Equations for a falling body

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Under normal earth-bound conditions, when objects move owing to a constant gravitational force a set of dynamical equations describe the resultant trajectories. 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 very much untrue over larger distances, such as spacecraft trajectories.

1 History
2 Overview
- 2.1 Gravitational potential
- 2.2 Acceleration relative to the rotating Earth
3 Notes
4 See also

[edit] History

Galileo 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 water².

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. For example, a person jumping headfirst from an airplane will never exceed a speed of about 200 mph due to air resistance. 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.

[edit] Overview

Near the surface of the Earth, use g = 9.8 m/s² (metres per second per second), approximately. For other planets, multiply g by the appropriate scaling factor. It is essential to use consistent units for g, d, t and v. Assuming SI units, g is measured in metres per second per second, so d must be measured in metres, t in seconds and v in metres per second. To convert metres per second to kilometres per hour (km/h) multiply by 3.6. In all cases the body is assumed to start from rest, and air resistance is neglected, or in other words, they assume constant acceleration. Generally, in Earth's atmosphere, this means all results below will be quite inaccurate after only 5 seconds of fall, and associated speeds of 160 feet per second. On an airless body like the moon or relatively airless body like Mars, with appropriate changes in g, these equations will yield accurate results over much longer times and much higher velocities.

Distance d travelled by an object falling for time t:	$\ d=\frac{1}{2}gt^2$
Time t taken for an object to fall distance d:	$\ t =\ \sqrt {2d/g}$
Instantaneous velocity v_i of a falling object after elapsed time t:	$\ v_i = gt$
Instantaneous velocity v_i of a falling object that has travelled distance d:	$\ v_i = \sqrt {2gd}\$
Average velocity v_a of an object that has been falling for time t (averaged over time):	$\ v_a =\frac{1}{2}gt$
Average velocity v_a of a falling object that has travelled distance d (averaged over time):	$\ v_a =\frac{ \sqrt {2gd}}{2} \$

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 metres; and so on.

NOTE for other astronomical bodies: 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 GM/r where G is the gravitational constant, M is the mass of the astronomical body, and r is the radius from the falling object to the center of the body. Values obtained are correct only in cases where the distance of fall d is small compared with r.

[edit] Gravitational potential

For any mass distribution there is a scalar field, the gravitational potential (a scalar potential), which is the gravitational potential energy per unit mass of a point mass, as function of position. It is

$- G \int{1 \over r} dm$

where the integral is taken over all mass. Minus its gradient is the gravity field itself, and minus its Laplacian is the divergence of the gravity field, which is everywhere equal to -4πG times the local density.

Thus when outside masses the potential satisfies Laplace's equation (i.e., the potential is a harmonic function), and when inside masses the potential satisfies Poisson's equation with, as right-hand side, 4πG times the local density.

[edit] Acceleration relative to the rotating Earth

The acceleration measured on the rotating surface of the Earth is not quite the same as the acceleration that is measured for a free-falling body because of the centrifugal force. In other words, 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.