Terminal velocity is the maximum velocity (speed) 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 (Fd) and the buoyancy is equal to the downward force of gravity (FG) acting on the object. Since the net force on the object is zero, the object has zero acceleration.
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. 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 position is about . 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% 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 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
Using mathematical terms, terminal speed—without considering buoyancy effects—is given by
where
represents terminal velocity, is the mass of the falling object, is the acceleration due to gravity, is the drag coefficient, is the density of the fluid through which the object is falling, and 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: the mass has to be reduced by the displaced fluid mass , with the volume of the object. So instead of use the reduced mass 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 (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):