Free Fall Calculator

Background.

The free fall calculator computes the time of descent and impact velocity for an object released from rest or projected downward in a uniform gravitational field. It also reports the kinetic energy at impact if the object's mass is provided. The tool is grounded in the kinematic equations for constant acceleration, which Galileo Galilei first investigated systematically in the early seventeenth century. By rolling balls down inclined planes and measuring their positions at equal time intervals, Galileo demonstrated that the distance fallen is proportional to the square of the elapsed time, independently of the object's mass—a conclusion that overturned two millennia of Aristotelian physics.

The calculator requires three inputs: initial height, initial velocity, and local gravitational acceleration. For objects dropped from rest near Earth's surface, users can accept the default values of v₀ = 0 and g = 9.80665 m/s², the CODATA 2018 standard acceleration of gravity. The calculator then solves the quadratic equation h = v₀t + ½gt² for the positive root, giving the time to impact, and substitutes this time into v = v₀ + gt to obtain the impact velocity. These calculations assume vacuum conditions; air resistance is neglected, which means the results are exact for lunar or Martian drop tests and are excellent approximations for dense, compact objects falling moderate distances in Earth's atmosphere.

The educational applications of this calculator are extensive. High school physics students use it to verify laboratory measurements of falling bodies and to develop intuition about the parabolic relationship between height and fall time. University engineering students apply it to estimate the descent characteristics of payloads, the drop testing of consumer electronics, and the design of fall-arrest systems for construction and mountaineering. Safety engineers use the impact velocity output to assess the severity of falls from height, converting kinetic energy into equivalent g-forces to evaluate helmet and harness performance. The tool's ability to adjust g makes it applicable to extraterrestrial scenarios: a tool dropped on the Moon falls for 2.45 times as long as it would on Earth for the same height, because lunar gravity is 1.625 m/s².

Galileo's insight that all bodies fall at the same rate in a vacuum was confirmed with extraordinary precision by the Apollo 15 mission in 1971, when astronaut David Scott dropped a hammer and a feather simultaneously on the lunar surface. Both struck the ground together, verifying the equivalence principle that underlies general relativity. The calculator encodes this universality: the mass of the falling object does not appear in the equations for time or velocity, though it does enter the kinetic energy calculation. This is why a bowling ball and a marble dropped from a tower reach the ground simultaneously in vacuum, even though the bowling ball carries vastly more destructive energy at impact.

The assumption of constant g is valid for heights small compared to Earth's radius. At the top of the Burj Khalifa (828 m), the local gravitational acceleration is only 0.003% less than at sea level, producing a negligible error in fall time. For sub-orbital trajectories and satellite re-entry, however, g varies significantly with altitude and the simple kinematic equations no longer apply. The calculator includes a tolerance hint appropriate to classroom and engineering desk-check precision, and users working with atmospheric drag can treat the vacuum result as an upper bound on impact speed.

What is free fall calculator?

Free fall is the motion of an object under the influence of gravitational force alone, with no other forces such as air resistance or thrust acting upon it. In a uniform gravitational field, free fall produces constant acceleration directed toward the center of the gravitating body. Near Earth's surface, this acceleration is denoted by g and has the standard value 9.80665 m/s² as defined by CODATA.

The term "free fall" applies strictly to motion in vacuum, but it is commonly used more loosely to describe any unpowered descent where gravity is the dominant force. In physics education, free fall problems typically assume a point mass, a uniform gravitational field, and an inertial reference frame fixed to the Earth's surface. Under these assumptions, the motion is one-dimensional with constant acceleration, and the full apparatus of kinematic equations applies.

Free fall is a direct manifestation of the equivalence principle, which states that gravitational mass and inertial mass are identical. This equivalence means that the acceleration of a falling body does not depend on its composition, shape, or mass, provided that non-gravitational forces are absent. It is one of the most precisely tested principles in physics, with modern torsion-balance experiments confirming it to better than one part in 10¹⁵.

How to use this calculator.

1. 01Enter the initial height above the landing surface in meters.
2. 02Enter the initial downward velocity in meters per second, or leave as 0 for a dropped object.
3. 03Enter the local gravitational acceleration in m/s², or accept the default Earth value.
4. 04The calculator solves h = v₀t + ½gt² for the positive time root.
5. 05The calculator computes impact velocity using v = v₀ + gt.
6. 06If mass is provided, the calculator computes kinetic energy at impact.
7. 07Review outputs, noting that air resistance is neglected in the calculation.

The formula.

The free fall equations are special cases of the general kinematic equations for motion with constant acceleration. For one-dimensional vertical motion with acceleration g, the position as a function of time is y(t) = y₀ + v₀t + ½gt². Setting the origin at the release point and measuring downward as positive, the displacement h after time t is h = v₀t + ½gt². This is a quadratic equation in t; solving with the quadratic formula and selecting the positive root gives t = (−v₀ + √(v₀² + 2gh)) / g. For the common case of an object dropped from rest, v₀ = 0 and the expression simplifies to t = √(2h/g).

The velocity equation v = v₀ + gt follows from integrating the constant acceleration with respect to time. For a dropped object, this becomes v = gt, and substituting t = √(2h/g) yields the alternative form v = √(2gh). This result shows that impact velocity depends on the square root of height, not on height linearly. Doubling the fall height increases the impact speed by a factor of √2 ≈ 1.414, not 2. This square-root relationship explains why falls from modest heights can already produce dangerous impact speeds: a 2-meter fall yields 6.26 m/s (22.5 km/h), while a 20-meter fall yields 19.8 m/s (71.3 km/h).

The kinetic energy at impact, K = ½mv² = mgh, reveals the work-energy theorem in action: the gravitational potential energy lost during the fall, mgh, is converted entirely into kinetic energy (in the absence of air resistance). This energy equivalence is independent of the path taken and depends only on the vertical height difference, a consequence of gravity being a conservative force. The formula K = mgh is widely used in safety engineering to estimate the energy that fall-protection equipment must dissipate. A 70-kilogram worker falling 2 meters releases 1,373 joules, which a shock-absorbing lanyard must convert to harmless deformation and heat over a stopping distance of a few decimeters.

The constancy of g used in these equations is an approximation valid when the fall height is much smaller than the radius of the gravitating body. The exact Newtonian expression for gravitational acceleration at distance r from the center is g(r) = GM/r², which decreases with altitude. For a fall from 10 kilometers, the variation in g is 0.3% and is usually neglected. For sub-orbital hops and planetary re-entry, the variable-g correction becomes essential, and numerical integration replaces the closed-form kinematic solution.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Galilei, G. (1638). Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Leiden: Elsevier. Day 3: Uniformly Accelerated Motion.
2. [2]Halliday, D., Resnick, R., and Walker, J. (2021). Fundamentals of Physics, 11th ed. Hoboken, NJ: Wiley. Chapter 2: Motion Along a Straight Line.
3. [3]NIST (2024). CODATA Recommended Values of the Fundamental Physical Constants: 2018.
4. [4]BIPM (2019). The International System of Units (SI Brochure), 9th ed.
5. [5]Taylor, J.R. (2005). Classical Mechanics. Sausalito, CA: University Science Books. Chapter 1: Newton's Laws.
6. [6]Will, C.M. (2014). The Confrontation between General Relativity and Experiment. Living Reviews in Relativity 17:4. doi:10.12942/lrr-2014-4