Terminal velocity calculator
Terminal velocity is where gravity and drag balance. The formula is simple; one of its inputs is not a number you can look up, and this page is mostly about that. Nothing you type here leaves your browser — it is worked out on your own device, so there is nothing to send and nothing to log.
A sphere does not have a drag coefficient
Every terminal velocity calculator asks you for Cd as though it were a property of the shape, like its area. NASA publish the drag coefficient of a sphere as 0.07 to 0.5 — a range spanning a factor of seven — and say why:
“The drag coefficient for a sphere is given
with a range of values because the drag on a sphere is highly
dependent on Reynolds number.”
— NASA Glenn Research Center, Shape Effects on Drag
Reynolds number depends on velocity. Velocity is what you are trying to calculate. The input is a function of the answer. A sphere at walking pace and the same sphere at terminal velocity are in different flow regimes with different drag coefficients, and choosing one number for both is choosing which answer you want.
Across NASA's published range for a sphere, the terminal velocity comes out a factor of 2.67 apart. That is not a rounding question. That is the difference between a hailstone that hurts and one that does not.
And the square root hides it
v = √(2mg / ρACd), so velocity goes as Cd to the power of minus a half. Halving or doubling the drag coefficient moves the answer by only 29%:
| Cd wrong by | Answer wrong by |
|---|---|
| 10% | 4.7% |
| 25% | 10.6% |
| 100% | 29.3% |
| 200% | 42.3% |
This is the worst combination available: the input you are least sure of is the one the output is least sensitive to. The result is a calculator that cannot be obviously wrong and cannot be right either — it will always produce a plausible number, for any drag coefficient you feed it.
That is why this page shows what the answer would be at double and half the drag coefficient, rather than printing one figure to five significant places and letting you believe it.
The shapes NASA give
| Flat plate | 1.28 |
|---|---|
| Wedge-shaped prism | 1.14 |
| Model rocket | 0.75 |
| Sphere | 0.07 to 0.5 |
| Bullet | 0.295 |
| Airfoil | 0.045 |
Drag coefficients from NASA Glenn Research Center, Beginner's Guide to Aeronautics, Shape Effects on Drag. A work of the United States government, in the public domain.
A flat plate and a good airfoil differ by a factor of nearly thirty, which is the entire reason aircraft are shaped the way they are.