What free fall means in physics
Free fall means motion under gravity alone. No thrust, no lift, no drag — just weight. That restriction is what makes the arithmetic so clean: the acceleration is the same at every instant, so the constant-acceleration equations apply exactly, and the mass drops out of every result that describes the motion.
The mass cancelling is the famous part. A 1 kg steel ball and a 10 kg steel ball released together in a vacuum hit the ground at the same moment, because although the heavier one is pulled by ten times the force, it also has ten times the inertia resisting that force. Newton's second law gives a = F/m = mg/m = g, and the mass has vanished. That is why this calculator asks for a mass only when you want the impact energy: mass changes what the landing does, never how long it takes or how fast it arrives.
In real air the restriction is violated the moment the object starts moving, because drag grows roughly with the square of speed. For a short drop of a dense object — a dropped spanner, a stone off a bridge, a load slipping from a hoist at low height — the vacuum answer is within a few per cent and perfectly usable. For a sheet of paper, a parachutist, or anything falling for more than a handful of seconds, it is not a prediction at all, only an upper bound.
Where the three free-fall equations come from
Start with the definition of acceleration. Velocity grows at a constant rate g, so after time t from rest the speed is simply v = gt. That is the first equation and it needs no derivation beyond the definition.
Distance follows from the average speed. Because the speed rises linearly from 0 to gt, the average speed over the fall is half the final speed, gt/2. Multiply by the duration and you get h = (gt/2) × t = ½gt². Invert that to get the fall time from a known height: t = √(2h/g).
The third equation eliminates time. Substitute t = v/g into h = ½gt² and you get h = v²/(2g), which rearranges to v = √(2gh). This is the form to remember, because it is the energy statement in disguise: multiply through by ½m and you have ½mv² = mgh. All the gravitational potential energy the object had at the top has become kinetic energy at the bottom, which is why energy at impact equals mgh and you never need the speed to find it.
The square root is the reason falling gets slow returns on height. Doubling the drop height does not double the impact speed; it multiplies it by √2, about 1.41. To double the impact speed you need four times the height — and four times the impact energy, since energy is proportional to height directly.
Worked example: a tool dropped from 12 m of scaffold
A 2.5 kg wrench slips off a working platform 12 m above the deck. How long does anyone below have, how fast is it moving on arrival, and how much energy does it carry?
- Fall time. t = √(2h/g) = √(2 × 12 ÷ 9.80665) = √2.44730 = 1.5644 s. That is the whole warning period.
- Impact speed. v = gt = 9.80665 × 1.5644 = 15.34 m/s, which is 55.2 km/h. Cross-check with the other form: √(2 × 9.80665 × 12) = √235.36 = 15.34 m/s. The two agree, as they must.
- Impact energy. KE = mgh = 2.5 × 9.80665 × 12 = 294.2 J. Or from the speed: ½ × 2.5 × 15.34² = 0.5 × 2.5 × 235.4 = 294.2 J.
- Momentum. p = mv = 2.5 × 15.34 = 38.4 kg·m/s. If the wrench stops in 5 ms against a hard surface, the average force is 38.4 ÷ 0.005 ≈ 7,700 N — an illustration of why short stopping times are what make falls dangerous, not speed on its own.
Now change one thing: raise the platform to 24 m. The fall time goes to 2.212 s, only 41% longer. The impact speed goes to 21.70 m/s, also 41% higher. But the energy goes to 588.4 J — exactly double, because energy tracks height directly while speed tracks its square root.
Reading the result honestly, and where drag takes over
Treat every figure this calculator produces as a vacuum value. The question that decides whether it is usable is how the object's drag compares with its weight at the speeds involved. Drag depends on frontal area, shape and air density, and it rises steeply with speed, so a compact dense object falling a short way stays close to the vacuum answer while a light or bulky one departs from it almost immediately.
Every object in air has a terminal velocity, the speed at which drag exactly balances weight and the acceleration falls to zero. Below roughly a quarter of that speed the vacuum equations are a good approximation; as the object approaches it, they overstate the speed badly and the energy worse, because energy goes as the square of the error in speed. The calculator raises a note once the computed speed passes 20 m/s for exactly this reason.
The number people most often want from this page is the fall time, and it is the one least sensitive to drag over short drops, since the object spends the early part of the fall at low speed where drag is small. Impact energy is the most sensitive. If you are sizing a barrier or an arrest system, using the vacuum energy is conservative — it errs on the side of too much energy — which is usually the right way to be wrong.
Fall time and impact speed by drop height (vacuum, g = 9.80665 m/s²)
| Drop height (m) | Fall time (s) | Impact speed (m/s) | Impact speed (km/h) |
|---|---|---|---|
| 1 | 0.452 | 4.43 | 15.9 |
| 2 | 0.639 | 6.26 | 22.5 |
| 3 | 0.782 | 7.67 | 27.6 |
| 5 | 1.010 | 9.90 | 35.7 |
| 10 | 1.428 | 14.00 | 50.4 |
| 20 | 2.020 | 19.81 | 71.3 |
| 50 | 3.193 | 31.32 | 112.7 |
| 100 | 4.516 | 44.29 | 159.4 |
| 200 | 6.387 | 62.63 | 225.5 |
Read the pattern: quadrupling the height doubles the impact speed and doubles the fall time, because both depend on √h.
This is a physics tool, not a fall-safety design tool
Fall arrest, dropped-object protection and rigging clearance are governed by regulations and by manufacturer data for the specific equipment, none of which is a matter of kinematics alone. Real arrest calculations must account for lanyard stretch, deceleration distance, anchor location, swing fall and the free-fall distance permitted by the standard in force. Use this page to understand the physics and to sanity-check magnitudes; use the applicable standard and the equipment maker's instructions to design anything that a person depends on.
Assumptions and the mistakes they cause
- Air resistance is ignored entirely. The results are exact in a vacuum and optimistic in air. The lighter and bulkier the object, the sooner the departure becomes large.
- The object starts from rest. If it is thrown down or up, this page does not apply — use the final velocity calculator with a non-zero initial velocity, or the projectile motion calculator if there is a horizontal component too.
- Gravity is treated as uniform. Over any drop you can stage from, that is excellent; g falls by only about 0.03% per kilometre of altitude near the surface.
- Height is measured to the point of impact, not to the ground. For a falling person or a load, the relevant height is to whatever they strike first.
- Impact force is not the same as impact energy. Energy is fixed by mgh; the force depends on how far the object travels while stopping, which is a property of what it lands on, not of the fall.
- Entering mass changes nothing about the timing. If you expected a heavier object to fall faster, that is drag you are thinking of, and drag is not modelled here.
Related calculations and when to use them instead
If the object has any initial velocity, the free-fall equations no longer apply and you want the general constant-acceleration set in the final velocity calculator, entering gravity as the acceleration. If it also has a sideways velocity — anything thrown, launched or driven off an edge — the vertical motion is still free fall but the horizontal motion is not, and the projectile motion calculator handles both axes at once.
For the energy side of the problem, the gravitational potential energy calculator gives you mgh directly for a mass held at height, and the kinetic energy calculator converts a known impact speed back into joules or foot-pounds. The equality of those two figures at the ground is the clearest demonstration of energy conservation available in a first course. If you want the force of the impact rather than its energy, the linear momentum calculator turns the momentum change into an average force once you supply a stopping time.
Historically, this is the calculation that overturned Aristotle. The claim that heavy bodies fall faster in proportion to their weight survived for nearly two thousand years because in air it looks true. Galileo's insight was that the difference is caused by the medium, not by gravity, and that in the limit of no medium every object falls identically — a prediction confirmed spectacularly on the Moon in 1971, when Apollo 15 commander David Scott dropped a hammer and a falcon feather together and they landed at the same instant.