How this instrument works
Free-fall velocity is the speed an object has picked up the instant it lands, having started at rest and fallen a known height with nothing but gravity acting on it — no motor, no push, no drag holding it back. The formula v = √(2gh) comes straight out of energy conservation: the potential energy it had at the top, mgh, converts completely into kinetic energy at the bottom, ½mv². Set those equal, cancel the mass — it appears on both sides — and solve for v, and the mass has vanished from the answer entirely.
That cancellation is why a marble and a cannonball, dropped from the same height in a vacuum, hit the ground at the identical speed. It is also why the relationship is a square root rather than a straight line: velocity depends on the square root of height, so quadrupling the drop — from 20 m to 80 m — only doubles the impact speed, from about 19.81 m/s to 39.61 m/s. Gravity itself enters as a fixed constant, g = 9.80665 m/s², the standard value fixed by international convention rather than a locally measured figure.
The formula describes an idealization: motion with only gravity acting, which is close enough for dense, compact objects falling modest distances — a dropped tool, a diving-board jump, a fall-arrest check for a construction harness. It stops being accurate once air resistance becomes significant relative to weight, which happens sooner for light or spread-out shapes; a feather or a sheet of paper never reaches the speed this equation predicts, because drag caps its velocity long before impact. For a skydiver or a falling raindrop, the real ceiling is terminal velocity, a separate balance between drag and weight that this instrument does not model.
- Enter the drop distance in the Height fallen field, in metres or centimetres — this is how far the object falls before landing.
- Leave out mass, shape, or weight; free-fall velocity depends only on height and gravity, not on what is falling.
- Read the result in Impact velocity — the speed at the instant of landing, before any bounce or ground contact.
- Switch the Impact velocity unit menu to km/h or mph to compare the figure against everyday speeds, like a motorway speed limit.
Worked example — dropped from 20 metres
Drop an object from 20 m — roughly the height of a six-storey building — and set Height fallen to 20. The instrument computes v = √(2 × 9.80665 × 20) = √392.266 = 19.8057062485 m/s, which reads out as 19.81 m/s. Multiply by 3.6 and that lands at about 71 km/h, comparable to a car merging onto a motorway, reached in barely two seconds of falling, entirely from gravity.
The number does not change if the object is a dropped wrench, a bag of sand, or, absent air resistance, a bowling ball — mass canceled out of the derivation, so height alone sets the speed. This is exactly why fall-arrest system designers and packaging engineers use the same equation: a stated maximum free-fall distance in a harness standard converts directly into an impact speed the equipment has to absorb, independent of what is attached to the line.
Questions
Does the falling object's mass change the impact velocity?
No. Mass cancels out of the derivation — it appears on both sides of the energy-conservation equation (½mv² = mgh) and divides away, leaving v = √(2gh) with no mass term at all. A coin and a cannonball dropped from the same height reach the same speed, a result Galileo is credited with demonstrating, and one that only breaks down once air resistance matters.
Why does doubling the height not double the impact velocity?
Because velocity depends on the square root of height, not on height directly. Doubling v would require multiplying the drop distance by four: fall 20 m and you land at 19.81 m/s; fall 80 m, four times as far, and you land at 39.61 m/s, exactly double. The square-root relationship is built into the energy equation, where kinetic energy scales with velocity squared.
Does this formula account for air resistance?
No — it assumes gravity is the only force acting, which is the free-fall idealization. Real falling objects also feel drag, which grows with speed and eventually balances weight at terminal velocity, a speed this equation cannot predict. For a dense, compact object falling a modest distance — a tool, a stone, a person in the first second or two of a fall — drag stays small enough that the formula holds up well.
What value of g does the calculator use?
Standard gravity, 9.80665 m/s², the value fixed by international agreement in 1901 and still the reference figure used in engineering and metrology worldwide. Local gravity varies slightly with latitude and altitude, roughly 9.78 m/s² near the equator to 9.83 m/s² near the poles, but for a general-purpose instrument the standard value is the correct default.
Can I use this for a fall that starts with some initial speed, like a thrown ball?
No — this instrument assumes the object starts from rest, which is what free fall means here. A thrown or launched object needs the initial speed added inside the square root, as v = √(v0² + 2gh), a different formula with an extra input. Using this calculator for a non-zero starting speed will understate the impact velocity.
How is this different from terminal velocity?
Terminal velocity is the speed at which drag exactly cancels weight, so the object stops accelerating; free-fall velocity from this formula keeps growing without limit as height increases, because it ignores drag entirely. For a human skydiver, terminal velocity, around 55 m/s belly-to-earth, is reached after roughly 12 to 15 seconds and several hundred metres, well before this equation's implied speed would apply.