How Terminal Velocity Is Reached

When an object is released, its weight pulls downward. If it begins from rest, air resistance is initially small because the object is not moving quickly through the air. The downward force is larger than the upward resistance, so there is a downward resultant force.

A resultant force causes acceleration. The object's speed increases, but its mass and weight remain almost unchanged during an ordinary short fall. What changes strongly is the air resistance acting against the motion.

Air resistance comes from interactions with air particles and pressure differences around the object. It depends on speed, shape, surface area and air density. A broad flat object usually experiences more drag than a compact object of similar mass. A denser atmosphere can also produce more drag at the same speed, while thinner air usually produces less.

As the object falls faster, it collides with and redirects more air each second. The upward drag force increases. The gap between weight and drag becomes smaller, so the downward resultant force and acceleration also become smaller.

Eventually drag becomes equal in size to weight. The forces are balanced, the resultant force is zero and acceleration stops. The object is still moving downward, but its velocity is now constant. Balanced forces do not mean the object must be stationary.

This constant falling speed is terminal velocity. It is not one universal value. A raindrop, a skydiver and a sheet of paper have different terminal velocities because their mass, shape and surface area differ. Terminal velocity is therefore a balance created by a particular object in particular conditions.

A skydiver can alter drag by changing body position. Spreading arms and legs increases the area facing the airflow and lowers terminal velocity. Tucking into a compact shape reduces drag and allows a higher speed.

Opening a parachute creates a much larger surface area. Drag suddenly becomes greater than weight, producing an upward resultant force that slows the downward motion. As speed decreases, drag falls until it balances weight again at a new, lower terminal velocity.

Follow the force story:

• •Weight acts downward throughout the fall.
• •Drag acts upward against the motion.
• •Increasing speed produces increasing drag.
• •Equal weight and drag give zero acceleration.
• •A larger area usually gives a lower terminal velocity.

Terminal velocity depends heavily on size because weight and drag grow at different rates. If you double every dimension of an object while keeping its shape and material the same, its volume and mass increase by a factor of eight, but the area facing the air only increases by a factor of four. Weight grows faster than the surface that creates drag, so a larger object must fall faster before drag can catch up. This is the same scaling rule that biologists use to explain differences between small and large animals.

It explains why an insect or a mouse can fall a long way and walk off, while a larger animal falling the same distance could be badly hurt. The small animal reaches a low terminal velocity quickly, and it also has relatively little mass to stop when it lands. The same idea applies to raindrops. Tiny cloud droplets fall so slowly that rising air can keep them suspended, while large drops fall much faster. Very large drops tend to flatten and break apart, which limits how big a falling raindrop can become.

A common belief is that heavier objects naturally fall faster. In air, a hammer does reach the ground before a feather, but that is because of drag, not because gravity favours heavy things. Without air, every object near a planet's surface accelerates at the same rate. In 1971 the Apollo 15 astronaut David Scott tested this on the Moon, which has almost no atmosphere. He dropped a geological hammer and a falcon feather together on camera, and they landed at the same moment. On Earth, the feather's terminal velocity is tiny compared with the hammer's.

Air density matters too, which explains some record-breaking jumps. In 2012 the Austrian skydiver Felix Baumgartner stepped from a balloon capsule roughly 39 kilometres above New Mexico. The air there is so thin that drag stays small even at very high speeds, so his terminal velocity was enormous, and he became the first person to exceed the speed of sound in free fall without a vehicle. As he dropped into denser air lower down, drag increased and slowed him well before he opened his parachute. His fall showed terminal velocity changing continuously with altitude.

At skydiving speeds, drag grows roughly with the square of speed, so doubling your speed makes the drag about four times larger. That rule explains some familiar numbers. A skydiver falling belly-down usually settles at around 200 kilometres per hour, reached after roughly twelve seconds of falling. Diving head-first cuts the area facing the air dramatically, and speeds well above 250 kilometres per hour become possible. Under an open parachute the area is so large that drag balances weight at only around five metres per second, similar to the landing speed after jumping off a wall a little over a metre high.

You can watch terminal velocity happen in a school laboratory with a tall tube of thick oil or glycerol. Drop in a small steel ball bearing and it speeds up only briefly before moving steadily downwards, because the liquid resists motion far more strongly than air. Students mark bands on the tube with rubber rings, start timing below the top band so the ball has already reached its steady speed, and check that equal distances take equal times. Repeating with larger balls shows them settling at higher speeds, the same scaling effect, now measured in centimetres per second rather than kilometres per hour.

For very small objects, such as dust specks or mist droplets, drag behaves differently from the skydiving rule. The air flows smoothly around them, and drag rises in simple proportion to speed rather than with its square. The physicist George Stokes worked out the formula in the nineteenth century. One consequence is that a tiny sphere's terminal velocity depends on the square of its radius, so halving a droplet's size quarters its falling speed. Turn that round and something useful appears: if you can time a droplet's steady fall, you can calculate how big it is.

Robert Millikan, working with his student Harvey Fletcher, used exactly this trick in experiments begun in 1909. He sprayed fine oil droplets between two metal plates and watched individual drops through a microscope, timing them between marked lines as they fell at terminal velocity. Their speed gave their size and therefore their mass. He then switched on a voltage across the plates and adjusted the electric force to hold drops still or drive them upwards, which revealed the charge each drop carried. The charges always came in whole multiples of one tiny value, the charge of a single electron. Millikan received the Nobel Prize in Physics in 1923.

Terminal velocity occurs when air resistance balances an object's weight. Before that point, the object accelerates at a decreasing rate as drag grows. Track the two opposing forces rather than speed alone, and the stages of a fall become a clear example of how resultant force controls acceleration.