# Free fall **Free fall** is motion in which [[Gravity|gravity]] is the only [[Force|force]] acting, so that the body's acceleration is the same whatever it is made of and however heavy it is. Near the surface of the [[Earth|Earth]] that acceleration is g = 9.8 m/s², and the kinematics reduce to v = g·t and s = ½·g·t².[^b077-kin] The difficulty is experimental rather than conceptual: a body dropped from a table is on the floor in under half a second, which no seventeenth-century clock could resolve. [[Galileo_Galilei|Galileo]]'s answer was to slow the fall without changing its character, and that is what the microsim below does. The reader tilts an [[Inclined_plane|inclined plane]] from 1° to 90° and watches a ball roll under a = g·sin(θ): the distances covered in successive equal ticks stay in the ratio 1 : 3 : 5 : 7 at every angle, so the odd-number rule that holds on a gentle slope is the same rule that holds in vertical fall. A [[Friction|friction]] toggle switches the acceleration to a = g·(sin(θ) − μ_k·cos(θ)), and with wood on wood the block refuses to start below 26.6° and refuses to stop above 16.7°. On the [[Physics|Physics]] flagship this article serves Part I — History at *Scientific Revolution* (row P5), one step after [[Aristotelian_physics|Aristotelian physics]] on the same spine and using the same sim family. The pairing is the point: Aristotle's rule that speed is set by weight over resistance is a statement about the steady state, and free fall is the case with no resistance at all, where there is no steady state and the whole content of the motion is in the transient. ## History For nearly two thousand years fall was described as motion at a speed proportional to weight, and the observation that a body speeds up as it descends was handled by saying that it grows more eager as it nears its natural place. The medieval correction came in two stages: [[John_Philoponus|Philoponus]] in the sixth century denied that heavier bodies fall much faster, and the fourteenth-century Latin schools produced the kinematics of uniformly accelerated motion — the Merton mean speed theorem and Nicole Oresme's proof of it by area — without applying it to falling bodies.[^clagett] Joining the two took another two centuries. ### Domingo de Soto In his commentary on Aristotle's *Physics*, printed in 1551, the Spanish Dominican Domingo de Soto stated that a body falling freely is *uniformiter difformis* with respect to time: its speed increases in equal amounts in equal times. That is exactly the modern statement, arrived at before any measurement could have suggested it, and it carries with it the Merton theorem's consequence that distance grows as the square of the time.[^soto] Soto offered no experiment and does not seem to have regarded the claim as remarkable; he applied the schoolmen's kinematics to the one case they had left alone. His priority was recovered only in the twentieth century, and it complicates the usual story in which the [[Modern_physics|modern]] account of fall begins in 1638. ### Galileo Galilei Galileo's contribution was not the law but the measurement. In the *Discorsi* of 1638 he describes a beam about twelve cubits long with a straight, smooth channel cut along it, raised at one end by one or two cubits, down which a bronze ball was released while water running into a vessel measured the time.[^galileo] The device is a clock amplifier. On a plane of inclination θ the acceleration along the slope is a = g·sin(θ), so the run time over a fixed length is longer than the free-fall time over that length by a factor 1/√(sin θ). At one cubit in twelve, sin θ = 0.083 and the dilation factor is 3.46: a seven-metre run that would take 1.20 s in free fall takes 4.14 s on the plane (derived).[^b077-kin] That is a time a water clock can divide. This is what the microsim reproduces. Its one control is the tilt angle θ from 1° to 90°, and its HUD carries `a = g*sin(theta) ; s = 0.5*a*t^2`. Markers drop on the track at equal time ticks, and the intervals between them stay in the ratio 1 : 3 : 5 : 7 : 9 whatever the tilt, because s ∝ t² makes the increment between tick n and tick n+1 proportional to (n+1)² − n² = 2n + 1. Galileo's odd-number rule is therefore not a fact about gentle slopes that has to be extrapolated to the vertical; it is the signature of constant acceleration, and the vertical case at θ = 90° is simply the end of the slider where a = g. The reader can also check that the ratio of run times between any two angles is √(sin θ₂/sin θ₁), independent of the length of the track. The friction toggle adds what Galileo could only minimise. With the empirical laws f_s ≤ μ_s·N and f_k = μ_k·N and N = m·g·cos(θ), a block on the plane stays put until tan(θ) exceeds μ_s and then accelerates at a = g·(sin(θ) − μ_k·cos(θ)); once moving it keeps moving until tan(θ) falls below μ_k.[^b077-fric] For wood on wood the book's table gives μ_s = 0.5 and μ_k = 0.3, so the start angle is arctan 0.5 = 26.57° and the stop angle arctan 0.3 = 16.70°, a hysteresis band almost ten degrees wide (derived). At 30° the sliding acceleration is 9.8·(0.500 − 0.3 × 0.866) = 2.35 m/s², against 4.90 m/s² for the same tilt with the toggle off (derived). The displayed equations for the incline were lost in the extraction of Portal Book 077 and the forms used here are the standard ones fixed by the book's own stated force components, to be checked against the printed pages.[^b077-fric] ## Examples The clearest modern examples are not falls at all. An astronaut aboard a station orbiting 420 km up is in free fall continuously: the local gravitational acceleration there is still 8.64 m/s², 88 per cent of its value at the ground, and the orbital speed is 7.66 km/s with a period of 92.8 minutes (all derived).[^b077-grav] Nothing has cancelled gravity; the station and everything in it are falling together, so nothing presses on anything else. "Weightlessness" names the absence of contact forces, not the absence of weight, and [[Newton's_cannonball|Newton's cannonball]] is the argument in its original form: fire the ball fast enough and it falls all the way round. Terrestrial free fall is short. Because s = ½·g·t², seconds cost depth quadratically: one second needs 4.9 m, two seconds 19.6 m, five seconds 123 m and ten seconds 490 m (derived). Drop towers of the order of a hundred metres therefore buy under five seconds of it, which is why microgravity research also uses aircraft flying ballistic arcs — an aircraft that enters the arc with 100 m/s of upward speed and leaves it with 100 m/s downward is in free fall for 2v/g, about twenty seconds (derived). A skydiver is in free fall only for the first few seconds; after that [[Drag_(physics)|air drag]] has grown to match weight and the fall is steady rather than free, at roughly 200 km/h spread-eagle or 350 km/h head-first.[^b077-drag] ## Free fall in Newtonian mechanics Three cases have to be separated, and the microsim's slider only covers the first. In a uniform field with no resistance the motion is exactly parabolic in time; with resistance it is exponential or hyperbolic, depending on the drag law; and over distances comparable with the Earth's radius the field is not uniform and the problem becomes a degenerate [[Orbit|orbit]]. ### Uniform gravitational field without air resistance Taking down as positive and starting from rest, [[Newton's_laws_of_motion|Newton's second law]] gives m·dv/dt = m·g, in which the mass cancels — the single fact that makes free fall universal — leaving v = g·t and s = ½·g·t².[^b077-kin] Eliminating t gives v² = 2·g·s, so a body dropped 10 m arrives at 14.0 m/s after 1.43 s, and one dropped from 100 m at 44.3 m/s after 4.52 s (derived). The cancellation is the equality of inertial and gravitational mass, and it is an experimental fact rather than a theorem of the theory: Newtonian mechanics would be perfectly consistent if the two masses differed, and it is only in [[General_relativity|general relativity]] that their equality becomes structural. Galileo's odd-number rule is the same statement in difference form, and the [[Conservation_of_energy|conservation of energy]] gives a third: the loss of [[Potential_energy|gravitational potential energy]] m·g·h equals the gain of [[Kinetic_energy|kinetic energy]] ½·m·v², which again has m on both sides. ### Uniform gravitational field with air resistance With resistance the mass stops cancelling, and the whole of Aristotle's physics reappears as the steady state. For a small, slow body the drag is linear, m·dv/dt = m·g − k·v, whose solution from rest is v(t) = v_T·(1 − exp(−k·t/m)) with [[Terminal_velocity|terminal velocity]] v_T = m·g/k and time constant τ = m/k.[^b032-drag] For a large, fast body the drag goes as v², F_D = ½·C·ρ·A·v², and the terminal velocity is v_T = √(2·m·g/(ρ·C·A)); the solution from rest is v(t) = v_T·tanh(g·t/v_T).[^b077-drag] A worked case fixes the scale. A 2.5 g coin dropped 381 m would reach the ground in 8.8179 s at 86.4 m/s if there were no air; with the quadratic law and b = 7.5×10⁻⁵ kg/m it arrives in 22.4 s at 18.1 m/s, and the terminal velocity √(m·g/b) is 18.07 m/s.[^b085-penny] The fall is free for well under a second and steady for the remaining twenty-one. This is why the question "do heavy and light bodies fall together?" has no single answer at the bottom of an atmosphere, and why the decisive demonstration had to wait for a vacuum — or for the Moon, where David Scott dropped a hammer and a falcon feather side by side on 2 August 1971 and they landed together.[^apollo15] ### Inverse-square law gravitational field Over heights small compared with the Earth's radius R = 6,371 km the field is uniform to a fraction of a per cent, but over larger distances g must be replaced by [[Newton's_law_of_universal_gravitation|G·M/r²]].[^b077-grav] Radial fall from rest then has no elementary solution in time, but energy conservation gives the speed directly: v² = 2·G·M·(1/r − 1/r₀). Letting r₀ → ∞ gives the [[Escape_velocity|escape speed]] √(2·G·M/R) = 11.2 km/s at the Earth's surface, and the same expression run the other way says that a body released from rest very far away arrives at that speed (derived). The fall time is the degenerate limit of [[Kepler's_laws_of_planetary_motion|Kepler's third law]], an ellipse of zero width whose semi-major axis is r₀/2, giving t = (π/2)·√(r₀³/(2·G·M)). Released from rest at the [[Moon|Moon]]'s mean distance of 384,400 km, a body would reach the Earth in 4.85 days (derived).[^b080-idema] A circular [[Orbit|orbit]] grazing the surface would take 84.3 minutes and require 7.9 km/s — the numbers that separate falling down from falling around. ## In general relativity [[Albert_Einstein|Einstein]] took the cancellation of mass in Newtonian free fall and made it the foundation of the theory. If every body falls the same way, then a frame falling with them is indistinguishable, over a small enough region and a short enough time, from a frame far from all gravitation: this is the [[Equivalence_principle|equivalence principle]], which Einstein called the happiest thought of his life and first published in 1907.[^einstein] In general relativity a freely falling body is therefore not accelerated at all. It follows a [[Geodesic|geodesic]] of curved [[Spacetime|spacetime]], the straightest available path, and what an accelerometer reads in free fall is zero. The body standing on the ground is the accelerated one, pushed upward off its geodesic by the floor. Gravity does not disappear from the falling frame entirely, because the equivalence holds only locally. Two bodies falling side by side toward the same centre converge, and two separated along the fall direction draw apart; that relative acceleration is [[Geodesics_in_general_relativity|geodesic deviation]], it is proportional to the tidal curvature, and no choice of frame removes it. Time also runs differently at different heights: Pound and Rebka measured the predicted shift of gamma-ray frequency over 22.5 m of a Harvard tower in 1960, to within about ten per cent of the prediction.[^pound-rebka] Because the whole structure rests on the universality of free fall, testing it has been a continuing programme of [[Tests_of_general_relativity|precision experiments]] — torsion balances from Eötvös onward, lunar laser ranging, and the MICROSCOPE satellite, which compared two test masses of platinum and titanium in orbital free fall and found their accelerations equal to about one part in 10¹⁵.[^microscope] Free fall began as the hardest motion to time and has become one of the most precisely tested statements in [[Classical_physics|physics]]. ## See also - [[Inclined_plane]] - [[Galileo's_Leaning_Tower_of_Pisa_experiment]] - [[Newton's_laws_of_motion]] - [[Terminal_velocity]] - [[Aristotelian_physics]] - [[Equivalence_principle]] - [[Escape_velocity]] ## References [^b077-kin]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. Chapter 3, "Motion Along a Straight Line", pp. 109–158 (constant acceleration, v = g·t and s = ½·g·t², free fall as the case a = g; page to pin). Portal Book 077. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^b077-fric]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, Chapter 6, "Applications of Newton's Laws", pp. 278–287 (f_s ≤ μ_s·N and f_k = μ_k·N as empirical laws; Table 6.1 of coefficients, wood on wood 0.5 and 0.3; the incline with N = m·g·cos θ; page to pin). The displayed incline equations on pp. 283 and 287 were lost in text extraction; a = g·(sin θ − μ_k·cos θ) and μ_k = tan θ are quoted here as the standard forms fixed by the book's own stated force components, to be verified against the PDF pages. [^b077-drag]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, Chapter 6, pp. 297–304 (drag; the quadratic law for large, fast bodies and Stokes' linear law for small, slow ones; skydiver speeds of roughly 350 km/h head-first and 200 km/h spread-eagle; page to pin). The drag equations on these pages were lost in extraction and F_D = ½·C·ρ·A·v² and v_T = √(2·m·g/(ρ·C·A)) are quoted as standard forms. [^b077-grav]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, Chapter 13, "Gravitation", pp. 611–664 (the inverse-square field, escape speed and circular orbital speed; page to pin), with Appendix D "Astronomical Data", pp. 885–886, for the Earth and Moon parameters used in the derived numbers. [^b080-idema]: Idema, Timon (2018). *Mechanics and Relativity*. Part I, "Classical mechanics", pp. 14–119 (Kepler orbits and the radial limit; page to pin). Portal Book 080. https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^b032-drag]: Trench, William F. (2013). *Elementary Differential Equations with Boundary Value Problems*. Section 4.3, "Elementary Mechanics", pp. 160–162 (the linear-drag equation, its exponential solution and v_T = m·g/k; g as 9.8 m/s²). Portal Book 032. https://open.umn.edu/opentextbooks/textbooks/elementary-differential-equations-with-boundary-value-problems [^b085-penny]: Downey, Allen B. (2021). *Physical Modeling in MATLAB*, version 4.0. Chapter 12, pp. 125–131 (a 2.5 g coin dropped 381 m: 8.8179 s and 86.4153 m/s with no air; 22.4 s and 18.1 m/s with b = 7.5×10⁻⁵ kg/m; √(m·g/b) = 18.07 m/s). Portal Book 085. https://open.umn.edu/opentextbooks/textbooks/physical-modeling-in-matlab [^galileo]: Galilei, Galileo (1638). *Discorsi e dimostrazioni matematiche intorno a due nuove scienze*. Leiden: Elzevir. Third Day, on naturally accelerated motion: the description of the grooved beam, the water clock, and the rule that distances in successive equal times are as the odd numbers (page to pin). [^soto]: Wallace, William A. (1968). "The Enigma of Domingo de Soto: *Uniformiter difformis* and Falling Bodies in Late Medieval Physics." *Isis* 59 (4) (pages to pin). On Soto's 1551 statement that free fall is uniformly accelerated with respect to time. [^clagett]: Clagett, Marshall (1959). *The Science of Mechanics in the Middle Ages*. Madison: University of Wisconsin Press (the Merton mean speed theorem, Oresme's geometrical proof, and Philoponus's denial that fall speed is proportional to weight; page to pin). [^einstein]: Einstein, Albert (1907). "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen." *Jahrbuch der Radioaktivität und Elektronik* 4 (pages to pin). The first published statement of the equivalence of a uniformly accelerated frame and a uniform gravitational field. [^pound-rebka]: Pound, R. V.; Rebka, G. A. (1960). "Apparent Weight of Photons." *Physical Review Letters* 4 (7): 337–341. [^microscope]: Touboul, P., et al. (MICROSCOPE Collaboration) (2022). "MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle." *Physical Review Letters* 129 (article and pages to pin). Equality of the free-fall accelerations of platinum and titanium test masses to about one part in 10¹⁵. [^apollo15]: National Aeronautics and Space Administration (1972). *Apollo 15 Preliminary Science Report*, NASA SP-289 (the lunar surface activities of 30 July – 2 August 1971, including the hammer-and-feather demonstration; page to pin). ## External links - [*University Physics Volume 1*](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1), OpenStax — Portal Book 077, chapter 3 "Motion Along a Straight Line" and chapter 6 "Applications of Newton's Laws" - [*Mechanics and Relativity*](https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity), Idema — Portal Book 080 - [*Physical Modeling in MATLAB*](https://open.umn.edu/opentextbooks/textbooks/physical-modeling-in-matlab), Downey — Portal Book 085, the falling-coin model quoted above <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Free_fall.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Free fall* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Free_fall.html" data-title="Free fall"></div> *Built from `MICROSIM_GUIDE/specs/sims/Free_fall.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Free_fall) : [Wikitube](https://en.wikitube.io/wiki/Free_fall) · pinned revision [1370236774](https://en.wikipedia.org/w/index.php?oldid=1370236774) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P5 · sim pending (matter/Free_fall).*