Free Fall Calculator
The free fall calculator starts with a drop of 20 m from rest, with down taken as positive and the acceleration set to exactly 1 g. Change the height, or give it a starting speed if it was thrown rather than dropped.
Time t
- Time t From s = ut + ½at²: ½ × 9.80665 × t² + 0 × t − 20 = 0, so t = [−u ± √(u² + 2as)] / a = [0 ± √392.3] / 9.80665 → t = -2.02 or 2.02 s.
- 2.02 s
- Final velocity v From v = u + at: v = 0 + 9.80665 × 2.01962 = 19.81 m/s.
- 19.81 m/s
- Both roots of the quadratic Squaring t makes a second answer appear. The negative one is the time the same motion, run backwards, would have passed that point before the clock started. It satisfies the algebra but is not physical, so 2.02 s is the answer.
- t = -2.02 s (v = -19.81 m/s); t = 2.02 s (v = 19.81 m/s)
- Knowns used (SI) Everything is converted to meters, seconds and meters per second before the equations run, so mixed units cannot leak into the arithmetic.
- s = 20 m, u = 0 m/s, a = 9.807 m/s²
- Average velocity ½(u + v). Under constant acceleration, and only then, the average velocity is the mean of the start and end velocities, which is why s = ½(u + v)t works.
- 9.903 m/s
- Acceleration in g Divided by standard gravity, 9.80665 m/s². A useful sense check: a hard road-car launch is a fraction of 1 g, and free fall is exactly 1 g.
- 1 g
- Final velocity in other units
- 19.81 m/s 71.3 km/h 44.3 mph 64.98 ft/s
| Equation | Leaves out | Left side | Right side | Used here |
|---|---|---|---|---|
| v = u + at | s | 19.81 | 19.81 | yes |
| s = ut + ½at² | v | 20 | 20 | yes |
| v² = u² + 2as | t | 392.3 | 392.3 | |
| s = ½(u + v)t | a | 20 | 20 | |
| s = vt − ½at² | u | 20 | 20 |
- The t and v boxes are ignored: those are the two being solved for.
- These equations assume the acceleration is constant for the whole interval. If it changes (a car shifting gear, anything with real air resistance), split the motion into pieces that are each roughly constant.
- Results are shown to four significant figures. Round the answer to the precision of your least precise measurement.
How the Kinematics Calculator works
Free fall is constant-acceleration motion where the acceleration is gravity, so it is the SUVAT equations with a fixed: a = g. This calculator uses standard gravity, g = 9.80665 m/s², which is a defined value rather than a measurement. It was fixed in 1901 and is the same constant behind the pound-force and the kilogram-force. Real surface gravity varies by a fraction of a percent with latitude and altitude, from roughly 9.78 m/s² at the equator to 9.83 m/s² at the poles, which is far smaller than the error from ignoring air.
The two formulas. Dropped from rest (u = 0), the distance fallen after time t is s = ½gt², and the speed after falling a height h is v = √(2gh). Rearrange the first and the fall time from height h is t = √(2h / g).
Worked example, the one above: a 20 m drop. t = √(2 × 20 / 9.80665) = √4.0789 = 2.020 s, and the impact speed is v = √(2 × 9.80665 × 20) = 19.81 m/s, about 71 km/h. Notice what the square root does: doubling the height does not double the time. A 40 m drop takes 2.856 s, only 41% longer, and lands at 28.01 m/s.
The distances in each second grow in the ratio 1 : 3 : 5 : 7. In the first second an object falls 4.903 m, by the end of two seconds 19.61 m, by three 44.13 m. That pattern is Galileo's, and it is the signature of constant acceleration.
The calculator solves s = ut + ½at² for t, which is a quadratic, so it shows two roots. Dropped from rest they are equal and opposite, and the negative one is discarded. It describes a different story that fits the same equation: a ball thrown up from 20 m below at 19.81 m/s, which would rise to the drop point and stop there 2.020 s before the clock starts. Give the object a starting speed and the two roots stop being mirror images.
Signs. This calculator takes down as positive, so the height and g are both positive. If you prefer up as positive, which most textbooks do for thrown objects, enter the displacement as −20 m and the acceleration as −1 g; the answers are the same with the velocity negative.
What it ignores: air resistance, completely. That is accurate for a dense object over a short drop and badly wrong for anything light or long. Drag grows with the square of speed, so a falling object accelerates less and less until drag balances its weight and it stops speeding up. That speed is its terminal velocity. A skydiver, a raindrop and a feather all reach theirs, the feather almost at once. For those, treat the result here as an upper bound on the speed and a lower bound on the time.
Kinematics Calculator FAQ
How long does it take something to fall?
From rest, with air resistance ignored, t = √(2h / g). With g = 9.80665 m/s², a 5 m fall takes 1.010 s, 20 m takes 2.020 s, and 100 m takes 4.516 s. Fall time grows with the square root of the height, so four times the height only doubles the time.
How fast does a falling object hit the ground?
v = √(2gh) from rest. A 20 m drop lands at 19.81 m/s (about 71 km/h). The speed does not depend on the mass, which is the point of Galileo's result: a heavy and a light object dropped together land together when air resistance is negligible.
Why 9.80665 and not 9.8 or 9.81?
9.80665 m/s² is standard gravity, a defined constant used for weights and unit definitions. 9.8 and 9.81 are roundings of it. Actual gravity at a given place varies slightly, but by less than the effect of air on most real falls.