Terminal velocity and free fall are two related concepts that tend to get confusing because they depend on whether or not a body is in empty space or in a fluid (e.g., and atmosphere or even water). Take a look at the definitions and equations of the terms, how they are related, and how fast a body falls in free fall or at terminal velocity under different conditions.
Terminal Velocity Definition
Terminal velocity is defined as the highest velocity that can be achieved by an object that is falling through a fluid, such as air or water. When terminal velocity is reached, the downward force of gravity is equal to the sum of the object's buoyancy and the drag force. An object a terminal velocity has zero net acceleration.
Terminal Velocity Equation
There are two particularly useful equations for finding terminal velocity. The first is for terminal velocity without taking into account buoyancy:
Vt = (2mg/ρACd)1/2
- Vt is the terminal velocity
- m is the mass of the object that is falling
- g is acceleration due to gravity
- Cd is the drag coefficient
- ρ is the density of the fluid through which the object is falling
- A is the cross-sectional area projected by the object
In liquids, in particular, it's important to account for the buoyancy of the object. Archimedes' principle is used to account for the displacement of volume (V) by the mass. The equation then becomes:
Vt = 2(m - ρV)g/ρACd1/2
Free Fall Definition
The everyday use of the term "free fall" is not the same as the scientific definition. In common usage, a sky diver is considered to be in free fall upon achieving terminal velocity without a parachute. In actuality, the weight of the sky diver is supported by a cushion of air.
Free fall is defined either according to Newtonian (classical) physics or in terms of general relativity. In classical mechanics, free fall describes the motion of a body when the only force acting upon it is gravity. The direction of the movement (up, down, etc.) is unimportant. If the gravitational field is uniform, it acts equally on all parts of the body, making it "weightless" or experiencing "0 g". Although it might seem strange, an object can be in free fall even when moving upward or at the top of its motion. A skydiver jumping from outside the atmosphere (like a HALO jump) very nearly achieves true terminal velocity and free fall.
In general, as long as air resistance is negligible with respect to an object's weight, it can achieve free fall. Examples include:
- A spacecraft in space without a propulsion system engaged
- An object thrown upward
- An object dropped from a drop tower or into a drop tube
- A person jumping up
In contrast, objects not in free fall include
- A flying bird
- A flying aircraft (because the wings provide lift)
- Using a parachute (because it counters gravity with drag and in some cases may provide lift)
- A skydiver not using a parachute (because the drag force equals his weight at terminal velocity)
In general relativity, free fall is defined as the movement of a body along a geodesic, with gravity described as space-time curvature.
Free Fall Equation
If an object is falling toward the surface of a planet and the force of gravity is much greater than the force of air resistance or else its velocity is much less than terminal velocity, the vertical velocity of free fall may be approximated as:
vt = gt + v0
- vt is the vertical velocity in meters per second
- v0 is the initial velocity (m/s)
- g is the acceleration due to gravity (about 9.81 m/s2 near Earth)
- t is the elapsed time (s)
How Fast Is Terminal Velocity? How Far Do You Fall?
Because terminal velocity depends on drag and an object's cross section, there is no one speed for terminal velocity. In general, a person falling through the air on Earth reaches terminal velocity after about 12 seconds, which covers about 450 meters or 1500 feet.
A skydiver in the belly-to-earth position reaches a terminal velocity of about 195 km/hr (54 m/s or 121 mph). If the skydiver pulls in his arms and legs, his cross section is decreased, increasing terminal velocity to about 320 km/hr (90 m/s or just under 200 mph). This is about the same as the terminal velocity achieved by a peregrine falcon diving for prey or for a bullet falling down after having been dropped or fired upward. The world record terminal velocity was set by Felix Baumgartner, who jumped from 39,000 meters and reached a terminal velocity of 134 km/hr (834 mph).
References and Further Reading
- Huang, Jian (1999). "Speed of a Skydiver (Terminal Velocity)". The Physics Factbook. Glenn Elert, Midwood High School, Brooklyn College.
- All About the Peregrine Falcon". U.S. Fish and Wildlife Service. December 20, 2007. (archived)
- The Ballistician (March 2001). "Bullets in the Sky". W. Square Enterprises, 9826 Sagedale, Houston, Texas 77089.