Example 1
A splash is heard 2 s after release, at Earth’s g = 9.81 m/s². This is the reference example on this page: ½·9.81·4 = 19.62 m.
h = 19.62 m
Free Fall Calculator
Enter the fall time in seconds. The calculator finds the distance from rest with h = ½gt² and also shows the speed v = gt and the distance at double the time. An empty g field means 9.81 m/s². After 2 s the distance is 19.62 m. The model does not include air drag or a nonzero initial speed.
Result
Enter the fall time. The distance h, the speed v = gt, and the distance-versus-time chart will appear here.
You know the height and want the time? Compute t = √(2h/g) in the free-fall time calculator. You have an initial speed: vertical throw.
How it works
t²4.00 s²
g·t²39.24 m
× ½19.62
h19.62 m
Free fall from rest is motion in which the initial speed is zero and the acceleration is constant and equal to g. The distance covered in time t follows h = ½gt². Textbooks often write the same distance as s. This calculator reports h. For 2 s and g = 9.81 m/s² the square of the time is 4 s², and ½·9.81·4 = 19.62 m. The speed at the end of that time is v = gt = 19.62 m/s, which is 70.6 km/h.
The calculator uses the numbers you enter: the time t in seconds and the acceleration g. An empty g field means 9.81 m/s². That is the classroom value for Earth, not the standard acceleration 9.80665 m/s². For the same 2 s the standard value would show 19.61 m on the panel, so an empty field stays at 19.62 m. A comma and a period are the same decimal separator. One minute is 60 seconds, not the number 1 in the time field.
What the result panel shows. Beside the distance, the calculator reports the speed v = gt, the average speed from rest v/2 (9.81 m/s for 2 s), the square of the time, the distance at double the time, and the entered t and g. After 4 s, double the reference time of 2 s, the distance is 78.48 m. At 0.5 s, 1 s, 1.5 s, and 2 s the same formula gives 1.23 m, 4.91 m, 11.04 m, and 19.62 m. The extra distance in equal time steps follows 1 : 3 : 5 : 7. The chart draws the distance at 0.5 s, 1 s, 1.5 s, 2 s, 2.5 s, and 3 s for the g you entered.
Limits of the model. The formula assumes v₀ = 0, constant g along the whole path, and no air resistance. Drag is not subtracted, and there is no air-resistance mode. Terminal speed is computed by the v = √(2mg/(ρCS)) calculator. A throw with a nonzero v₀ is computed by the vertical-throw calculator. Time from a known height is computed by the t = √(2h/g) calculator. Mass does not enter the formula: in this model each object covers the same distance in the same time.
The unit switch in the header changes distance to feet and g to ft/s², and it converts a number already entered in the g field. Time stays in seconds. Sentences on the page stay in SI units as the reference.
The examples use the same formula. 2 s at g = 9.81 m/s² gives 19.62 m. 0.45 s gives 0.99 m. 1.4 s at g = 1.62 m/s² gives 1.59 m. The button “Fill this example” enters those numbers and refreshes the panel and the chart. The fields start empty.
Formula
h = ½ · g · t²
[h] = (m/s²)·s² = m; v = g·t; vavg = v/2; h(2t) = 4h
Model: uniformly accelerated motion from rest (Galileo, Discorsi, 1638; in classroom notation h = ½gt², often also written s = ½gt²). The rearrangement t = √(2h/g) follows from the same formula, but this calculator finds only h. Order of operations: t², then ½gt² in meters, then v = gt.
Theory, derivation, and domain
Definition of distance in a fall from rest
Free fall is motion under gravity alone. Near Earth’s surface that force gives each object the same acceleration g, independent of mass. When the body starts from rest, speed grows linearly, v = gt, and distance grows with the square of time, h = ½gt². The letter s in s = ½gt² is the same distance. This calculator reports that distance as h.
Galileo and the ratio 1 : 3 : 5 : 7
In Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638) Galileo described distances covered in successive equal times as the odd numbers. That is the classroom form of the square law. For g = 9.81 m/s² and steps of 0.5 s the formula gives cumulative distances of 1.23 m, 4.91 m, 11.04 m, and 19.62 m. Galileo did not write the constant g in today’s notation. Current textbooks (Halliday, Resnick, and Walker; Young and Freedman) derive h = ½gt² from constant acceleration with v₀ = 0.
Speed, the average, and double the time
For 2 s and g = 9.81 m/s² the final speed is 19.62 m/s, or 70.6 km/h. The average speed over the whole path, from zero to v, is half the final speed: 9.81 m/s. Doubling the time multiplies the distance by four, because time is squared: after 4 s the distance is 78.48 m. After 1 s the same formula gives 4.91 m, not half of 19.62 m.
What an empty g field uses
An empty g field means exactly 9.81 m/s². Standard gravity, adopted by the CGPM in 1901, is 9.80665 m/s². For 2 s the panel would then show 19.61 m. The calculator does not switch to that value on its own. The lunar 1.62 m/s² and the Martian 3.71 m/s² come from NASA fact sheets and are entered in the g field. A value of g from a planet’s mass and radius is computed on the separate g = GM/r² page.
Air resistance is a limit, not a result
In air the drag force grows with speed and with cross-sectional area, and the speed approaches a terminal value. This calculator does not compute that speed and does not shorten the distance for drag. For a short time and a compact body the difference can be small. For a sheet of paper, a leaf, or a long fall, h = ½gt² is an upper bound on the distance in the given time, not a measurement in the atmosphere. Terminal speed, where weight balances quadratic drag, is computed by a separate calculator.
How to use
- Enter the fall time in seconds, for example 2. That is the time from release, not the height.
- Leave g empty for 9.81 m/s², or enter another value, for example 1.62 for the Moon. Do not enter 9.80665 when the problem states 9.81.
- Click Calculate or wait for the live result. For 2 s and an empty g the calculator shows 19.62 m.
- Read the panel: the speed v = gt, the average speed, t², and the distance at double the time. The chart is the curve h(t).
- Clearing the form removes the entered numbers and hides the panel. Time from a height is computed on the separate t = √(2h/g) page.
Symbol glossary
h is the distance fallen from rest, t is the time, and g is the acceleration. For 2 s and g = 9.81 m/s² the result is 19.62 m.
- h
- Fall distance [m], h = ½gt². For 2 s and 9.81 m/s² the panel shows 19.62 m. This is not the peak height of an upward throw.
- s
- The same distance in the textbook writing s = ½gt². The calculator displays it as h.
- t
- Fall time [s], the first field. Zero seconds gives h = 0. A negative value is rejected.
- g
- Acceleration [m/s²]. An empty field means 9.81, not 9.80665. It enters the formula linearly.
- v
- Speed at the end of the time [m/s], v = gt. For 2 s it is 19.62 m/s (70.6 km/h).
- v₀
- Initial speed. Here it is 0. A nonzero v₀ belongs on the vertical-throw page.
- ½
- The factor in the distance formula for constant acceleration starting from rest.
Worked examples
Example 2
The same Earth value and a time of 1 s. The time is half the reference example, and the distance is one quarter, because t is squared.
h = 4.91 m
Example 3
An object slipped from a hand and fell for 0.45 s. The calculator finds the distance from that time. It does not guess the height of a table.
h = 0.99 m
Example 4
From a gutter to a windowsill takes 0.8 s. The result is the distance in the no-drag model, at g = 9.81 m/s².
h = 3.14 m
Example 5
From a bounce to the water surface takes 1.4 s. At Earth’s g the distance is 9.61 m. Air resistance is not subtracted.
h = 9.61 m
Example 6
An object slipped and fell for 0.4 s. A short time gives a short distance, still from h = ½gt².
h = 0.78 m
Example 7
A classroom problem: 0.7 s of fall from a branch, starting from rest, with g = 9.81 m/s².
h = 2.40 m
Example 8
The same 1.4 s, but at the lunar g = 1.62 m/s² from the NASA fact sheet. The Apollo 15 demonstration showed a hammer and a feather falling together. This calculator does not distinguish their masses, and it does not claim that a stopwatch on the Moon read exactly 1.4 s.
h = 1.59 m
Example 9
A time of 2 s at the Martian g = 3.71 m/s². That is the same two seconds as the reference example, but the smaller g gives a shorter distance.
h = 7.42 m
Example 10
A child dropped a ball and the ball fell for 1.1 s. The mass of the ball does not enter the formula.
h = 5.94 m
Example 11
A full 3 s of fall at g = 9.81 m/s². The panel shows 44.14 m. This is still the no-drag model, so in air the real distance in that time would be smaller.
h = 44.14 m
Common questions
What does this calculator compute, and from which formula?
The calculator finds the distance fallen from rest with h = ½gt². It squares the time, multiplies by g, and divides by two. For 2 s and g = 9.81 m/s² the result is 19.62 m. Textbooks sometimes write the same distance as s = ½gt².
Which units should I enter for time and g?
Enter time in seconds and g in meters per second squared. The distance is in meters. After you switch the header to US units, the g field accepts ft/s² and the distance is shown in feet, with meters in parentheses. Time stays in seconds. A comma and a period are the same decimal separator.
What happens if I leave g empty?
The calculator uses 9.81 m/s². It does not insert the standard acceleration 9.80665 m/s². For 2 s the panel difference is 19.62 m versus 19.61 m. The lunar 1.62 and the Martian 3.71 have to be entered by hand.
Why does twice the time give four times the distance?
Because time is squared in the formula. After 1 s the distance is 4.91 m, and after 2 s it is 19.62 m. After 4 s, double the 2 s example, the distance is 78.48 m. The panel shows that value on its own.
Does the object’s mass change the distance?
Not in this model. The acceleration g is the same for each object, so there is no mass field. Differences between a feather and a hammer on Earth come from air resistance, which this calculator does not compute.
What speed does the panel show?
The panel reports v = gt and the average speed v/2. For 2 s and g = 9.81 m/s² the final speed is 19.62 m/s (70.6 km/h), and the average is 9.81 m/s (35.3 km/h). That is not a terminal speed.
Can I enter an initial speed?
No. The start is from rest, so v₀ = 0. An upward or downward throw with a nonzero v₀ uses h = v₀t + ½gt² on the vertical-throw page. A drop from a ledge with a horizontal speed is on the horizontal-throw page.
Does the result include air resistance?
No. Air resistance is a limit of the model, not a separate result. The calculator does not compute a drag force, does not shorten the distance, and does not report a terminal speed. That speed, from v = √(2mg/(ρCS)), is on a separate page. On a long fall in air the distance in a given time is smaller than h = ½gt².
I know the height. How do I find the time?
This calculator goes only from time to distance. The inverse problem, t = √(2h/g), is solved by the free-fall time calculator. There is no height field here.
What do a time of zero and a negative number mean?
At t = 0 the distance is 0 m and the speed is 0 m/s, because nothing has fallen yet. A negative time is rejected. An empty time field leaves the panel blank, with no number. A value of g that is zero or negative is rejected too.
What do the panel and the chart show?
The panel reports the distance, the speed v = gt, the average speed, t², the distance at double the time, and the entered t and g. The chart draws h at 0.5 s, 1 s, 1.5 s, 2 s, 2.5 s, and 3 s for that same g. The button “Fill this example” enters an example and refreshes the panel and the chart.
How do I enter g for the Moon or Mars?
Enter 1.62 for the Moon or 3.71 for Mars. For 1.4 s on the Moon the distance is 1.59 m. For 2 s on Mars the distance is 7.42 m. Both values of g are on NASA fact sheets. The calculator does not check whether the body has an atmosphere.
How do I calculate distance with h = ½gt²?
Multiply half of g by the square of time. At g = 9.81 m/s² and t = 2 s: ½·9.81·4 = 19.62 m.
What is the speed after 2 s of free fall from rest?
v = g·t = 9.81·2 = 19.62 m/s. Average speed on that interval is half of that value, 9.81 m/s.
Why does doubling the time quadruple the distance?
The formula has t². At 4 s the distance is ½·9.81·16 = 78.48 m, four times the distance at 2 s.
How is this page different from the free-fall time calculator?
Here you enter time and get distance and speed. The free-fall time companion takes height h and computes t = √(2h/g).
Can I enter a downward initial speed?
No. The model assumes v₀ = 0. A downward or upward throw belongs on the vertical-throw calculators.
Sources
The formula, the constant g, and the units come from mechanics textbooks and from metrology sources. This page does not invent publication identifiers.
- Wikipedia — Free fall
- Wikipedia — Two New Sciences (Galileo, Discorsi, 1638)
- Wikipedia — Standard gravity (9.80665 m/s²)
- BIPM — The International System of Units (SI Brochure)
- GUM — Polish Central Office of Measures
- NASA — Moon Fact Sheet
- NASA — Mars Fact Sheet
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics, chapter on motion with constant acceleration.
- Young, H. D.; Freedman, R. A., University Physics, chapter on motion along a straight line with constant acceleration.
Page updated November 3, 2026.