Terminal velocity is the approximately constant speed reached by a falling object when the upward resistive force, usually aerodynamic drag, balances the downward gravitational force, causing the net force and therefore acceleration to become approximately zero. For an object falling through Earth's atmosphere, gravity initially accelerates the object downward. As its speed increases, aerodynamic drag generally increases, causing the net downward force to decrease. Eventually, under suitable conditions, drag becomes equal in magnitude to the object's weight.
At that point Newton's second law gives a net force of approximately zero and the object's acceleration approaches zero. The object can then continue falling at a nearly constant speed called terminal velocity. Terminal velocity is not a universal constant. It depends on the object's mass, shape, cross sectional area, orientation, drag coefficient and the density and properties of the surrounding fluid.
It can also change during a single fall if the object's shape or orientation changes or if the surrounding atmospheric density changes with altitude. Terminal velocity therefore describes a force balance under particular conditions rather than an intrinsic speed possessed by a particular object everywhere.
Aerodynamic drag is commonly modelled at ordinary high speeds using the equation F drag equals one half rho C d A v squared, where rho is fluid density, C d is the drag coefficient, A is reference area and v is speed relative to the fluid. Gravitational force near Earth's surface is approximately m g. Terminal velocity occurs when these magnitudes are equal, giving m g equals one half rho C d A v t squared.
Solving for terminal speed gives v t equal to the square root of 2 m g divided by rho C d A. This equation shows why heavier objects with the same shape and area can have greater terminal velocities, while larger cross sectional area generally reduces terminal velocity. It also shows why a denser surrounding fluid produces a lower terminal velocity. The formula is an approximation because drag does not always follow the simple square law, especially at low Reynolds numbers or under conditions where fluid compressibility, turbulence or other effects become important. Nevertheless, it provides an effective model for many familiar atmospheric falling problems.
A falling object does not normally begin at terminal velocity. If it is released from rest, its initial downward acceleration is close to g because aerodynamic drag is initially small. As the object speeds up, drag increases and the net force decreases. The acceleration therefore decreases continuously rather than remaining exactly constant. Eventually the drag force approaches the object's weight and acceleration approaches zero. The speed then approaches a limiting value.
Mathematically, the approach to terminal velocity can be described by differential equations because drag depends on velocity. With a linear drag approximation, the velocity approaches terminal speed exponentially. With quadratic drag, which is more common for many macroscopic objects moving rapidly through air, the approach has a different functional form. In either case, terminal velocity is generally an asymptotic or near equilibrium state rather than a speed instantly reached at a particular moment. The time and distance required to approach it depend on the object's properties and the surrounding fluid.
Shape and orientation have a major influence on terminal velocity. A person falling belly first presents a much larger effective cross sectional area than the same person falling head first, producing substantially different drag and therefore different terminal speeds. Skydivers deliberately alter body position to control descent speed. Spreading the arms and legs increases area and changes aerodynamic flow, reducing terminal speed, while adopting a streamlined posture reduces drag and permits a higher speed.
A parachute produces a much larger effective area and usually a high drag coefficient compared with a person without a parachute, dramatically reducing terminal velocity. The parachute does not remove gravity; it increases the upward aerodynamic force until it balances the person's weight at a lower speed. Changes in body or parachute configuration therefore change the terminal condition. This principle is used in parachute design, aerial delivery systems, atmospheric probes and safety equipment.
Terminal velocity varies with altitude because atmospheric density changes. At higher altitudes, air density generally decreases, reducing drag for a given speed and therefore increasing the speed required for drag to balance weight under the simple model. A skydiver descending through the atmosphere can therefore experience changes in terminal speed as air density changes. Atmospheric temperature, pressure and composition also influence density and aerodynamic behaviour.
For very high speed objects, compressibility and shock waves become important, so the simple incompressible drag equation becomes inadequate. At sufficiently high altitudes, the atmosphere becomes so thin that ordinary aerodynamic terminal velocity may no longer be a useful concept. In other fluids, terminal velocity can arise through viscous resistance. Small particles settling through water or oil can reach a terminal settling speed when viscous drag balances effective weight. Thus terminal velocity is a general consequence of a driving force balanced by velocity dependent resistance and is not limited to falling objects in air.
Terminal velocity should be distinguished from the maximum possible speed of an object in all circumstances. It is a condition specific to a particular environment and configuration. If the object changes shape, opens a parachute, enters a denser fluid or encounters a different atmospheric region, the terminal velocity changes. An object can also temporarily accelerate again after reaching a terminal speed if the conditions change. Furthermore, terminal velocity does not necessarily mean that every force on the object has disappeared.
Gravity and drag can both remain substantial while their vector sum is approximately zero. The object therefore continues moving despite having zero or near zero acceleration. This is an important example of Newton's first and second laws. Constant velocity does not imply zero forces; it implies zero net force. If an object is descending at terminal speed, its weight is balanced by aerodynamic resistance, assuming no other significant forces.
Terminal velocity has important applications beyond parachuting. Engineers use drag and terminal settling calculations in designing sedimentation tanks, particle separators, aerosol systems, dust collectors and fluidised processes. Meteorology uses the fall speeds of raindrops, snowflakes and ice particles to model precipitation and atmospheric transport. The terminal speeds of raindrops depend on their size and shape, and large drops can deform or fragment rather than simply behaving as perfect spheres.
In environmental science, the settling velocity of particles influences sediment transport, water quality and pollutant movement. In biology, organisms and seeds can experience terminal settling behaviour in air or water, affecting dispersal. In aerospace engineering, atmospheric entry involves gravitational acceleration, aerodynamic drag, heating and rapidly changing atmospheric conditions, so terminal velocity may occur only in particular regimes rather than throughout the trajectory. These applications show that terminal velocity is a general dynamical concept involving the balance between an applied driving force and velocity dependent resistance.
Terminal velocity is therefore the speed at which an object's resistive force balances the force driving its motion, producing approximately zero net force and approximately zero acceleration. For an object falling through air, the driving force is gravity and the resisting force is aerodynamic drag. The terminal speed depends on mass, gravity, air density, drag coefficient, cross sectional area and object orientation, and it can change as environmental or physical conditions change.
An object generally approaches terminal velocity rather than instantly acquiring it. The concept is useful for understanding skydiving, parachutes, falling particles, precipitation, fluid mechanics, atmospheric entry and many engineering systems. It also illustrates the broader physical principle that an object can continue moving at constant velocity while substantial opposing forces act, provided those forces balance so that the net force is zero.
