Free Fall Time Calculator

Definition of free fall and the time formula

Free fall is the motion of a body acted on by gravity alone; near the Earth’s surface that force gives every body the same acceleration g, independent of mass. A body released from rest moves with uniform acceleration: its speed grows linearly, v = g·t, and its distance quadratically, h = ½gt². The time to fall a height h is therefore the square root t = √(2h/g).

Two properties follow from the form of the formula, and both clash with everyday intuition. First, time grows with the square root of height: a building four times taller gives a fall only twice as long. Second, time does not depend on mass — in a vacuum a lead ball and a feather take the same time. On Earth the differences between objects come from air resistance, which this model ignores.

Historical note: Aristotle, Galileo, and the Apollo 15 hammer

For almost two thousand years Aristotle’s view prevailed: heavier bodies fall faster, with speed proportional to weight. Galileo Galilei undermined it first with a thought experiment (two bodies tied together would have to fall both faster and slower than each alone) and then with measurements. Because a vertical drop was too fast for the clocks of his day, Galileo slowed the motion by rolling balls down an inclined plane and timing them with a water clock.

In Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638) he stated the law: distances covered in successive equal intervals of time stand in the ratio of the odd numbers 1 : 3 : 5 : 7, which is equivalent to h ∝ t². The most famous demonstration of the principle remains David Scott’s experiment on Apollo 15 (1971): a hammer and a falcon feather released together on the airless Moon struck the ground at the same instant.

Deriving t = √(2h/g) from the equations of motion

Step 1. Take an axis pointing downward with its origin at the release point and initial speed v₀ = 0. The acceleration is constant and equal to g, so the speed at time t is v(t) = g·t. Step 2. Distance is the integral of speed over time: h(t) = ∫₀ᵗ g·τ dτ = ½gt². The same result is the area of the triangle under the v(t) graph with base t and height g·t.

Step 3. Solve h = ½gt² for time: t² = 2h/g, and since time is non-negative, t = √(2h/g). Step 4. The speed just before the ground follows from v = g·t = √(2gh); energy conservation, mgh = ½mv², gives the same value with the mass canceling. For h = 10 m and g = 9.81 m/s²: t = √(2.039) = 1.43 s, v = 14.0 m/s. The time to the halfway point is t/√2 ≈ 0.707·t, because h/2 = ½g(t/√2)².

Model assumptions: rest, constant g, no drag

The formula t = √(2h/g) is exact only under three assumptions. First: zero initial speed, meaning the object is released, not thrown. A downward throw shortens the time and an upward throw lengthens the total flight; both cases are covered by the more general equation h = v₀t + ½gt², handled by a separate vertical throw calculator. Second: the acceleration g is constant along the path. For heights of meters to kilometers this is an excellent approximation, because g decreases by only about 0.3 % per 10 km of altitude.

The third and, in practice, most important assumption is the absence of air resistance. Drag grows with speed, so the longer the fall, the more the real motion departs from the model. For a compact, dense object — a stone, a key, a ball — dropped from a few to a few tens of meters, the time differs by no more than a few percent. For light, extended objects (a sheet of paper, a leaf, a balloon) the model breaks down within a fraction of a second.

Air resistance and terminal velocity

At speeds typical of a fall, the drag force is approximately proportional to the square of the speed: F = ½·C·ρ·A·v², where C is a shape coefficient, ρ the air density (about 1.2 kg/m³), and A the cross-sectional area. When drag equals the weight m·g, the acceleration drops to zero and the body continues at a constant terminal velocity v_t = √(2mg/(C·ρ·A)).

For a skydiver in a belly-down position the terminal velocity is about 50–55 m/s (110–120 mph), reached after roughly 12 seconds and 450 m of fall; the vacuum model would predict 118 m/s by then. A 2 mm raindrop reaches only about 6.5 m/s. On the other hand, a steel ball dropped from 10 m hits the ground at a speed only slightly below the 14 m/s the formula predicts. The practical rule for this page: the larger the ratio of mass to cross-sectional area and the lower the height, the better the result describes reality.

The value of g: Earth, other worlds, and measurement

The standard acceleration of gravity g_n = 9.80665 m/s² was adopted by the 3rd General Conference on Weights and Measures in 1901 as a conventional value. The real g at the Earth’s surface ranges from about 9.780 m/s² at the equator to 9.832 m/s² at the poles, mainly because of the planet’s flattening and the centrifugal effect of its rotation; across the contiguous United States it is close to 9.80 m/s², so the default of 9.81 changes a 10 m fall time by about a thousandth of a second. Local deviations of the order of 10⁻⁵ m/s² are measured with gravimeters and used in geodesy and mineral exploration.

On other bodies of the Solar System g is different, and with it the fall time, which scales as 1/√g. On the Moon (1.62 m/s²) a 2 m drop takes 1.57 s instead of 0.639 s on Earth — hence the slow-motion look of Apollo footage. On Mars (3.71 m/s²) the same drop takes 1.04 s, and on Jupiter, if it had a surface (24.79 m/s²), only 0.40 s. The g for any mass and radius comes from the g = GM/r² calculator; the result can be entered into the g field on this page.

Sources

The definition, derivation, and historical note come from general physics textbooks and metrology sources: G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Leiden 1638 (English: Dialogues Concerning Two New Sciences, Crew and de Salvio translation); D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapter on motion along a straight line; H. D. Young, R. A. Freedman, University Physics, chapter “Motion Along a Straight Line”; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, chapters 8 (Motion) and 9 (Newton’s Laws of Dynamics); BIPM, The International System of Units (SI), 9th edition, and the 3rd CGPM resolution (1901) on g_n; NASA, Apollo 15 Lunar Surface Journal, the hammer-and-feather demonstration. This page is for educational purposes.