Terminal velocity

The downward force of gravity (F_g) equals the restraining force of drag (F_d) plus the buoyancy. The net force on the object then, is zero, and the result is that the velocity of the object remains constant.

Terminal velocity is the highest velocity attainable by an object as it falls through a fluid (air is the most common example). It occurs when the sum of the drag force (F_d) and the buoyancy is equal to 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, 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 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 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, 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 bullet.

Examples

Based on wind resistance, for example, the terminal velocity of a skydiver in a belly-to-earth (i.e., face down) free-fall position is about 195 km/h (122 mph or 54 m/s).^[2] This velocity is the asymptotic limiting value of the velocity, and the forces acting 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 .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.^[4]

Competition speed skydivers fly in a head-down 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), though he achieved this velocity at high altitude, where extremely thin air presents less drag force.

The biologist J. B. S. Haldane wrote,

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.^[5]

Physics

Using mathematical terms, terminal velocity—without considering buoyancy effects—is given by

V_{t}={\sqrt {{\frac {2mg}{\rho AC_{d}}}}}

where

$V_{t}$ represents terminal velocity,
$m$ is the mass of the falling object,
$g$ is the acceleration due to gravity,
$C_{d}$ is the drag coefficient,
$\rho$ is the density 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 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 $m$ has to be reduced by the displaced fluid mass $\rho V$ , with $V$ the volume of the object. So instead of $m$ use the reduced mass $m_{r}=m-\rho V$ in this and subsequent formulas.

The terminal velocity 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 80 metres (260 ft) (see barometric formula). For objects falling through the atmosphere, for every 160 metres (520 ft) of fall, 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

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):

F_{{net}}=ma=mg-{1 \over 2}\rho v^{2}AC_{{\mathrm {d}}}

At equilibrium, the net force is zero (F = 0);

mg-{1 \over 2}\rho v^{2}AC_{{\mathrm {d}}}=0

Solving for v yields

v={\sqrt {\frac {2mg}{\rho AC_{{\mathrm {d}}}}}}

Derivation of the solution for the velocity v as a function of time t

The drag equation is

ma=m{\frac {{\mathrm {d}}v}{{\mathrm {d}}t}}=mg-{\frac {1}{2}}\rho v^{2}AC_{{\mathrm {d}}}

Although this is a Riccati equation 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 k = ¹⁄₂ρAC_d.

Dividing both sides by m gives

{\frac {{\mathrm {d}}v}{{\mathrm {d}}t}}=g-{\frac {kv^{2}}{m}}

The equation can be re-arranged into

dt={\frac {{\mathrm {d}}v}{g-{\frac {kv^{2}}{m}}}}

Taking the integral of both sides yields

\int _{0}^{t}{{\mathrm {d}}t^{\prime }}=\int _{0}^{v}{\frac {{\mathrm {d}}v^{\prime }}{g-{\frac {kv^{{\prime 2}}}{m}}}}={1 \over g}\int _{0}^{v}{\frac {{\mathrm {d}}v^{\prime }}{1-\alpha ^{2}v^{{\prime 2}}}}

where α = ( ^k⁄_mg )^¹⁄₂.

After integration, this becomes

t-0={1 \over g}\left[{\ln(1+\alpha v^{\prime }) \over 2\alpha }-{\frac {\ln(1-\alpha v^{\prime })}{2\alpha }}+C\right]_{{v^{\prime }=0}}^{{v^{\prime }=v}}={1 \over g}\left[{\ln {\frac {1+\alpha v^{\prime }}{1-\alpha v^{\prime }}} \over 2\alpha }+C\right]_{{v^{\prime }=0}}^{{v^{\prime }=v}}

or in simpler a form

t={1 \over 2\alpha g}\ln {\frac {1+\alpha v}{1-\alpha v}}

The inverse hyperbolic tangent is defined as:

{\frac {1}{2}}\ln {\frac {1+\alpha v}{1-\alpha v}}={\mathrm {arctanh}}(\alpha v)

So the solution of the integral is

t={\frac {{\mathrm {arctanh}}(\alpha v)}{\alpha g}}

or alternatively,

{\frac {1}{\alpha }}\tanh(\alpha gt)=v

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

v={\sqrt {{\frac {mg}{k}}}}\tanh \left({\sqrt {{\frac {k}{mg}}}}gt\right)

Next, after k = ¹⁄₂ρAC_d has been substituted, the velocity v is in the desired form:

v={\sqrt {\frac {2mg}{\rho AC_{d}}}}\tanh \left(t{\sqrt {{\frac {g\rho AC_{d}}{2m}}}}\right)

As time tends to infinity ( t → ∞ ), the hyperbolic tangent tends to 1, resulting in the terminal velocity

v={\sqrt {\frac {2mg}{\rho AC_{d}}}}

Terminal velocity in a creeping flow

Creeping flow past a sphere: streamlines, drag force F_d and force by gravity F_g

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, $Re\ll 1$ . The equation of motion for creeping flow (simplified Navier-Stokes equation) is given by

\nabla p=\mu \nabla ^{2}{{\mathbf v}}

where

${\mathbf {v} }$ is the fluid velocity vector field
$p$ is the fluid pressure field
$\mu$ is the liquid\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

\quad (6)\qquad D=3\pi \mu dV\qquad \qquad {\text{or}}\qquad \qquad C_{d}={\frac {24}{Re}}

where the Reynolds number, $Re={\frac {1}{\mu }}\rho dV$ . The expression for the drag force given by equation (6) is called Stokes' law.

When the value of $C_{d}$ is substituted in the equation (5), we obtain the expression for terminal velocity of a spherical object moving under creeping flow conditions:

V_{t}={\frac {gd^{2}}{18\mu }}\left(\rho _{s}-\rho \right)

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, parrafin, tar etc.

Terminal velocity in the presence of buoyancy force

Settling velocity W_s of a sand grain (diameter d, density 2650 kg/m³) in water at 20 °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

\quad (1)\qquad W=F_{b}+D

where

$W$ = weight of the object,
$F_{b}$ = buoyancy force acting on the object, and
$D$ = drag force acting on the object.

If the falling object is spherical in shape, the expression for the three forces are given below:

{\begin{aligned}\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{aligned}}

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,
$A={\frac {1}{4}}\pi d^{2}$ 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 (2–4) in equation (1) and solving for terminal velocity, $V_{t}$ to yield the following expression

\quad (5)\qquad V_{t}={\sqrt {{\frac {4gd}{3C_{d}}}\left({\frac {\rho _{s}-\rho }{\rho }}\right)}}

References

↑ "Terminal Velocity". NASA Glenn Research Center. Retrieved March 4, 2009.
1 2 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.
↑ Haldane, J. B. S. (March 1926). "On Being the Right Size". Harper's Magazine. Archived from the original on 2016-02-13.

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.

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Terminal velocity

Examples

Physics

Derivation for terminal velocity

Terminal velocity in a creeping flow

Applications

Terminal velocity in the presence of buoyancy force

See also

References

External links