres per second squared; which might be thought of as "metres per second, per second", or 32 ft/s² as "feet per second per second"), approximately. For other planets, multiply g by the appropriate scaling factor. It is essential to use a coherent set of units for g, d, t and v. Assuming SI units, g is measured in metres per second squared, so d must be measured in metres, t in seconds and v in metres per second.
In all cases, the body is assumed to start from rest, and air resistance is neglected. Generally, in Earth's atmosphere, this means all results below will 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). When a body is travelling through any atmosphere other than a perfect vacuum it will encounter a drag force induced by air resistance, this drag force increases with velocity. The object will reach a state where the drag force equals the gravitational force at this point the acceleration of the object becomes 0, the object now falls at a constant velocity. This state is called the terminal velocity.
The drag force is dependant on the density of the atmosphere, the coefficient of drag for the object, the velocity of the object (instantaneous) and the area presented to the airflow.
Apart from the last formula, these formulas also assume that g does not vary significantly with height during the fall (that is, they assume constant acceleration). For situations where fractional distance from the center of the planet varies significantly during the fall, resulting in significant changes in g, the last equation must be used for accuracy. This equation occurs in many applications of basic physics.
| Distance travelled by an object falling for time : | |
| Time taken for an object to fall distance : | |
| Instantaneous velocity of a falling object after elapsed time : | |
| Instantaneous velocity of a falling object that has travelled distance : | |
| Average velocity of an object that has been falling for time (averaged over time): | |
| Average velocity of a falling object that has travelled distance (averaged over time): | |
| Instantaneous velocity of a falling object that has travelled distance on a planet with mass, with the combined radius of the planet and altitude of the falling object being, this equation is used for larger radii where is smaller than standard at the surface of Earth, but assumes a small distance of fall, so the change in is small and relatively constant: | |
| Instantaneous velocity of a falling object that has travelled distance on a planet with mass and radius (used for large fall distances where can change significantly): |
Example: the first equation shows that, after one second, an object will have fallen a distance of 1/2 × 9.8 × 12 = 4.9 metres. After two seconds it will have fallen 1/2 × 9.8 × 22 = 19.6 metres; and so on.
We can see how the second to last, and the last equation change as the distance increases. If an object were to fall 10,000 metres to Earth, the results of both equations differ by only 0.08%. However, if the distance increases to that of geosynchronous orbit, which is 42,164 km, the difference changes to being almost 64%. At high values, the results of the second to last equation become grossly inaccurate.
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 G(M+m)/r² where G is the gravitational constant, M is the mass of the astronomical body, m is the mass of the falling body, and r is the radius from the falling object to the center of the body.
Removing the simplifying assumption of uniform gravitational acceleration provides more accurate results. We find from the formula for radial elliptic trajectories:
The time t taken for an object to fall from a height r to a height x, measured from the centers of the two bodies, is given by:
where is the sum of the standard gravitational parameters of the two bodies. This equation should be used whenever there is a significant difference in the gravitational acceleration during the fall.
Read more about this topic: Equations For A Falling Body