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Terminal velocity is the maximum speed attainable by an object as it falls through a
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
(
air An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
is the most common example). It is reached when the sum of the drag force (''Fd'') and the
buoyancy Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may be also be a parcel of fluid). In a column of fluid, pressure increases with depth as a result of the weight of t ...
is equal to the downward force of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
(''FG'') acting on the object. Since the
net force In mechanics, the net force is the sum of all the forces acting on an object. For example, if two forces are acting upon an object in opposite directions, and one force is greater than the other, the forces can be replaced with a single force tha ...
on the object is zero, the object has zero
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
. 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

Based on air resistance, for example, the terminal speed of a skydiver in a belly-to-earth (i.e., face down)
free fall 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. If the common definition of the word ...
position is about . This speed is the
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
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). In this case, the terminal speed increases to about , which is almost the terminal speed of the
peregrine falcon The peregrine falcon (''Falco peregrinus''), also known simply as the peregrine, is a Cosmopolitan distribution, cosmopolitan bird of prey (raptor) in the family (biology), family Falconidae renowned for its speed. A large, Corvus (genus), cro ...
diving down on its prey. The same terminal speed is reached for a typical .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. Competition speed skydivers fly in a head-down position and can reach speeds of . The current record is held by Felix Baumgartner who jumped from an altitude of and reached , 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. The biologist J. B. S. Haldane wrote,


Physics

For terminal velocity in falling through air , where
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
is negligible compared to the drag force, and without considering
buoyancy Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may be also be a parcel of fluid). In a column of fluid, pressure increases with depth as a result of the weight of t ...
effects, terminal velocity is given by V_t= \sqrt\frac where *V_t represents terminal velocity, *m is the
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
of the fluid through which the object is falling, and *A is the projected area of the object. 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
Archimedes' principle Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Archimedes' principle is a law of physics fun ...
: the mass m has to be reduced by the displaced fluid mass \rho V, with V the
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
of the object. So instead of m use the reduced mass m_r = m-\rho V 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 The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
. Air density increases with decreasing altitude, at about 1% per (see barometric formula). For objects falling through the atmosphere, for every 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 In fluid dynamics, the drag equation is a formula used to calculate the force of drag (physics), drag experienced by an object due to movement through a fully enclosing fluid. The equation is: F_\, =\, \tfrac12\, \rho\, u^2\, c_\, A where *F_ is ...
): F_\text = m a = m g - \frac \rho v^2 A C_d, with ''v''(''t'') the velocity of the object as a function of time ''t''. At equilibrium, the
net force In mechanics, the net force is the sum of all the forces acting on an object. For example, if two forces are acting upon an object in opposite directions, and one force is greater than the other, the forces can be replaced with a single force tha ...
is zero (''F''net = 0) and the velocity becomes the terminal velocity : m g - \rho V_t^2 A C_d = 0. Solving for ''V''''t'' yields: The drag equation is—assuming ''ρ'', ''g'' and ''C''''d'' to be constants: m a = m \frac = m g - \frac \rho v^2 A C_d. Although this is a
Riccati equation In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2( ...
that can be solved by reduction to a second-order linear differential equation, it is easier to separate variables. A more practical form of this equation can be obtained by making the substitution . Dividing both sides by ''m'' gives \frac = g \left( 1 - \alpha^2 v^2 \right). The equation can be re-arranged into \mathrmt = \frac. Taking the integral of both sides yields \int_0^t = \int_0^v \frac. After integration, this becomes t - 0 = \left - \frac + C \right^ = \left + C \right^ or in a simpler form t = \ln \frac = \frac, with artanh the inverse hyperbolic tangent function. Alternatively, \frac\tanh(\alpha g t) = v, 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 v = \sqrt\frac \tanh \left(t \sqrt\right). Using the formula for terminal velocity V_t = \sqrt\frac the equation can be rewritten as v = V_t \tanh \left(t \frac\right). As time tends to infinity (''t'' → ∞), the hyperbolic tangent tends to 1, resulting in the terminal speed V_t = \lim_ v(t) = \sqrt\frac. 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 creeping or Stokes flows and the condition to be satisfied for the flows to be creeping flows is the
Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
, Re \ll 1. The equation of motion for creeping flow (simplified Navier–Stokes equation) is given by: p = \mu \nabla^2 where: * \mathbf v is the fluid velocity vector field, * p is the fluid pressure field, * \mu is the liquid/fluid
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
. The analytical solution for the creeping flow around a sphere was first given by Stokes in 1851. From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac V. The expression for the drag force given by equation () is called
Stokes' law In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the S ...
. When the value of C_d is substituted in the equation (), we obtain the expression for terminal speed of a spherical object moving under creeping flow conditions: Originally published in 1879, the 6th extended edition appeared first in 1932. V_t = \frac \left(\rho_s - \rho \right), where \rho_s 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 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

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 where *W is the weight of the object, *F_b is the buoyancy force acting on the object, and *D 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: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). Substitution of equations (–) in equation () and solving for terminal velocity, V_t to yield the following expression In equation (), 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 Terminal ballistics is a sub-field of ballistics concerned with the behavior and effects of a projectile when it hits and transfers its energy to a target. This field is usually cited in forensic ballistics. Bullet design (as well as the veloci ...


References


External links


Terminal Velocity Interactive Tool
- NASA site, Beginners Guide to Aeronautics
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  mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com.

at all realistic Reynolds Numbers, by Heywood Tables approach. Falling Fluid dynamics Velocity