Sunday, February 15, 2009

A free falling object achieves its terminal velocity when the downward force of gravity (F_g) equals the upward force of drag (F_d). This causes the net force on the object to be zero, resulting in an acceleration of zero. Mathematically an object asymptotically approaches and can never reach its terminal velocity.

As the object accelerates (usually downwards due to gravity), the drag force acting on the object increases. At a particular speed, the drag force produced will equal the object's weight (mg). Eventually, it plummets at a constant speed called terminal velocity (also called settling velocity). Terminal velocity varies directly with the ratio of drag to weight. More drag means a lower terminal velocity, while increased weight means a higher terminal velocity. An object moving downward with greater than terminal velocity (for example because it was affected by a downward force or it fell from a thinner part of the atmosphere or it changed shape) will slow until it reaches terminal velocity.

e.g. Based on wind resistance, for example, the terminal velocity of a skydiver in a free-fall position with a semi-closed parachute is about 195 km/h (120 mph or 55 m/s). This velocity is the asymptotic limiting value of the acceleration process, since the effective forces on the body more and more closely balance each other 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 limbs. In this case, the terminal velocity increases to about 320 km/h (200 mph or 90 m/s) which is also the terminal velocity of the peregrine falcon diving down on its prey. And the same terminal velocity is reached for a typical 150 grain bullet travelling in the downward vertical direction — when it is returning to earth having been fired upwards, or perhaps just dropped from a tower — according to a 1920 U.S. Army Ordnance study.

Competition speed skydivers fly in the head down position reaching even higher speeds. The current world record is 614 mph (988 km/h) by Joseph Kittinger, set at high altitude where the lesser density of the atmosphere decreased drag.

An object falling toward the surface of the Earth will fall 9.81 meters per second faster every second (an acceleration of 9.81 m/s²). The reason an object reaches a terminal velocity is that the drag force resisting motion is directly proportional to the square of its speed. At low speeds, the drag is much less than the gravitational force and so the object accelerates. As it accelerates, the drag increases, until it equals the weight. Drag also depends on the projected area. This is why things with a large projected area, such as parachutes, have a lower terminal velocity than small objects such as cannon balls.

Mathematically, terminal velocity, without considering the buoyancy effects, is given by

where

V_t = terminal velocity,

m = mass of the falling object,

g = gravitational acceleration,

C_d = drag coefficient,

ρ = density of the fluid through which the object is falling, and

A = projected area of the object.

On Earth, 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.

For objects falling through the atmosphere, air density increases with decreasing altitude, ca. 1% per 80 m (see barometric formula). Therefore, for every 160 m of falling, the terminal velocity decreases 1%. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal velocity.

And according to The Physics Zone:

Terminal velocity is the velocity at which the driving forces are cancelled out by the resistive forces. Terminal velocity depends a great deal upon the shape of the object that is facing the direction it is moving. Once an object has reached terminal velocity, the object is not accelerating (a=0), therefore it is not speeding up or slowing down. It is a constant velocity unless the driving forces or the resistive forces change. Typically, Terminal Velocity is only a possibility when you are dealing with fluid friction as opposed to contact friction like static or kinetic friction.

Driving Forces are forces that try to cause motion: In the case of falling it would be gravity (weight). In the case of a car it would be the force from the engine (through the friction on tires). Sometimes an object can have multiple driving forces. (ie. an airplane in a dive, has the engine pushing it down at the same time gravity is pulling it down).

Resistive Forces are the forces that try to resist motion. The force of fluid friction (from a liquid or gas) plays the main role in creating terminal velocity.

Fluid Friction differs from contact friction because the amount of fluid friction depends on how fast the object is moving through the fluid. The greater the speed, the greater the friction. This can be felt if you are in a pool of water. Trying to walk from one side of the pool to the other is much easier than trying to run. That's because the faster you move, the harder the fluid pushes against you. When you examine contact friction you find that speed has no effect on the amount of kinetic friction.