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.
History
was the first to demonstrate and then formulate these equations. He used a ramp to study rolling balls, the ramp slowing the acceleration enough to measure the time taken for the ball to roll a known distance. He measured elapsed time with a water clock, using an "extremely accurate balance" to measure the amount of water.The equations ignore air resistance, which has a dramatic effect on objects falling an appreciable distance in air, causing them to quickly approach a terminal velocity. 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.
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.
Overview
Near the surface of the Earth, the acceleration due to gravity = 9.807 m/s2 approximately. A coherent set of units for,, and is essential. Assuming SI units, is measured in metres per second squared, so must be measured in metres, in seconds and 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, all results below will therefore be quite inaccurate after only 5 seconds of fall. Air resistance induces a drag force on any body that falls through any atmosphere other than a perfect vacuum, and this drag force increases with velocity until it equals the gravitational force, leaving the object to fall at a constant terminal velocity.
Terminal velocity depends on atmospheric drag, the coefficient of drag for the object, the velocity of the object, and the area presented to the airflow.
Apart from the last formula, these formulas also assume that negligibly varies with height during the fall. The last equation is more accurate where significant changes in fractional distance from the centre of the planet during the fall cause significant changes in. This equation occurs in many applications of basic physics.
The following equations start from the general equations of linear motion:
and equation for universal gravitation :
Equations
Example
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. If an object fell 10000 m to Earth, then the results of both equations differ by only 0.08%; however, if it fell from geosynchronous orbit, which is 42164 km, then the difference changes to almost 64%.Based on wind resistance, for example, the terminal velocity of a skydiver in a belly-to-earth free-fall position is about 195 km/h. This velocity is the asymptotic limiting value of the acceleration process, because the effective forces on the body balance each other more and more closely as the terminal velocity is approached. In this example, a speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on.
Higher speeds can be attained if the skydiver pulls in his or her limbs. In this case, the terminal velocity increases to about 320 km/h, which is almost the terminal velocity of the peregrine falcon diving down on its prey. The same terminal velocity is reached for a typical.30-06 bullet dropping downwards—when it is returning to earth having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study.
For astronomical bodies other than Earth, and for short distances of fall at other than "ground" level, in the above equations may be replaced by where is the gravitational constant, is the mass of the astronomical body, is the mass of the falling body, and is the radius from the falling object to the center of the astronomical body.
Removing the simplifying assumption of uniform gravitational acceleration provides more accurate results. We find from the formula for radial elliptic trajectories:
The time taken for an object to fall from a height to a height, 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.
Note that when this equation gives, as expected; and when it gives, which is the time to collision.