Editing Terminal velocity

{{Short description|Highest velocity attainable by a falling object}}
{{other uses}}
{{More citations needed|article|date=March 2012}}
[[File:Terminal velocity.svg|thumb|upright=0.7|The downward force of gravity (''F<sub>g</sub>'') equals the restraining force of drag (''F<sub>d</sub>'') plus the buoyancy. The net force on the object is zero, and the result is that the velocity of the object remains constant.]]

'''Terminal velocity''' is the maximum speed attainable by an object as it falls through a [[fluid]] ([[air]] is the most common example). It is reached when the sum of the [[Drag (physics)|drag]] force (''F<sub>d</sub>'') and the [[buoyancy]] is equal to the downward force of [[gravity]] (''F<sub>G</sub>'') acting on the object. Since the [[net force]] on the object is zero, the object has zero [[acceleration]].<ref>{{Cite web |title=6.4 Drag Force and Terminal Speed - University Physics Volume 1 {{!}} OpenStax |url=https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed |access-date=2023-07-15 |website=openstax.org |date=19 September 2016 |language=en}}</ref><ref>{{cite journal |last1=Riazi |first1=A. |last2=Türker |first2=U. | date=January 2019 |title=The drag coefficient and settling velocity of natural sediment particles |journal=Computational Particle Mechanics |volume=6 |issue=3 |pages=427–437 | doi=10.1007/s40571-019-00223-6|bibcode=2019CPM.....6..427R |s2cid=127789299 }}</ref> For objects falling through air at normal pressure, the buoyant force is usually dismissed and not taken into account, as its effects are negligible.

As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). At some speed, the drag or force of resistance will be equal to the gravitational pull on the object. At this point the object stops accelerating and continues falling at a constant speed called the terminal velocity (also called '''settling velocity'''). 

An object moving downward faster than the terminal velocity (for example because it was thrown downwards, it fell from a thinner part of the atmosphere, or it changed shape) will slow down until it reaches the terminal velocity. Drag depends on the [[projected area]], here represented by the object's cross-section or silhouette in a horizontal plane.

An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. In general, for the same shape and material, the terminal velocity of an object increases with size. This is because the downward force (weight) is proportional to the cube of the linear dimension, but the air resistance is approximately proportional to the cross-section area which increases only as the square of the linear dimension. 

For very small objects such as dust and mist, the terminal velocity is easily overcome by convection currents which can prevent them from reaching the ground at all, and hence they can stay suspended in the air for indefinite periods. Air pollution and fog are examples.

==Examples==
[[File:Graph of velocity versus time of a skydiver reaching a terminal velocity.svg|thumb|Graph of velocity versus time of a skydiver reaching a terminal velocity.]]

Based on air resistance, for example, the terminal speed of a [[skydiving|skydiver]] in a belly-to-earth (i.e., face down) [[free fall]] position is about {{convert|55|m/s|ft/s|round=5|abbr=on}}.<ref name="Huang1999">{{cite web
| url=https://hypertextbook.com/facts/1998/JianHuang.shtml
| title=Speed of a skydiver (terminal velocity)
| first=Jian
| last=Huang
| year=1998
| website=The Physics Factbook
| editor-last=Elert
| editor-first=Glenn
| accessdate=2022-01-25
}}</ref> This speed is the [[asymptotic]] limiting value of the speed, and the forces acting on the body balance each other more and more closely as the terminal speed is approached. In this example, a speed of 50.0% of terminal speed 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 their limbs (see also [[freeflying]]).<ref name="Huang1999" /> In this case, the terminal speed increases to about {{convert|90|m/s|ft/s|abbr=on|-1}},{{Citation needed|date=October 2024}} which is almost the terminal speed of the [[peregrine falcon]] diving down on its prey.<ref>{{cite web |url=http://www.fws.gov/endangered/recovery/peregrine/QandA.html |archive-url=https://web.archive.org/web/20100308202440/http://www.fws.gov/endangered/recovery/peregrine/QandA.html |title=All About the Peregrine Falcon |publisher=U.S. Fish and Wildlife Service |date=December 20, 2007|archive-date=March 8, 2010}}</ref> The same terminal speed is reached for a typical [[.30-06 Springfield|.30-06]] bullet dropping downwards—when it is returning to the ground having been fired upwards or dropped from a tower—according to a 1920 U.S. Army Ordnance study.<ref>{{cite web |url=http://www.loadammo.com/Topics/March01.htm |title=Bullets in the Sky |author=The Ballistician |publisher=W. Square Enterprises, 9826 Sagedale, Houston, Texas 77089 |date=March 2001 |url-status=dead |archive-url=https://web.archive.org/web/20080331192517/http://www.loadammo.com/Topics/March01.htm |archive-date=2008-03-31 }}</ref>

Competition [[Speed skydiving|speed skydivers]] fly in a head-down position and can reach speeds of {{convert|150|m/s|ft/s|abbr=on|sigfig=2}}.{{citation needed|date=January 2022}} The current record is held by [[Felix Baumgartner]] who jumped from an altitude of {{convert|127582|ft|m|abbr=on|order=flip}} and reached {{convert|380|m/s|ft/s|abbr=on|sigfig=2}}, though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force.<ref>{{cite journal|title=Physiological Monitoring and Analysis of a Manned Stratospheric Balloon Test Program|last1=Garbino|first1=Alejandro|last2=Blue|first2=Rebecca S.|last3=Pattarini|first3=James M.|last4=Law|first4=Jennifer|last5=Clark|first5=Jonathan B.|journal=[[Aviation, Space, and Environmental Medicine]]|date=February 2014|volume=85|issue=2|pages=177–178|doi=10.3357/ASEM.3744.2014|pmid=24597163 |doi-access=free}}</ref>
 
The biologist [[J. B. S. Haldane]] wrote, {{Quote|To the mouse and any smaller animal [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal's length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force.<ref>{{cite magazine |last=Haldane |first=J. B. S. |author-link=J. B. S. Haldane |date=March 1926 |title=On Being the Right Size |title-link=On Being the Right Size |magazine=Harper's Magazine |volume=March 1926 }}</ref>}}

==Physics==
For terminal velocity in falling through air , where [[viscosity]] is negligible compared to the drag force, and without considering [[buoyancy]] effects, terminal velocity is given by
<math display="block">V_t= \sqrt\frac{2 m g}{\rho A C_d} </math>
where
*<math>V_t</math> represents terminal velocity,
*<math>m</math> is the [[mass]] of the falling object,
*<math>g</math> is the [[Earth's gravity|acceleration due to gravity]],
*<math>C_d</math> is the [[drag coefficient]],
*<math>\rho</math> is the [[density]] of the fluid through which the object is falling, and
*<math>A</math> is the [[projected area]] of the object.<ref>{{cite book|url=https://books.google.com/books?id=OPsTDAAAQBAJ&pg=PA26|title=Dispersal in Plants: A Population Perspective|last1=Cousens|first1=Roger|last2=Dytham|first2=Calvin|last3=Law|first3=Richard|publisher=[[Oxford University Press]]|date=2008|pages=26–27|isbn=978-0-19-929911-9}}</ref>

In reality, an object approaches its terminal speed [[asymptotically]].

Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using [[buoyancy|Archimedes' principle]]: the mass <math>m</math> has to be reduced by the displaced fluid mass <math>\rho V</math>, with <math>V</math> the [[volume]] of the object. So instead of <math>m</math> use the reduced mass <math>m_r = m-\rho V</math> in this and subsequent formulas.

The terminal speed of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional [[surface area]].

Air density increases with decreasing altitude, at about 1% per {{convert|80|m|ft}} (see [[barometric formula]]). For objects falling through the atmosphere, for every {{convert|160|m|ft}} of fall, the terminal speed decreases 1%. After reaching the local terminal velocity, while continuing the fall, speed ''decreases'' to change with the local terminal speed.

Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the [[drag equation]]):

<math display="block">F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d,</math>

with ''v''(''t'') the velocity of the object as a function of time ''t''.

At [[List of types of equilibrium|equilibrium]], the [[net force]] is zero (''F''<sub>net</sub> = 0)<ref>{{cite book|url=https://books.google.com/books?id=ouzxCAAAQBAJ&pg=PA22|title=Fluid Mechanics for Marine Ecologists|last=Massel|first=Stanisław R.|publisher=[[Springer Science+Business Media]]|date=1999|page=22|doi=10.1007/978-3-642-60209-2|isbn=978-3-642-60209-2}}</ref> and the velocity becomes the terminal velocity {{math|1=lim{{sub|''t''→∞}} ''v''(''t'') = ''V''<sub>''t''</sub>}}:

<math display="block">m g - {1 \over 2} \rho V_t^2 A C_d = 0.</math>

Solving for ''V''<sub>''t''</sub> yields:

{{NumBlk|:|<math>V_t = \sqrt\frac{2mg}{\rho A C_d}.</math>|{{EquationRef|5}}}}

The drag equation is—assuming ''ρ'', ''g'' and ''C''<sub>''d''</sub> to be constants:
<math display="block"> m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d.</math>

Although this is a [[Riccati equation]] that can be solved by reduction to a second-order linear differential equation, it is easier to [[Separation of variables|separate variables]].

A more practical form of this equation can be obtained by making the substitution {{math|1=''α''<sup>2</sup> = {{sfrac|''ρAC''<sub>''d''</sub>|2''mg''}} }}.

Dividing both sides by ''m'' gives
<math display="block">\frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right).</math>

The equation can be re-arranged into
<math display="block">\mathrm{d}t = \frac{\mathrm{d}v}{g ( 1 - \alpha^2 v^2)}.</math>

Taking the integral of both sides yields
<math display="block">\int_0^t {\mathrm{d}t'} = {1 \over g}\int_0^v \frac{\mathrm{d}v'}{1-\alpha^2 v^{\prime 2}}.</math>

After integration, this becomes
<math display="block">t - 0 = {1 \over g}\left[{\ln(1 + \alpha v') \over 2\alpha} - \frac{\ln(1 - \alpha v')}{2\alpha} + C \right]_{v'=0}^{v'=v} ={1 \over g} \left[{\ln \frac{1 + \alpha v'}{1 - \alpha v'} \over 2\alpha} + C \right]_{v'=0}^{v'=v}</math>

or in a simpler form
<math display="block">t = {1 \over 2\alpha g} \ln \frac{1 + \alpha v}{1 - \alpha v} = \frac{\mathrm{artanh}(\alpha v)}{\alpha g},</math>
with artanh the [[inverse hyperbolic tangent]] function.

Alternatively,
<math display="block">\frac{1}{\alpha}\tanh(\alpha g t) = v,</math>
with tanh the [[hyperbolic tangent]] function. Assuming that ''g'' is positive (which it was defined to be), and substituting ''α'' back in, the speed ''v'' becomes
<math display="block">v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right).</math>

Using the formula for terminal velocity
<math display="block">V_t = \sqrt\frac{2mg}{\rho A C_d}</math> 
the equation can be rewritten as
<math display="block"> v = V_t \tanh \left(t \frac{g}{V_t}\right).</math>

As time tends to infinity (''t'' → ∞), the hyperbolic tangent tends to 1, resulting in the terminal speed
<math display="block"> V_t = \lim_{t \to \infty} v(t) = \sqrt\frac{2mg}{\rho A C_d}.</math>

[[Image:Stokes sphere.svg|thumb|upright|Creeping flow past a sphere: [[Streamlines, streaklines, and pathlines|streamlines]], drag force ''F''<sub>d</sub> and force by gravity ''F''<sub>g</sub>]]

For very slow motion of the fluid, the inertia forces of the fluid are negligible (assumption of massless fluid) in comparison to other forces. Such flows are called [[Stokes flow|creeping or Stokes flows]] and the condition to be satisfied for the flows to be creeping flows is the [[Reynolds number]], <math>Re \ll 1</math>. The equation of motion for creeping flow (simplified [[Navier–Stokes equation]]) is given by:

<math display="block">{\mathbf \nabla} p = \mu \nabla^2 {\mathbf v} </math>

where:

* <math>\mathbf v</math> is the fluid velocity vector field,
* <math>p</math> is the fluid pressure field,
* <math>\mu</math> is the liquid/fluid [[viscosity]].

The analytical solution for the creeping flow around a sphere was first given by [[George Gabriel Stokes|Stokes]] in 1851.<ref>{{cite journal |last1=Stokes |first1=G. G. |title=On the effect of internal friction of fluids on the motion of pendulums |journal=Transactions of the Cambridge Philosophical Society |date=1851 |volume=9, part ii |pages=8–106 |bibcode=1851TCaPS...9....8S |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015012112531;view=1up;seq=208}}  The formula for terminal velocity (''V'')] appears on p. [52], equation (127).</ref>
From Stokes' solution, the drag force acting on the sphere of diameter <math>d</math> can be obtained as
{{NumBlk|:|<math> D = 3\pi \mu d V \qquad \text{or} \qquad C_d = \frac{24}{Re} </math>|{{EquationRef|6}}}}

where the Reynolds number, <math>Re = \frac{\rho d}{\mu}  V</math>. The expression for the drag force given by equation ({{EquationNote|6}}) is called [[Stokes' law]].

When the value of <math>C_d</math> is substituted in the equation ({{EquationNote|5}}), we obtain the expression for terminal speed of a spherical object moving under creeping flow conditions:<ref>{{cite book | first=H. | last=Lamb | author-link=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th| isbn=978-0-521-45868-9  |pages=599}} Originally published in 1879, the 6th extended edition appeared first in 1932.</ref>

<math display="block">V_t = \frac{g d^2}{18 \mu} \left(\rho_s - \rho \right),</math>
where <math>\rho_s</math> is the density of the object.

== Applications ==
The creeping flow results can be applied in order to study the settling of sediments near the ocean bottom and the fall of moisture drops in the atmosphere. The principle is also applied in the [[Viscometer#Falling sphere viscometers|falling sphere viscometer]], an experimental device used to measure the viscosity of highly viscous fluids, for example oil, paraffin, tar etc.

==Terminal velocity in the presence of buoyancy force==
{{see also|Sediment_transport#Settling_velocity}}

[[Image:Settling velocity quartz.png|thumb|Settling velocity W<sub>s</sub> of a sand grain (diameter d, density 2650&nbsp;kg/m<sup>3</sup>) in water at 20&nbsp;°C, computed with the formula of Soulsby (1997).]]

When the buoyancy effects are taken into account, an object falling through a fluid under its own weight can reach a terminal velocity (settling velocity) if the net force acting on the object becomes zero. When the terminal velocity is reached the weight of the object is exactly balanced by the upward [[buoyancy force]] and drag force. That is
{{NumBlk|:|<math> W = F_b + D </math>|{{EquationRef|1}}}}
where
*<math>W</math> is the weight of the object,
*<math>F_b</math> is the buoyancy force acting on the object, and
*<math>D</math> is the drag force acting on the object.

If the falling object is spherical in shape, the expression for the three forces are given below:
{{NumBlk|:|<math> W = \frac{\pi}{6} d^3 \rho_s g,</math>|{{EquationRef|2}}}}
{{NumBlk|:|<math>F_b = \frac{\pi}{6} d^3 \rho g,</math>|{{EquationRef|3}}}}
{{NumBlk|:|<math> D = C_d \frac{1}{2} \rho V^2 A,</math>|{{EquationRef|4}}}}
where
*<math>d</math> is the diameter of the spherical object,
*<math>g</math> is the gravitational acceleration,
*<math>\rho</math> is the density of the fluid,
*<math>\rho_s</math> is the density of the object,
*<math>A = \frac{1}{4} \pi d^2</math> is the projected area of the sphere,
*<math>C_d</math> is the drag coefficient, and
*<math>V</math> is the characteristic velocity (taken as terminal velocity, <math>V_t </math>).

Substitution of equations ({{EquationNote|2}}–{{EquationNote|4}}) in equation ({{EquationNote|1}}) and solving for terminal velocity, <math>V_t</math> to yield the following expression
{{NumBlk|:|<math> V_t = \sqrt{\frac{4 g d}{3 C_d} \left( \frac{\rho_s - \rho}{\rho} \right)}. </math>|{{EquationRef|5}}}}

In equation ({{EquationNote|1}}), it is assumed that the object is denser than the fluid. If not, the sign of the drag force should be made negative since the object will be moving upwards, against gravity. Examples are bubbles formed at the bottom of a champagne glass and helium balloons. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up.

==See also==
* [[Stokes's law]]
* [[Terminal ballistics]]

==References==
{{reflist}}

==External links==
*[https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/termvel/ Terminal Velocity Interactive Tool] - NASA site, Beginners Guide to Aeronautics
*[http://io9.com/5893615/absolutely-mindblowing-video-shot-from-the-space-shuttle-during-launch Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere], from {{convert|2900|mph|Mach}} at 5:15 in the video, to 220&nbsp; mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com.
*[https://www.particles.org.uk/settling/index.htm Terminal settling velocity of a sphere] at all realistic Reynolds Numbers, by Heywood Tables approach.

[[Category:Falling]]
[[Category:Fluid dynamics]]
[[Category:Velocity]]