Acceleration due to gravity by free fall theory

Free fall is defined as the motion of an object subjected solely to the force of gravitational attraction, with all other external forces, such as air resistance, buoyant lift, and mechanical constraints, either absent or demonstrably negligible. In the vicinity of the Earth's surface, the gravitational force acting on a body of mass m is given by Newton's Universal Law of Gravitation as F_g = G M_E m / R_E², where G is the gravitational constant, M_E is the mass of the Earth, and R_E is the Earth's mean radius. Equating this force to Newton's Second Law of Motion, F = m a, reveals that the mass m cancels identically from both sides of the equation: m a = m (G M_E / R_E²), yielding a = g = G M_E / R_E². Consequently, every freely falling body experiences a constant downward acceleration g, independent of its mass, density, or internal structural composition. Because the acceleration is uniform (d²h/dt² = g), integrating the equation of motion once with respect to time yields the instantaneous velocity v(t) = v₀ + g t, where v₀ is the initial velocity at t = 0. Integrating a second time produces the displacement relation h(t) = h₀ + v₀ t + 1/2 g t². When the body is released from rest (v₀ = 0) at origin h₀ = 0, the distance fallen simplifies to h = 1/2 g t². Eliminating time between the velocity and displacement equations yields the Torricellian kinematic relationship v² = 2 g h. These fundamental equations form the bedrock of classical terrestrial kinematics.

The universality of free fall is one of the deepest and most profound principles in physics, known formally as the Weak Equivalence Principle (WEP). Newton originally formulated his laws of mechanics using two conceptually distinct definitions of mass: inertial mass m_i, which characterizes an object's resistance to any change in its state of motion (F = m_i a), and gravitational mass m_g, which determines the magnitude of the gravitational attraction it exerts and experiences in a gravitational field (F = m_g g_field). The equation governing the acceleration of a falling body is therefore a = (m_g / m_i) g_field. If the ratio m_g / m_i were dependent on the material composition, nuclear binding energy, or atomic structure of the object, different substances would fall at different rates. Galileo's early experiments with inclined planes and pendulums demonstrated that m_g / m_i is universal to within one part in a hundred. Newton improved this limit to one part in a thousand using hollow pendulum bobs filled with different materials. In the late nineteenth and early twentieth centuries, Loránd Eötvös utilized a high-precision torsion balance to verify equivalence to one part in 10⁹, and modern satellite experiments such as MICROSCOPE (2017) have validated the Weak Equivalence Principle to within a few parts in 10¹⁵. Einstein recognized that this exact proportionality between gravitational and inertial mass implies that gravitation is not an arbitrary mechanical force, but rather a geometric curvature of four-dimensional spacetime, laying the philosophical foundation for General Relativity.

In experimental physics, verifying a nonlinear theoretical relationship by directly fitting a curve to raw data points is often fraught with subjective bias and numerical ambiguity. The quadratic equation h = 1/2 g t² describes a parabola passing through the origin. While one could compute individual estimates of gravity from each trial using g_i = 2 h_i / t_i², this approach suffers from significant statistical drawbacks: it weights each data point arbitrarily, makes it difficult to detect systematic zero-offset errors, and conceals subtle nonlinear departures from the model. By transforming the independent time coordinate into its square, X = t², the physical relationship takes the standard linear slope-intercept form Y = m X + b, where Y = h, X = t², slope m = 1/2 g, and intercept b = 0. When an ordinary least-squares (OLS) linear regression is performed on the paired dataset {(t_i², h_i)}, the resulting best-fit slope minimizes the sum of squared vertical residuals: m = [N ∑(t_i² h_i) - ∑(t_i²) ∑(h_i)] / [N ∑(t_i⁴) - (∑ t_i²)²]. The acceleration due to gravity is directly obtained from the fitted slope as g = 2 · m. Furthermore, the fitted intercept b provides an indispensable diagnostic tool: if b is statistically indistinguishable from zero within its standard error, the experiment confirms that the bob was released with zero initial velocity and that the height scale has no uncalibrated zero offset. If b is significantly positive or negative, it immediately alerts the experimenter to an offset in height indexing or an uncorrected timing latency.

The measurement of sub-second time intervals requires specialized instruments capable of overcoming contact bounce and signal propagation delays. Historically, the Behr Free Fall Apparatus and Cenco spark timers solved this challenge using synchronized electrical sparks. In a spark-timing apparatus, a high-voltage induction coil discharge pulses at a known, highly stable frequency (typically 50 Hz or 60 Hz, regulated by an electrically driven tuning fork or synchronous AC line frequency). A cylindrical plummet falls vertically between two parallel conductors—one carrying high-voltage pulses and the other grounded—separated by a strip of dielectric paper treated with heat-sensitive or spark-sensitized chemical coating. At each spark pulse, an electric spark jumps across the narrow gap between the wire, through the plummet's metal skirt, perforating the paper strip and leaving a sharp, microscopic pinhole at the bob's instantaneous position. Because the time interval between successive sparks Δt is strictly constant (e.g., 20.0 ms for 50 Hz), the distance between successive pinholes provides an indelible, tamper-proof record of the motion. In modern laboratories, optical photogates replace spark gaps: an infrared light beam is focused across the fall axis onto a photodiode. When the leading edge of the falling plummet interrupts the light beam, the photodiode circuit switches states in less than a microsecond, halting a digital high-frequency quartz crystal counter. Both techniques decouple the timing measurement from human physiological reaction time, achieving timing uncertainties below 0.1 ms.

When an electromagnet is used to hold and drop a ferromagnetic projectile, Faraday's Law of Electromagnetic Induction dictates that the magnetic flux cannot collapse instantaneously when the supply switch is opened. The self-inductance L of the solenoid coil resists the decay of electric current: di/dt = -V_emf / L. In an unquenched circuit, an inductive voltage spike is generated, maintaining an electric arc across the switch contacts and sustaining a decaying magnetic field for several milliseconds after mechanical contact separation. Furthermore, the ferromagnetic core material exhibits magnetic hysteresis and residual remanence (coercivity), retaining a weak magnetic attraction even after the current has fallen to zero. As a result, the plummet experiences an unrecorded mechanical release delay τ_delay, typically between 2 ms and 15 ms, during which the timer may already have started while the bob remains momentarily suspended or dragging against the pole piece. If the timer is started by the switch interruption rather than direct optical detachment, all recorded fall times will be artificially prolonged by τ_delay: t_obs = t_true + τ_delay. Squaring this observed time yields t_obs² ≈ t_true² + 2 t_true τ_delay, which introduces a mild curvature into the h versus t² graph. To eliminate this systematic error, high-precision apparatuses utilize a fast-demagnetization transient suppressor (such as a zener-diode clamp or reverse-current pulse) and trigger the timer only when the bob physically breaks a low-mass mechanical contact foil or clears an optical photogate at the instant of release.

When an object descends through a real fluid such as air, it experiences an opposing aerodynamic drag force F_drag arising from the collision with air molecules and skin friction shear along its surface. For macroscopic bodies moving at typical laboratory velocities (1 m/s to 5 m/s), the flow is characterized by intermediate Reynolds numbers (Re ~ 10³ to 10⁴), where inertial form drag dominates over viscous Stokes drag. The aerodynamic drag force is modeled by Rayleigh's drag equation: F_drag = 1/2 C_d ρ_air A v², where C_d is the dimensionless drag coefficient (approximately 0.8 for a flat-ended cylinder and 0.4 for a streamlined cone), ρ_air is the density of ambient air (~1.20 kg/m³ at sea level and 20 °C), A is the frontal projected cross-sectional area, and v is the instantaneous fall velocity. Applying Newton's Second Law to the falling body yields m (dv/dt) = m g - 1/2 C_d ρ_air A v². As the velocity increases, the drag force grows quadratically until it exactly balances the gravitational weight m g, at which point the acceleration drops to zero and the object approaches its terminal velocity: v_t = sqrt[2 m g / (C_d ρ_air A)]. For a dense steel cylinder of diameter 25 mm and mass 150 g, the terminal velocity exceeds 60 m/s. Over a 1.0 m drop in the laboratory, the maximum impact velocity is only v = sqrt(2 · 9.81 · 1.0) ≈ 4.43 m/s, which is less than 8% of terminal velocity. Numerical integration of the drag equation shows that air resistance alters the fall time over a 1 m drop by less than 0.05%, well within typical random experimental error.

The standard reference value for gravitational acceleration, defined by the General Conference on Weights and Measures (CGPM) as g₀ = 9.80665 m s⁻², represents a conventional agreed standard rather than an invariant physical constant across the Earth's surface. In reality, local gravitational acceleration varies with geographic latitude φ and elevation h_elev above sea level due to two primary geophysical phenomena: the Earth's oblateness (equatorial bulge caused by planetary rotation) and the centrifugal acceleration of the rotating geoid. The International Gravity Formula (Somigliana equation, 1980 Geodetic Reference System) models this latitude dependence at sea level as g(φ) = 9.780327 · [1 + 0.0053024 sin²(φ) - 0.0000058 sin²(2φ)] m s⁻². At the equator (φ = 0°), sea-level gravity is approximately 9.7803 m s⁻², whereas at the geographic poles (φ = 90°), it reaches 9.8322 m s⁻²—a variation of roughly 0.53%. Furthermore, increasing elevation above sea level decreases the gravitational field according to the free-air correction rate: Δg_FA / Δh ≈ -3.086 × 10⁻⁶ m s⁻² per meter of altitude. In a high-altitude laboratory (e.g., Denver at 1600 m elevation), local gravity is reduced by approximately 0.005 m s⁻² relative to sea level. Precise gravimetry therefore requires comparing experimental results against the local theoretical gravity computed for the exact latitude and altitude of the testing laboratory.