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{{Short description|Motion of a body subject only to gravity}}
{{Other uses|Free fall (disambiguation)}}

In [[classical mechanics]], '''free fall''' is any motion of a [[physical object|body]] where [[gravity]] is the only [[force]] acting upon it.
A freely falling object may not necessarily be falling down in the [[vertical direction]]. An object moving upwards might not normally be considered to be falling, but if it is subject to only the force of gravity, it is said to be in free fall.  The [[Moon]] is thus in free fall around the [[Earth]], though its [[orbital speed]] keeps it in [[orbit of the Moon|very far orbit]] from the [[Earth's surface]].

In a roughly uniform [[gravitational field]] gravity acts on each part of a body approximately equally.  When there are no other forces, such as the [[normal force]] exerted between a body (e.g. an [[astronaut]] in orbit) and its surrounding objects, it will result in the sensation of [[weightlessness]], a condition that also occurs when the gravitational field is weak (such as when far away from any source of gravity).

The term "free fall" is often used more loosely than in the strict sense defined above. Thus, falling through an [[atmosphere]] without a deployed [[parachute]], or lifting device, is also often referred to as ''free fall''. The [[aerodynamic]] drag forces in such situations prevent them from producing full weightlessness, and thus a skydiver's "free fall" after reaching [[terminal velocity]] produces the sensation of the body's weight being supported on a cushion of air.

In the context of [[general relativity]], where gravitation is reduced to a [[General relativity|space-time curvature]], a body in free fall has no force acting on it.

==History==
{{Main|History of gravitational theory}}

In the Western world prior to the 16th century, it was generally assumed that the speed of a falling body would be proportional to its weight—that is, a 10&nbsp;kg object was expected to fall ten times faster than an otherwise identical 1&nbsp;kg object through the same medium. The ancient Greek philosopher [[Aristotle]] (384–322 BC) discussed falling objects in ''[[Physics (Aristotle)|Physics]]'' (Book VII), one of the oldest books on [[mechanics]] (see [[Aristotelian physics]]). Although, in the 6th century, [[John Philoponus]] challenged this argument and said that, by observation, two balls of very different weights will fall at nearly the same speed.<ref>{{cite book|editor-first1=Morris R. |editor-last1=Cohen |editor-first2=I. E. |editor-last2=Drabkin |year=1958 |title=A Source Book in Greek Science |page=220 |location=Cambridge, MA |publisher=Harvard University Press}}</ref>

In 12th-century Iraq, [[Abu'l-Barakāt al-Baghdādī]] gave an explanation for the [[gravitational acceleration]] of falling bodies. According to [[Shlomo Pines]], al-Baghdādī's theory of motion was "the oldest negation of Aristotle's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law of [[classical mechanics]] [namely, that a force applied continuously produces acceleration]."<ref>{{cite encyclopedia  | last = Pines  | first = Shlomo  | title = Abu'l-Barakāt al-Baghdādī , Hibat Allah  | encyclopedia = [[Dictionary of Scientific Biography]]  | volume = 1  | pages = 26–28   | publisher = Charles Scribner's Sons  | location = New York  | year = 1970  | isbn = 0-684-10114-9 }} <br> ([[cf.]] Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", ''Journal of the History of Ideas'' '''64''' (4), pp. 521–546 [528].)</ref>

=== Galileo Galilei ===
{{See also|Galileo Galilei#Falling bodies}}{{See also|Galileo's Leaning Tower of Pisa experiment}}
According to a tale that may be apocryphal, in 1589–1592 Galileo [[Galileo's Leaning Tower of Pisa experiment|dropped two objects of unequal mass from the Leaning Tower of Pisa]]. Given the speed at which such a fall would occur, it is doubtful that Galileo could have extracted much information from this experiment. Most of his observations of falling bodies were really of bodies rolling down ramps. This slowed things down enough to the point where he was able to measure the time intervals with [[water clock]]s and his own pulse (stopwatches having not yet been invented). He repeated this "a full hundred times" until he had achieved "an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse beat." In 1589–1592, Galileo wrote ''[[De Motu Antiquiora]]'', an unpublished manuscript on the motion of falling bodies.{{Citation needed|date=October 2019}}

==Examples==
{{original research|section=1|date=July 2020}}

Examples of objects in free fall include:
* A [[spacecraft]] (in space) with propulsion off (e.g. in a continuous orbit, or on a suborbital trajectory ([[ballistics]]) going up for some minutes, and then down).
* An object dropped at the top of a [[drop tube]].
* An object thrown upward or a person jumping off the ground at low speed (i.e. as long as air resistance is negligible in comparison to weight).
Technically, an object is in free fall even when moving upwards or instantaneously at rest at the top of its motion. If gravity is the only influence acting, then the acceleration<ref>{{cite web | url=https://www.feynmanlectures.caltech.edu/I_08.html#8-5_acceleration | title=The Feynman Lectures on Physics Vol. I Ch. 8: Motion }}</ref> is always downward and has the same magnitude for all bodies, commonly denoted <math>g</math>.

Since all objects fall at the same rate in the absence of other forces, objects and people will experience [[weightlessness]] in these situations.

Examples of objects not in free-fall:
* Flying in an aircraft: there is also an additional force of [[lift (force)|lift]].
* Standing on the ground: the gravitational force is counteracted by the [[normal force]] from the ground.
* Descending to the Earth using a parachute, which balances the force of gravity with an aerodynamic drag force (and with some parachutes, an additional lift force).

The example of a falling skydiver who has not yet deployed a parachute is not considered free fall from a physics perspective, since they experience a [[drag force]] that equals their weight once they have achieved [[terminal velocity]] (see below).

[[File:Drop time.jpg|thumb|Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of <math>\sqrt{2h/g}</math>, where h is the height and g is the free-fall acceleration due to gravity.]]

Near the surface of the Earth, an object in free fall in a vacuum will accelerate at approximately 9.8&nbsp;m/s<sup>2</sup>, independent of its [[mass]]. With air resistance acting on an object that has been dropped, the object will eventually reach a terminal velocity, which is around 53&nbsp;m/s (190&nbsp;km/h or 118&nbsp;mph<ref name="Greenharbor"/>) for a human skydiver. The terminal velocity depends on many factors including mass, [[drag coefficient]], and relative surface area and will only be achieved if the fall is from sufficient altitude. A typical skydiver in a spread-eagle position will reach terminal velocity after about 12 seconds, during which time they will have fallen around 450&nbsp;m (1,500&nbsp;ft).<ref name="Greenharbor">{{cite web | url=http://www.greenharbor.com/fffolder/speedtime.pdf | title=Free fall graph | publisher=Green Harbor Publications | date=2010 | access-date=14 March 2016}}</ref>

Free fall was demonstrated on the Moon by astronaut [[David Scott]] on August 2, 1971. He simultaneously released a hammer and a feather from the same height above the Moon's surface. The hammer and the feather both fell at the same rate and hit the surface at the same time. This demonstrated Galileo's discovery that, in the absence of air resistance, all objects experience the same acceleration due to gravity. On the Moon, however, the [[gravitational acceleration]] is approximately 1.63&nbsp;m/s<sup>2</sup>, or only about <sup>1</sup>⁄<sub>6</sub> that on Earth.

==Free fall in Newtonian mechanics==
{{Main|Newtonian mechanics}}

=== Uniform gravitational field without air resistance ===
This is the "textbook" case of the vertical motion of an object falling a small distance close to the surface of a planet. It is a good approximation in air as long as the force of gravity on the object is much greater than the force of air resistance, or equivalently the object's velocity is always much less than the terminal velocity (see below).

[[File:Free-fall.gif|right|100px|Free-fall]]
:<math>v(t)=v_{0}-gt\,</math>

:<math>y(t)=v_{0}t+y_{0}-\frac{1}{2}gt^2</math>

where
:<math>v_{0}\,</math> is the initial vertical component of the velocity (m/s).
:<math>v(t)\,</math> is the vertical component of the velocity at <math>t\,</math>(m/s).
:<math>y_{0}\,</math> is the initial altitude (m).
:<math>y(t)\,</math> is the altitude at <math>t\,</math>(m).
:<math>t\,</math> is time elapsed (s).
:<math>g\,</math> is the acceleration due to [[gravity]] (9.81 m/s<sup>2</sup> near the surface of the earth).
If the initial velocity is zero, then the distance fallen from the initial position will grow as the square of the elapsed time. Moreover, because [[square number#Properties|the odd numbers sum to the perfect squares]], the distance fallen in successive time intervals grows as the odd numbers. This description of the behavior of falling bodies was given by Galileo.<ref>{{Cite book|last1=Olenick|first1=Richard P.|url=https://books.google.com/books?id=xMWwTpn53KsC&pg=PA18|title=The Mechanical Universe: Introduction to Mechanics and Heat|last2=Apostol|first2=Tom M.|last3=Goodstein|first3=David L.|year=2008|publisher=Cambridge University Press|isbn=978-0-521-71592-8|page=18|language=en}}</ref>

=== Uniform gravitational field with air resistance ===
[[File:MeteorAccGraph.jpg|thumb|Acceleration of a small meteoroid when entering the Earth's atmosphere at different initial velocities]]

This case, which applies to skydivers, parachutists or any body of mass, <math>m</math>, and cross-sectional area, <math>A</math>, with [[Reynolds number]] well above the critical Reynolds number, so that the air resistance is proportional to the square of the fall velocity, <math>v</math>, has an equation of motion
:<math>m\frac{\mathrm{d}v}{\mathrm{d}t}=mg - \frac{1}{2} \rho C_{\mathrm{D}} A v^2 \, ,</math>
where <math>\rho</math> is the [[Density of air|air density]] and <math>C_{\mathrm{D}}</math> is the [[drag coefficient]], assumed to be constant although in general it will depend on the Reynolds number.

Assuming an object falling from rest and no change in air density with altitude, the solution is:
: <math>v(t) = v_{\infty}\tanh\left(\frac{gt}{v_{\infty}}\right),</math>

where the [[terminal speed]] is given by
:<math>v_{\infty}=\sqrt{\frac{2mg}{\rho C_D A}} \, .</math>

The object's speed versus time can be integrated over time to find the vertical position as a function of time:

:<math>y = y_0 - \frac{v_{\infty}^2}{g} \ln \cosh\left(\frac{gt}{v_\infty}\right).</math>

Using the figure of 56&nbsp;m/s for the terminal velocity of a human, one finds that after 10 seconds he will have fallen 348 metres and attained 94% of terminal velocity, and after 12 seconds he will have fallen 455 metres and will have attained 97% of terminal velocity. However, when the air density cannot be assumed to be constant, such as for objects falling from high altitude, the equation of motion becomes much more difficult to solve analytically and a numerical simulation of the motion is usually necessary. The figure shows the forces acting on meteoroids falling through the Earth's upper atmosphere. [[High-altitude military parachuting#High Altitude Low Opening – HALO|HALO jump]]s, including [[Joe Kittinger]]'s and [[Felix Baumgartner]]'s record jumps, also belong in this category.<ref>An analysis of such jumps is given in {{cite journal|title=High altitude free fall|journal= American Journal of Physics |volume= 64 |issue= 10 |page= 1242 |author1=Mohazzabi, P. |author2=Shea, J. |doi=10.1119/1.18386|bibcode=1996AmJPh..64.1242M|url=http://www.jasoncantarella.com/downloads/AJP001242.pdf|year= 1996 }}</ref>

=== Inverse-square law gravitational field ===
It can be said that two objects in space orbiting each other in the absence of other forces are in free fall around each other, e.g. that the Moon or an artificial satellite "falls around" the Earth, or a planet "falls around" the Sun. Assuming spherical objects means that the equation of motion is governed by [[Newton's law of universal gravitation]], with solutions to the [[gravitational two-body problem]] being [[elliptic orbits]] obeying [[Kepler's laws of planetary motion]]. This connection between falling objects close to the Earth and orbiting objects is best illustrated by the thought experiment, [[Newton's cannonball]].

The motion of two objects moving radially towards each other with no [[angular momentum]] can be considered a special case of an elliptical orbit of [[Orbital eccentricity|eccentricity]] {{nowrap|''e'' {{=}} 1}} ([[radial elliptic trajectory]]). This allows one to compute the [[free-fall time]] for two point objects on a radial path. The solution of this equation of motion yields time as a function of separation:
:<math>t(y)= \sqrt{ \frac{ {y_0}^3 }{2\mu} } \left(\sqrt{\frac{y}{y_0}\left(1-\frac{y}{y_0}\right)} + \arccos{\sqrt{\frac{y}{y_0}}}
 \right),</math>

where
:<math>t</math> is the time after the start of the fall
:<math>y</math> is the distance between the centers of the bodies
:<math>y_0</math> is the initial value of <math>y</math>
:<math>\mu = G(m_1 + m_2)</math> is the [[standard gravitational parameter]].

Substituting <math> y = 0</math> we get the [[free-fall time]]

:<math>t_{\text{ff}}=\pi\sqrt{y^3_0/(8\mu)}~.</math>

The separation can be expressed explicitly as a function of time <ref>{{cite journal |last1=Obreschkow |first1=Danail |title=From Cavitation to Astrophysics: Explicit Solution of the Spherical Collapse Equation |journal=Phys. Rev. E |date=7 June 2024 |volume=109 |issue=6 |page=065102 |doi=10.1103/PhysRevE.109.065102 |pmid=39021019 |url=https://link.aps.org/doi/10.1103/PhysRevE.109.065102|arxiv=2401.05445 |bibcode=2024PhRvE.109f5102O }}</ref>

:<math>y(t)=y_0~Q\left(1-\frac{t}{t_{\text{ff}}};\frac{3}{2},\frac{1}{2}\right) ~,</math>

where <math>Q(x;\alpha,\beta)</math> is the quantile function of the [[Beta distribution]], also known as the [[inverse function]] of the [[regularized incomplete beta function]] <math>I_x(\alpha,\beta)</math>.

This solution can also be represented exactly by the analytic power series

:<math> y( t ) = \sum_{n=1}^{ \infty }
\left[
 \lim_{ r \to 0 } \left(
  {\frac{ x^{ n }}{ n! }}
   \frac{\mathrm{d}^{\,n-1}}{\mathrm{ d } r ^{\,n-1}} \left[
    r^n \left( \frac{ 7 }{ 2 } ( \arcsin( \sqrt{ r } ) - \sqrt{ r - r^2 } ) 
   \right)^{ - \frac{2}{3} n }
  \right] \right)
 \right].
</math>

Evaluating this yields:<ref>{{cite journal|doi=10.1088/0143-0807/29/5/012|title=From Moon-fall to motions under inverse square laws|journal=European Journal of Physics|volume=29|issue=5|pages=987–1003|year=2008|last1=Foong|first1=S K|bibcode=2008EJPh...29..987F|s2cid=122494969 |doi-access=free}}</ref><ref>{{cite journal|doi=10.1119/1.3246467|url=https://apps.dtic.mil/sti/pdfs/ADA534896.pdf|title=Radial Motion of Two Mutually Attracting Particles|journal=The Physics Teacher|volume=47|issue=8|pages=502–507|year=2009|last1=Mungan|first1=Carl E.|bibcode=2009PhTea..47..502M}}</ref>
:<math>y(t)=y_0 \left( x - \frac{1}{5} x^2 - \frac{3}{175}x^3 
 - \frac{23}{7875}x^4 - \frac{1894}{3031875}x^5 - \frac{3293}{21896875}x^6 - \frac{2418092}{62077640625}x^7 - \cdots \right) \ , 
</math>
where

<math> x = \left[\frac{3}{2} \left( \frac{\pi}{2}- t \sqrt{ \frac{2\mu}{ {y_0}^3 } } \right) \right]^{2/3}. </math>

==In general relativity==
{{further|General relativity}}

In general relativity, an object in free fall is subject to no force and is an inertial body moving along a [[geodesics in general relativity|geodesic]]. Far away from any sources of space-time curvature, where [[spacetime]] is flat, the Newtonian theory of free fall agrees with general relativity. Otherwise the two disagree; e.g., only general relativity can account for the [[precession]] of orbits, the [[orbital decay]] or inspiral of compact binaries due to [[gravitational waves]], and the relativity of direction ([[geodetic precession]] and [[frame dragging]]).

The experimental observation that all objects in free fall accelerate at the same rate, as noted by Galileo and then embodied in Newton's theory as the equality of gravitational and inertial masses, and later confirmed to high accuracy by modern forms of the [[Eötvös experiment]], is the basis of the [[equivalence principle]], from which basis Einstein's theory of general relativity initially took off.

== See also ==
* [[Equations for a falling body]]
* [[G-force]]
* [[High-altitude military parachuting]]
* [[Reduced-gravity aircraft]]
* [[Terminal velocity]]
* [[Weightlessness]]

==References==
{{Reflist}}

==External links==
*[http://www.fxsolver.com/solve/share/ktnuXhnUCqwPW0o1bU1XyQ==/ Freefall formula calculator]
*[http://www.phy6.org/stargaze/Sfall.htm The Way Things Fall] an educational website

{{Commons category}}

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