Physics
How To Calculate Terminal Velocity
A falling object stops accelerating when the air pushing up matches its weight. Terminal velocity is the speed where those two forces balance.
Quick Answer
v = sqrt(2 x m x g / (rho x Cd x A))
- v
- The terminal velocity, the speed where acceleration stops
- m
- The mass of the falling object
- rho
- The density of the air it falls through
- Cd x A
- The drag coefficient times the presented area
Set weight equal to drag and solve for speed. An 80 kg body presenting 0.7 square metres with a drag coefficient of 1 in sea level air reaches 42.7763 metres per second, which is 153.9947 kilometres per hour.
What Is Terminal Velocity?
Terminal velocity is the steady speed a falling object settles at. Gravity keeps pulling downwards, but drag grows with speed until the two forces are equal and the net acceleration becomes zero.
Drag on a falling body at these speeds is quadratic in velocity. The force is half the air density, times the speed squared, times the drag coefficient, times the presented area.
Setting that equal to weight gives the formula directly. Two times mass times gravity, divided by the density, the drag coefficient and the area, all under a square root.
Heavier objects fall faster. Terminal velocity grows with the square root of mass, so doubling the mass while everything else stays fixed makes the speed about 1.41 times higher.
Area works the other way. Terminal velocity falls with the square root of the presented area, which is exactly why spreading out slows a fall and why a parachute works.
The drag coefficient captures the shape. A smooth sphere is around 0.47, a flat plate facing the flow is about 1.28, and a human body in a spread position is usually modelled near 1.
Air density matters and is not a constant in reality. Sea level air at 15 degrees is 1.225 kilograms per cubic metre, but at 3000 metres it is closer to 0.909, which raises the terminal velocity noticeably.
That is why the model gives a speed rather than a single true value. A skydiver falling from altitude starts out fast in thin air and slows as the air thickens below.
A belly to earth skydiver at 80 kilograms with an effective drag area near 0.45 square metres reaches about 53 metres per second. Tucking into a head down position shrinks the area further and pushes the speed past 80 metres per second.
The approach is asymptotic. An object gets close to terminal velocity without ever exactly reaching it, which is why the time constant matters more than the speed for short falls.
The time constant is the terminal velocity divided by gravity. For the default case that is 4.362 seconds, and about three time constants, roughly 13 seconds, gets you to 95 percent of the final speed.
Distance covered during the approach is substantial. A skydiver reaches most of terminal velocity within about 450 metres of fall, which is a little over twelve seconds.
Small animals survive falls that would kill a person. A cat presents a lot of area for very little mass, so its terminal velocity is low and the impact is survivable.
Raindrops have their own terminal velocity of roughly 9 metres per second, which is why they do not arrive like bullets. Hailstones are denser and larger, and they fall far faster.
A parachute works by multiplying the area. The same 80 kilogram mass under a canopy of 40 square metres with a drag coefficient near 1.4 lands at around 6 metres per second, slow enough to walk away from.
Formula
v = sqrt(2 x m x g / (rho x Cd x A))
Formula page →Weight equals quadratic drag, solved for speed.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| v | Terminal velocity | metres per second | The steady falling speed. |
| m | Mass | kilograms | The mass of the falling object. |
| rho | Air density | kg per cubic metre | The density of the fluid being fallen through. |
| Cd | Drag coefficient | unitless | A shape factor, about 1 for a human body. |
| A | Presented area | square metres | The cross sectional area facing the flow. |
F = 0.5 x rho x v^2 x Cd x A
Formula page →Quadratic drag, the force that grows until it matches weight.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| F | Drag force | newtons | The upward force from the air. |
| rho | Air density | kg per cubic metre | The density of the fluid. |
| v | Speed | metres per second | The current falling speed. |
| A | Presented area | square metres | The area facing the flow. |
tau = v / g
Formula page →The characteristic time of the approach, about three of which reach 95 percent.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| tau | Time constant | seconds | The characteristic time of the approach. |
| v | Terminal velocity | metres per second | The steady falling speed. |
| g | Gravity | metres per second squared | The acceleration of gravity. |
How To Calculate Terminal Velocity
- 1
Multiply the weight terms
Two times 80 kilograms times 9.80665 is 1569.064. This is twice the weight, and it sits in the numerator of the formula.
- 2
Multiply the drag terms
Air density times the drag coefficient times the area is 1.225 times 1 times 0.7, which is 0.8575. This is the denominator.
- 3
Divide and take the square root
1569.064 divided by 0.8575 is 1829.8122, and the square root of that is 42.7763 metres per second.
- 4
Confirm the forces balance
At that speed drag is 784.532 newtons, which is exactly the weight of an 80 kilogram mass. The balance confirms the speed is terminal.
Examples
Example 1: An 80 kg body in a spread position at sea level
- Mass
- 80
- Air density
- 1.225
- Drag coefficient
- 1
- Area
- 0.7
| Step | Calculation | Result |
|---|---|---|
| Twice the weight | 2 x 80 x 9.80665 | 1569.064 |
| Density times drag coefficient times area | 1.225 x 1 x 0.7 | 0.8575 |
| Ratio under the root | 1569.064 / 0.8575 | 1829.8122 |
| Terminal velocity | sqrt(1829.8122) | 42.7763 |
Result: The force balance gives 42.7763 metres per second, which is 153.9947 kilometres per hour or 95.6879 miles per hour. Drag at that speed is 784.532 newtons, matching the weight exactly.
Example 2: The same body tucked into a head down position
- Mass
- 80
- Air density
- 1.225
- Drag coefficient
- 1
- Area
- 0.18
| Step | Calculation | Result |
|---|---|---|
| Density times drag coefficient times area | 1.225 x 1 x 0.18 | 0.2205 |
| Ratio under the root | 1569.064 / 0.2205 | 7115.9347 |
| Terminal velocity | sqrt(7115.9347) | 84.356 |
| In kilometres per hour | 84.356 x 3.6 | 303.6816 |
Result: Tucking into a head down posture cuts the presented area to 0.18 square metres and raises the terminal velocity to 84.356 metres per second, about 303.6816 kilometres per hour.
Calculator
Terminal velocity
42.7763
- In kilometres per hour
- 153.9947
- In miles per hour
- 95.6879
- Time constant
- 4.362
- Drag force at that speed
- 784.532
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Terminal Velocity calculator page.
Common Mistakes
Using a linear drag model
At human falling speeds drag grows with the square of velocity. A linear model understates the force badly and gives a terminal velocity that is far too high.
Mixing up mass and weight
The numerator is weight, mass times gravity. Entering 80 as a force instead of a mass understates the numerator by a factor of 9.80665.
Using the wrong area
The area is the cross section facing the flow, not the total surface area. A spread body and a tucked body differ by a factor of four in area and 1.97 in speed.
Treating air density as fixed
Air at 3000 metres is about three quarters as dense as at sea level. Terminal velocity rises as the object descends rather than staying constant.
Assuming the speed is reached instantly
The approach takes about three time constants. With a time constant of 4.362 seconds, a fall shorter than about 13 seconds never gets close.
Taking the drag coefficient as universal
Cd depends on shape and on the Reynolds number. A sphere sits near 0.47, a flat plate near 1.28, and the value for a body changes with posture and clothing.
FAQ
What is the formula for terminal velocity?
The square root of two times mass times gravity, divided by air density, drag coefficient and presented area.
What is the terminal velocity of a human?
Around 53 metres per second, roughly 190 kilometres per hour, in a stable belly to earth position. A head down posture with less area can exceed 80 metres per second.
Does a heavier object fall faster?
At terminal velocity, yes. The speed grows with the square root of mass, so twice the mass is about 1.41 times the speed, all else being equal.
How long does it take to reach terminal velocity?
About three time constants, where the time constant is the terminal velocity divided by gravity. For the default case that is roughly 13 seconds and about 450 metres of fall.
Why does a parachute make such a difference?
Because terminal velocity falls with the square root of area. Forty square metres of canopy instead of 0.7 drops the speed from tens of metres per second to about six.
Is terminal velocity the same at every altitude?
No. Thinner air at altitude means less drag, so the speed is higher up high and decreases as the object descends into denser air.
References
- [1]Wikipedia, Terminal velocity — https://en.wikipedia.org/wiki/Terminal_velocity
- [2]Wikipedia, Drag equation — https://en.wikipedia.org/wiki/Drag_equation
- [3]Wikipedia, Drag coefficient — https://en.wikipedia.org/wiki/Drag_coefficient