Equations for a falling body - AbsoluteAstronomy.com

Under normal earth-bound conditions, when objects move owing to a constant gravitation

Gravitation

Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

al force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

a set of dynamical equations describe the resultant trajectories. For example, Newton's law of universal gravitation

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

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.
Please note that in this article any resistance from air (drag

Drag (physics)

In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

) is neglected.

History

Galileo

Galileo Galilei

Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

was the first to demonstrate and then formulate these equations. He used a ramp

Inclined plane

The inclined plane is one of the original six simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. By moving an object up an inclined plane rather than completely vertical, the amount of force required is reduced, at the expense of increasing the...

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

Water clock

A water clock or clepsydra is any timepiece in which time is measured by the regulated flow of liquid into or out from a vessel where the amount is then measured.Water clocks, along with sundials, are likely to be the oldest time-measuring instruments, with the only exceptions...

, using an "extremely accurate balance" to measure the amount of water.See the works of Stillman Drake

Stillman Drake

Stillman Drake was a Canadian historian of science best known for his work on Galileo Galilei . Drake published over 131 books, articles, and book chapters on Galileo. Drake received his first academic appointment in 1967 as full professor at the University of Toronto after a career as a...

, for a comprehensive study of Galileo

Galileo Galilei

and his times, the Scientific Revolution

Scientific revolution

The Scientific Revolution is an era associated primarily with the 16th and 17th centuries during which new ideas and knowledge in physics, astronomy, biology, medicine and chemistry transformed medieval and ancient views of nature and laid the foundations for modern science...

.

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

Terminal velocity

In fluid dynamics an object is moving at its terminal velocity if its speed is constant due to the restraining force exerted by the fluid through which it is moving....

. 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

David Scott

David Randolph Scott is an American engineer, test pilot, retired U.S. Air Force officer, and former NASA astronaut and engineer, who was one of the third group of astronauts selected by NASA in October 1963...

demonstrated by dropping a hammer and a feather on the surface of the Moon

Moon

The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

.)

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

Coriolis effect

In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...

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, 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

International System of Units

The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of 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). For a body encountering a thick atmosphere like the Earth's near sea level, terminal velocity

Terminal velocity

In fluid dynamics an object is moving at its terminal velocity if its speed is constant due to the restraining force exerted by the fluid through which it is moving....

is reached exponentially between 8 and 15 seconds, after which a steady velocity of very approximately 100 m/s is maintained for compact objects with densities between those of water and common metals.

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.

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.

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 meters. 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 meters to Earth, the results of both equations differ by only 0.08%. However, if the distance increases to that of geosynchronous orbit

Geosynchronous orbit

A geosynchronous orbit is an orbit around the Earth with an orbital period that matches the Earth's sidereal rotation period...

, 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

Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

, 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 parameter

Standard gravitational parameter

In astrodynamics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the body.\mu=GM \ The SI units of the standard gravitational parameter are m3s−2....

s 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

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 centripetal force

Centripetal force

Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

. 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.

History

Overview

Acceleration relative to the rotating Earth

See also