A set of equations describe the resultant trajectories when objects move owing to a constant gravitational force under normal Earth-bound conditions. For example, Newton's law of universal gravitation simplifies to F = mg, where m is the mass of the body. This assumption is reasonable for objects falling to earth over the relatively short vertical distances of our everyday experience, but is untrue over larger distances, 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 gravityg = 9.807 m/s2 approximately. A coherent set of units for g, d, t and v is essential. Assuming SI units, g is measured in meters per second squared, so d must be measured in meters, t in seconds and v in meters 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 g negligibly varies with height during the fall. The last equation is more accurate where significant changes in fractional distance from the center of the planet during the fall cause significant changes in g. This equation occurs in many applications of basic physics.
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 meters. After two seconds it will have fallen 1/2 × 9.8 × 22 = 19.6 meters; and so on. The second to last equation becomes grossly inaccurate at great distances. If an object fell 10,000 meters to Earth, then the results of both equations differ by only 0.08%; however, if it fell from geosynchronous orbit, which is 42,164 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. Competition speed skydivers fly in the head down position and reach even higher speeds. The current world record is 1,357.6 km/h by Felix Baumgartner who skydived from 38,969.4 m above earth on 14 October 2012. The record was set due to the high altitude where the lesser density of the atmosphere decreased drag. 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/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 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 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.
Acceleration relative to the rotating Earth
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.
