Hong Life CalculatorsUnderstanding the free fall — concept and principle
Free fall is motion under gravity alone, starting from rest. From a drop height, this calculator finds how long an object takes to reach the ground and how fast it is moving at impact, in m/s and km/h. It is used in physics exercises, for checking intuition about heights, and as a first estimate before air resistance is considered.
How it is calculated
- Fall time: t = √(2h ÷ g).
- Impact speed: v = √(2gh), equivalently v = g × t.
- Speed in km/h: v × 3.6, because 1 m/s = 3,600 m per hour = 3.6 km/h.
h is the drop height in metres and g the gravitational acceleration in m/s². The default 9.80665 m/s² is standard gravity; the real value ranges from about 9.78 at the equator to 9.83 at the poles.
The principle
Starting from rest with constant acceleration g, the distance fallen grows with the square of time: h = ½gt². Solving for t gives t = √(2h/g). Speed grows linearly, v = gt, and substituting t gives v = √(2gh). The same result follows from energy conservation: potential energy mgh turns into kinetic energy ½mv², and the mass cancels. That is why, without air, a feather and a hammer fall together, as Galileo argued and astronauts demonstrated on the Moon.
Worked example
- Inputs: h = 45 m, g = 9.80665 m/s².
- 2h ÷ g = 90 ÷ 9.80665 ≈ 9.1774, so t = √9.1774 ≈ 3.0294 s.
- 2gh = 2 × 9.80665 × 45 = 882.60, so v = √882.60 ≈ 29.7086 m/s.
- Check: g × t = 9.80665 × 3.0294 ≈ 29.71 m/s.
- v = 29.7086 × 3.6 ≈ 106.95 km/h.
| g (m/s²) | Fall time (s) |
|---|---|
| 2.45 | 6.05886302 |
| 4.9 | 4.284263128 |
| 7.35 | 3.498083818 |
| 9.81 | 3.029429966 |
| 12.3 | 2.709605916 |
| 14.7 | 2.473517107 |
| 19.6 | 2.142130472 |
How to read the result
Because time depends on the square root of height, quadrupling the height only doubles the fall time and the impact speed: from 180 m the object would fall for about 6.06 s and hit at about 59.4 m/s. A useful rule of thumb: in the first second an object falls about 4.9 m, after two seconds about 19.6 m, after three about 44 m. For light or fast objects, real values are lower because drag grows with speed until it balances weight at terminal velocity.
Common mistakes
- Using height in feet with g in m/s² (use 32.17 ft/s² for feet).
- Expecting the impact speed to depend on mass; in a vacuum it does not.
- Applying the formula to an object thrown downward or upward; then the initial speed must be included (use the kinematics equations).
Frequently asked questions
How high can something fall in a given time?
Use h = ½gt²; in 2 s that is ½ × 9.80665 × 4 ≈ 19.6 m.
Does g change with height?
Slightly. It decreases with altitude, but for heights of a few kilometres the change is well under 1% and can be ignored here.
Can I use this to estimate the depth of a well by dropping a stone?
Roughly, but the time also includes the sound returning to you, so the computed depth comes out a little too large for deep wells.
Sources: Halliday, D., Resnick, R. & Walker, J. Fundamentals of Physics (Wiley). · BIPM, The International System of Units (SI Brochure) — standard acceleration of gravity 9.80665 m/s².