What is the formula for terminal velocity?

Terminal velocity occurs when aerodynamic drag equals gravitational weight (Fd = m·g): v_term = √(2·m·g / [ρ·Cd·A]), where m is mass, g is gravity, ρ is air density, Cd is drag coefficient, and A is frontal cross-sectional area.

What is the velocity equation as a function of time during free fall with air resistance?

Dropping from rest under quadratic air resistance, downward velocity is: v(t) = v_term · tanh(g·t / v_term), asymptotically approaching terminal velocity.

What is the fall distance equation as a function of time with air resistance?

The vertical distance fallen after time t is: y(t) = (v_term² / g) · ln(cosh(g·t / v_term)).

What is the velocity of a falling object as a function of drop height with drag?

Downward velocity as a function of distance fallen y is: v(y) = v_term · √(1 - e^(-2·g·y / v_term²)).

How does a human skydiver's terminal velocity vary between belly-to-earth and head-down positions?

In a standard belly-to-earth spread position (A ≈ 0.7 m², Cd ≈ 1.0), terminal velocity is approximately 54 m/s (120 mph • 194 km/h), while in a streamlined head-down dive (A ≈ 0.18 m², Cd ≈ 0.7), terminal velocity increases to roughly 90 m/s (200 mph • 324 km/h).

How does air density (altitude) affect terminal velocity and free fall speed?

Because atmospheric air density ρ decreases exponentially with altitude, terminal velocity is significantly higher at high altitudes (e.g., Felix Baumgartner exceeded supersonic speed in the stratosphere where air density was negligible).

How does free fall with air resistance differ from vacuum free fall?

In a vacuum, acceleration is constant (a = g) and velocity increases indefinitely (v = g·t). With air resistance, acceleration decreases continuously (a(t) = g·[1 - (v/v_term)²]) until reaching zero at terminal velocity.

Why do heavy objects fall faster than light objects of the same shape in the atmosphere?

Terminal velocity scales with √(m / A). A heavier object has greater gravitational downward force relative to its drag cross-section, requiring a higher speed before drag balances weight.

How do ballistics engineers, parachute designers, and meteorologists use this calculator?

Parachute engineers size canopy surface area A to achieve safe landing touchdown speeds (v_term ≈ 5 m/s), ballistics analysts calculate bullet drop penalties, and meteorologists model raindrop and hail terminal velocities.

⚡ 100% Free • Terminal Velocity & Quadratic Drag Solver

Free Fall with Air Resistance Calculator

Calculate fall time, impact velocity, and distance with aerodynamic quadratic drag, terminal velocity (v_term = √(2mg/ρCdA)), and air resistance with the free Free Fall with Air Resistance Calculator.

Aerodynamic Drag Presets:

Object Mass (m) 80.0 kg (176.4 lbs)

Drop Elevation / Height (h) 1000.0 m (3280.8 ft)

Cross-Section Area ($A$ in $\text{m}^2$)

Drag Coeff ($C_d$)

Air Density ($\rho$ $\text{kg/m}^3$)

Gravity Field

Terminal Velocity ($v_{\text{term}}$) 53.6 m/s (120 mph)

Actual Fall Duration 21.8 s

Terminal Velocity ($v_{\text{term}}$)

53.6 m/s

120.0 mph • 193.1 km/h • Impact Speed: 53.4 m/s (99.6% of v_term)

Impact Speed with Drag

53.4 m/s

119.5 mph

Fall Time with Drag

21.8 s

Actual flight duration

Vacuum Fall Time

14.3 s

+7.5s air drag penalty

Vacuum Impact Speed

140.1 m/s

313.3 mph in vacuum

Peak Drag Force ($F_d$)

780.2 N

99.4% of object weight

Impact Kinetic Energy

114.1 kJ

Mechanical energy

Step-by-Step Terminal Velocity & Quadratic Drag Kinematics Derivation

What is Free Fall with Air Resistance in Aerodynamics and Physics?

In atmospheric physics and fluid mechanics, free fall with air resistance describes the downward trajectory of an object subjected to both gravitational attraction ($F_g = m g$) and opposing aerodynamic fluid drag ($F_d$). In high-speed subsonic regimes, aerodynamic resistance scales with the square of velocity (quadratic drag):

$$F_d = \frac{1}{2} \rho C_d A v^2$$

As the falling body accelerates, drag force grows until it exactly equals gravitational weight ($F_d = m g$). At this equilibrium, net acceleration drops to zero ($a = 0$), and the body coasts downward at a constant speed known as Terminal Velocity ($v_{\text{term}}$).

Core Mathematical Equations for Quadratic Air Drag Free Fall

Kinematic Metric	Formula	Description
Terminal Velocity ($v_{\text{term}}$)	$$v_{\text{term}} = \sqrt{\frac{2 m g}{\rho C_d A}}$$	Maximum asymptotic speed in $\text{m/s}$
Velocity vs. Time $v(t)$	$$v(t) = v_{\text{term}} \tanh\left(\frac{g t}{v_{\text{term}}}\right)$$	Instantaneous speed after $t$ seconds
Distance Fallen $y(t)$	$$y(t) = \frac{v_{\text{term}}^2}{g} \ln\left(\cosh\left(\frac{g t}{v_{\text{term}}}\right)\right)$$	Total vertical distance fallen after $t$
Velocity vs. Distance $v(y)$	$$v(y) = v_{\text{term}} \sqrt{1 - e^{-2 g y / v_{\text{term}}^2}}$$	Impact speed from drop height $y$
Fall Time from Height $h$	$$t(h) = \frac{v_{\text{term}}}{g} \operatorname{arcosh}\left(e^{g h / v_{\text{term}}^2}\right)$$	Elapsed flight time to reach ground

Human Skydiving: Belly-to-Earth vs. Head-Down Streamlined Dive

How skydivers control terminal velocity by adjusting body geometry:

Belly-to-Earth Arch ($A \approx 0.70\text{ m}^2, C_d \approx 1.0$)

Maximizes surface drag, giving a stable terminal velocity of $\approx 54\text{ m/s}$ ($120\text{ mph} \approx 194\text{ km/h}$). Reaches $95\%$ of terminal velocity in $\approx 8\text{ seconds}$ over a $300\text{-meter}$ drop.

Head-Down Dive ($A \approx 0.18\text{ m}^2, C_d \approx 0.70$)

Minimizes cross-sectional area, surging terminal velocity up to $\approx 90\text{ m/s}$ ($200\text{ mph} \approx 324\text{ km/h}$).

Problems Solved by the Free Fall with Air Resistance Calculator

1. Parachute Canopy Sizing

Determines the exact cross-sectional area ($A$) required to slow payload descents to safe touchdown speeds ($v_{\text{term}} \le 5\text{ m/s}$).

2. Realistic Impact Kinetics

Replaces inaccurate vacuum approximations ($v = \sqrt{2gh}$) with true drag-limited impact velocities and kinetic energies.

3. UAV & Rocket Recovery

Models emergency drone parachute recovery descent profiles and spent rocket booster atmospheric re-entry deceleration curves.

Key Features of the Free Fall with Air Resistance Calculator

Analytical Hyperbolic Solvers

Computes instantaneous speeds via $v(t) = v_{\text{term}}\tanh(gt/v_{\text{term}})$ with zero numerical rounding errors.

Vacuum Comparison Delta

Displays side-by-side time and velocity differences between atmospheric fall and ideal vacuum free fall.

Altitude & Planetary Gravity

Adjusts fluid density ($\rho$) and gravity ($g$) across Earth sea level, high altitude, Moon, Mars, and Jupiter.

Stratospheric Supersonic Free Fall: The Physics of Extreme High-Altitude Jumps

Why stratospheric skydivers break the sound barrier before slowing down:

When Felix Baumgartner stepped out of his capsule at $38,969\text{ meters}$ ($127,852\text{ ft}$), atmospheric density was less than $1.5\%$ of sea level ($\rho \approx 0.018\text{ kg/m}^3$). With negligible air resistance, his terminal velocity soared to over $1,350\text{ km/h}$, allowing him to reach a peak velocity of $1,357.6\text{ km/h}$ ($\text{Mach } 1.25$), breaking the speed of sound in free fall. As he descended into the denser troposphere, increasing air density progressively lowered his terminal velocity to $\approx 200\text{ km/h}$ long before parachute deployment.

Frequently Asked Questions

Comprehensive answers to common questions about free fall with air resistance, terminal velocity formulas, quadratic aerodynamic drag, and vacuum comparisons.