Terminal velocity

Terminal velocity is the maximum speed attainable by an object as it falls through a fluid (air is the most common example).

[1][2] For objects falling through air at normal pressure, the buoyant force is usually dismissed and not taken into account, as its effects are negligible.

[citation needed] 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.

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.

Based on air resistance, for example, the terminal speed of a skydiver in a belly-to-earth (i.e., face down) free fall position is about 55 m/s (180 ft/s).

[citation needed] The current record is held by Felix Baumgartner who jumped from an altitude of 38,887 m (127,582 ft) and reached 380 m/s (1,200 ft/s), 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.

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.

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.

Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass

Air density increases with decreasing altitude, at about 1% per 80 metres (260 ft) (see barometric formula).

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):

Solving for Vt yields: The drag equation is—assuming ρ, g and Cd to be constants:

Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes

[10] From Stokes' solution, the drag force acting on the sphere of diameter

The expression for the drag force given by equation (6) is called Stokes' law.

is substituted in the equation (5), we obtain the expression for terminal speed of a spherical object moving under creeping flow conditions:[11]

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.

to yield the following expression In equation (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.