Terminal velocity
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Terminal velocity is the highest velocity attainable by an object as it falls through air. It occurs once the sum of the drag force (F_{d}) and buoyancy equals the downward force of gravity (F_{G}) acting on the object. Since the net force on the object is zero, the object has zero acceleration.^{[1]}
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.
As the speed of an object increases, the drag force acting on the object, resultant of the substance (e.g., air or water) it is passing through, increases. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). At this point the object ceases to accelerate and continues falling at a constant speed called terminal velocity (also called settling velocity). An object moving downward with greater than terminal velocity (for example because it was thrown downwards or it fell from a thinner part of the atmosphere or it changed shape) will slow down until it reaches terminal velocity. Drag depends on the projected area, and this is why objects with a large projected area relative to mass, such as parachutes, have a lower terminal velocity than objects with a small projected area relative to mass, such as bullets.
Contents
Examples
Based on wind resistance, for example, the terminal velocity of a skydiver in a bellytoearth (i.e., face down) freefall position is about 195 km/h (122 mph or 54 m/s).^{[2]} This velocity is the asymptotic limiting value of the acceleration process, because the effective forces on the body balance each other more and more closely as the terminal velocity is approached. In this example, a speed of 50% of terminal velocity 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 his or her limbs (see also freeflying). In this case, the terminal velocity increases to about 320 km/h (200 mph or 90 m/s),^{[2]} which is almost the terminal velocity of the peregrine falcon diving down on its prey.^{[3]} The same terminal velocity is reached for a typical .3006 bullet dropping downwards—when it is returning to earth having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study.^{[4]}
Competition speed skydivers fly in a headdown position and can reach speeds of 530 km/h (330 mph); the current record is held by Felix Baumgartner who jumped from a height of 128,100 feet (39,000 m) and reached 1,342 km/h (834 mph), however not at sealevel air pressure.
Physics
Mathematically, terminal velocity—without considering buoyancy effects—is given by
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_t= \sqrt{\frac{2mg}{\rho A C_d }}}
where
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_t} is terminal velocity,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m} is the mass of the falling object,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle g} is the acceleration due to gravity,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle C_d} is the drag coefficient,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho} is the density of the fluid through which the object is falling, and
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle A} is the projected area of the object.
In reality, an object approaches its terminal velocity asymptotically.
Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m} has to be reduced by the displaced fluid mass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho\mathcal{V}} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{V}} the volume of the object. So instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m} use the reduced mass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m_r=m\rho\mathcal{V}} in this and subsequent formulas.
On Earth, the terminal velocity of an object changes due to the properties of the fluid, the mass of the object and its projected crosssectional surface area.
Air density increases with decreasing altitude, ca. 1% per 80 metres (260 ft) (see barometric formula). For objects falling through the atmosphere, for every 160 metres (520 ft) of falling, the terminal velocity decreases 1%. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal velocity.
Derivation for terminal velocity
Mathematically, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle F_{net} = m a = m g  {1 \over 2} \rho v^2 A C_\mathrm{d}}
At equilibrium, the net force is zero (F = 0);
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m g  {1 \over 2} \rho v^2 A C_\mathrm{d} = 0}
Solving for v yields
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v = \sqrt\frac{2mg}{\rho A C_\mathrm{d}}}
Derivation of the solution for the velocity v as a function of time t 

The drag equation is
Although this is a Riccati equation that can be solved by reduction to a secondorder linear differential equation, it is easier to separate variables. A more practical form of this equation can be obtained by making the substitution k = ^{1}⁄_{2}ρAC_{d}. Dividing both sides by m gives
The equation can be rearranged into
Taking the integral of both sides yields
where α = ( ^{k}⁄_{mg} )^{1⁄2}. After integration, this becomes
or in simpler a form
The inverse hyperbolic tangent is defined as:
So the solution of the integral is
or alternatively,
with tanh the hyperbolic tangent function. Assuming that g is positive (which it was defined to be), and substituting α back in, the velocity v becomes
Next, after k = ^{1}⁄_{2}ρAC_{d} has been substituted, the velocity v is in the desired form:
As time tends to infinity ( t → ∞ ), the hyperbolic tangent tends to 1, resulting in the terminal velocity

Terminal velocity in creeping flow
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 flows and the condition to be satisfied for the flows to be creeping flows is the Reynolds number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Re \ll 1} . The equation of motion for creeping flow (simplified NavierStokes equation) is given by
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \nabla p = \mu \nabla^2 {\mathbf v} }
where
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {\mathbf v}} is the fluid velocity vector field
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p} is the fluid pressure field
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mu} is the fluid viscosity
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 can be obtained as
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \quad (6) \qquad D = 3\pi \mu d V \qquad \qquad \text{or} \qquad \qquad C_d = \frac{24}{Re} }
where the Reynolds number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Re = \frac{1}{\mu} \rho d V} . The expression for the drag force given by equation (6) is called Stokes' law.
When the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle C_d} is substituted in the equation (5), we obtain the expression for terminal velocity of a spherical object moving under creeping flow conditions:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_t = \frac{g d^2}{18 \mu} \left(\rho_s  \rho \right)}
Applications
The creeping flow results can be applied in order to study the settling of sediment particles 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 (sticky) fluids, for example oil, parrafin, tar etc.
Finding the terminal velocity when the drag coefficient is not known
In principle one doesn't know beforehand whether to apply the creeping flow solution, or what coefficient of drag to use, because the coefficient depends on the speed. What one can do in this situation is to calculate the product of the coefficient of drag and the square of the Reynolds number:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle C_d \mathrm{Re}^2 = \frac{mgD^2}{A\rho\nu^2}}
where ν is the kinematic viscosity, equal to μ/ρ. This product is a function of Reynolds number, and one can consult a graph of C_{d} versus Re to find where along the curve the product attains the correct value (a qualitative example of such a graph for spheres is found at this NASA site: [1]) From this one knows the coefficient of drag and one can then use the formula given above to find the terminal velocity.
For a spherical object, the abovementioned product can be simplified:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle C_d \mathrm{Re}^2 = \frac{4mg}{\pi\rho\nu^2}}
We can see from this that the regime and the drag coefficient depend only on the sphere's weight and the fluid properties. There are three regimes: creeping flow, intermediateReynolds number Newton's Law (almost constant drag coefficient), and a highReynolds number regime.^{[5]} In the latter regime the boundary layer is everywhere turbulent (see Golf ball#Aerodynamics). These regimes are given in the following table. The weight range for each regime is given for water and air at 1 atm pressure and 25 °C. Note though that the weight (given in terms of mass in standard gravity) is the weight in the fluid, which is less than the mass times the local gravity because of buoyancy.
Regime  Range of Reynolds number  Range of C_{d}Re^{2}  Range of weight in water  Range of weight in air 

Creeping flow  Quite accurate up to 0.3  Up to 7.2  Up to 0.00058 mg_{f} (5.7 nN)  Up to 0.00017 mg_{f} (1.7 nN) 
C_{d} between 0.4 and 0.5  1000 to 200000  500000 to 2×10^{10}  40 mg_{f} (0.39 mN) to 1.6 kg_{f} (16 N)  11 mg_{f} (0.11 mN) to 470 g_{f} (4.6 N) 
C_{d} between 0.1 and 0.2  Over 400000  Over 1.6×10^{10}  Over 1.3 kg_{f} (13 N)  Over 375 g_{f} (3.68 N) 
Between the first two regimes there is a smooth transition. But notice that there is overlap between the ranges of C_{d}Re^{2} for the last two regimes. Spheres in this weight range have two stable terminal velocities. A rough surface, such as of a dimpled golf ball, allows transition to the lower drag coefficient at a lower Reynolds number.
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
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \quad (1) \qquad W = F_b + D }
where
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle W} = weight of the object,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle F_b} = buoyancy force acting on the object, and
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle D} = drag force acting on the object.
If the falling object is spherical in shape, the expression for the three forces are given below:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \begin{align} \quad &(2) \qquad& W &= \frac{\pi}{6} d^3 \rho_s g \\ \quad &(3) \qquad& F_b &= \frac{\pi}{6} d^3 \rho g \\ \quad &(4) \qquad& D &= C_d \frac{1}{2} \rho V^2 A \end{align}}
where
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d} is the diameter of the spherical object
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle g} is the gravitational acceleration,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho} is the density of the fluid,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho_s} is the density of the object,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle A = \frac{1}{4} \pi d^2} is the projected area of the sphere,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle C_d} is the drag coefficient, and
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} is the characteristic velocity (taken as terminal velocity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_t } ).
Substitution of equations (2–4) in equation (1) and solving for terminal velocity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V_t} to yield the following expression
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \quad (5) \qquad V_t = \sqrt{\frac{4 g d}{3 C_d} \left( \frac{\rho_s  \rho}{\rho} \right)} }
See also
References
 ↑ "Terminal Velocity". NASA Glenn Research Center. Retrieved March 4, 2009.
 ↑ ^{2.0} ^{2.1} Huang, Jian (1999). "Speed of a Skydiver (Terminal Velocity)". The Physics Factbook. Glenn Elert, Midwood High School, Brooklyn College.
 ↑ "All About the Peregrine Falcon (archived)". U.S. Fish and Wildlife Service. December 20, 2007. Archived from the original on March 8, 2010.
 ↑ The Ballistician (March 2001). "Bullets in the Sky". W. Square Enterprises, 9826 Sagedale, Houston, Texas 77089.
 ↑ Robert H. Perry; Cecil Chilton (eds.). Chemical Engineer's Handbook (fifth ed.). pp. 5–62. ISBN 9780070494787.
External links
 Terminal Velocity  NASA site
 Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from 2,900 miles per hour (Mach 3.8) 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.