How Fast Do You Fall? Terminal Velocity

The drag-free answer, and how long it stays true

Drop something and, for the first moment, its speed is set by one equation: v = g·t. With g ≈ 9.81 m/s² (standard gravity is defined as exactly 9.80665 m/s², the value the calculators on this site use), that is about 9.8 m/s after one second, 19.6 m/s after two and 29.4 m/s after three — roughly 35, 71 and 106 km/h. Distance follows d = ½·g·t², so those same three seconds cover 4.9 m, 19.6 m and 44.1 m.

Notice what the equation does not contain: mass. In a vacuum a hammer and a feather accelerate identically — Galileo's argument, and the reason the Apollo 15 crew could demonstrate it on the Moon. Nothing about being heavy makes you fall faster, until air enters the problem.

Air enters the problem quickly. The drag-free numbers are excellent for the first second or two, adequate for a dropped tool or a short stumble, and increasingly wrong after that. A body that kept accelerating at g would pass 193 km/h in about 5.5 seconds and keep going. Real bodies do not.

Why you stop speeding up

Air resistance is not a fixed penalty — it grows with the square of speed. Double your speed and the drag force quadruples. Weight, meanwhile, is the same all the way down. So the two forces are on a collision course: one constant, one climbing steeply, and there is exactly one speed at which they cancel.

At that speed the net force is zero, acceleration stops, and the fall continues at a steady rate. That is terminal velocity — not a limit imposed from outside, but the equilibrium your own shape negotiates with the air.

Setting drag equal to weight and solving gives the relationship worth carrying around: v_t = √(2mg / (ρ·A·Cd)). Mass m pushes terminal velocity up; air density ρ, frontal area A and drag coefficient Cd push it down. Every one of them sits under a square root, which is why terminal velocity is so stubborn: to fall twice as fast you need four times the mass, or a quarter of the drag area.

What the 120 mph figure actually means

The number attached to skydiving almost everywhere is 120 mph, and it is worth being precise about what it is. 120 mph is 193 km/h, or 53.6 m/s — and it is a terminal velocity, not a free-fall reading. It is the speed at which a stable belly-to-earth skydiver stops accelerating: the equilibrium, not a milestone passed on the way down.

A stable belly-to-earth position sits roughly in the 190–200 km/h band, which is where the 120 mph shorthand comes from. Run the equation backwards and it yields the one quantity that is otherwise hard to pin down: for an 80 kg jumper at sea-level air density (ρ = 1.225 kg/m³), holding 53.6 m/s requires an effective drag area A·Cd of about 0.45 m².

That combined A·Cd is the honest way to describe a falling human. Frontal area and drag coefficient are not independently measurable for a person in a jumpsuit — they are inferred together from observed fall rates. Where a source quotes a confident area and a confident drag coefficient separately, treat both as estimates and the product as the only meaningful number.

How long it takes to get there

Terminal velocity is approached, never quite reached: the closer you get, the smaller the remaining net force. Under the standard model — drag proportional to v², with body position and air density held constant — speed follows v(t) = v_t · tanh(g·t / v_t), which gives a clean sense of the timescale.

For the 53.6 m/s case that is about 3.0 seconds to reach half of terminal velocity, 8.0 seconds to reach 90% and 10.0 seconds to reach 95%, by which point roughly 342 m have gone by. Reaching 99% takes about 14.5 seconds and 575 m. The last few per cent consume most of the altitude.

Compare the drag-free fantasy: with no air, 53.6 m/s would arrive after 5.5 seconds and only 147 m. Real air more than doubles both the time and the distance, then refuses to allow any more speed at all. The gap between 147 m and 342 m is the entire practical contribution of drag.

The four dials, and who they favour

Mass raises terminal velocity as √m, weakly: a jumper 21% heavier than another in the same body position falls only about 10% faster. Area and drag coefficient work the same way in reverse — halve the effective drag area and terminal velocity rises by a factor of √2, about 41%. Apply that to the 193 km/h belly-to-earth case and you land near 273 km/h, which is why head-down and tracking positions are so much faster than belly-to-earth.

The same square root explains why small animals survive falls that would kill a person: terminal velocity scales with the square root of the mass-to-area ratio, and small bodies carry far more surface per unit mass. It is also why a parachute works — multiplying canopy area by a large factor divides terminal velocity by the square root of that factor, which is enough to bring a landing down to a survivable walking-pace speed.

Air density is the dial most often forgotten. ρ falls with altitude, so the same body in the same position falls measurably faster high up than it does near the ground. High-altitude jumps reach far greater speeds not because gravity is stronger up there, but because there is less air to push against.

What the speed does when you arrive

Speed does not injure; stopping does. The energy carried into the ground is ½mv², so it scales with the square of impact speed — double the speed and four times as much energy has to go somewhere. That is why fall height matters so disproportionately, and why the drag-free rule of thumb is the one worth knowing for short falls, where drag has barely engaged.

The force delivered on landing then depends on stopping distance. The same kinetic energy spread over a longer deceleration — bent knees, a crash mat, a proper landing roll — produces a smaller peak force. That is the principle behind every piece of impact protection ever designed, and it is why the stopping distance, not the speed, is the number engineers actually control.

This is educational content, not safety or professional advice: it explains why the numbers behave as they do, and qualifies nobody to jump off anything.

When the fall is sideways as well as down

Not everything falls straight down. Anything thrown, launched or driven off an edge is falling and travelling at the same time, and the two motions are independent — gravity changes only the vertical component. That is why a bullet fired horizontally and one simply dropped from the same height reach the ground together, air effects aside.

That independence is what makes projectile problems tractable: solve a free-fall problem vertically, a constant-speed problem horizontally, then combine them. Drag couples the two again in the real world, shortening ranges most for light, fast objects, but the ideal solution remains the correct starting point for any estimate — and the correct thing to be sceptical about afterwards.