Free Fall with Air Resistance Calculator
Enter the mass, drop altitude, drag coefficient, cross-sectional area, and air density of your object to get the terminal velocity, total fall time, maximum velocity reached, and the drag force at any speed. The calculator uses the standard quadratic drag model used in introductory physics and aerospace engineering. Results update instantly as you type.
How air resistance changes a free fall
In a perfect vacuum, every object accelerates at exactly g regardless of its mass or shape. In real air, a drag force proportional to the square of the object's speed acts upward, opposing the fall. As the object speeds up, this drag force grows until it exactly balances gravity. At that point, acceleration drops to zero and the object continues at a constant speed called terminal velocity. The formula for that speed is v_t = sqrt(m * g / k), where m is mass, g is gravitational acceleration, and k is the air resistance coefficient that bundles together the object's drag coefficient, frontal area, and the air density. A skydiver in belly-down position has a terminal velocity around 55 m/s (120 mph); in a streamlined head-down dive it is roughly 80-90 m/s (180-200 mph).
The air resistance coefficient k
The air resistance coefficient k combines three properties into one convenient constant: k = (rho * A * Cd) / 2. Here rho is the air density (1.225 kg/m^3 at sea level and 15 degrees C), A is the cross-sectional area of the object perpendicular to the direction of motion, and Cd is the dimensionless drag coefficient that captures the shape and surface texture of the object. A higher k means more drag and a lower terminal velocity. Doubling the frontal area or doubling the air density each doubles k, which cuts terminal velocity by a factor of about 1.41 (the square root of 2). A parachute is effective because it multiplies both A and Cd simultaneously.
Fall time and velocity over time
The velocity at any moment during the fall follows the formula v(t) = v_t * tanh(t * g / v_t), where tanh is the hyperbolic tangent function. This curve starts at zero, rises steeply at first when drag is small, then flattens asymptotically toward terminal velocity. The total time to fall a height h is t = (v_t / g) * acosh(exp(h * k / m)), where acosh is the inverse hyperbolic cosine. For short drops the object may impact long before reaching terminal velocity; for tall drops (hundreds of metres) it arrives at or very near terminal speed. The chart on this page shows how velocity builds second by second for your specific inputs.
Drag force and its practical meaning
At any given speed v, the drag force is F_drag = k * v^2. This is the upward force the air exerts on the falling object at that instant. Below terminal velocity the drag force is less than the weight (m * g), so the object still accelerates. At terminal velocity the two forces are equal and opposite. Above terminal velocity (if launched downward) the drag force exceeds gravity and the object decelerates. Engineers use this relationship to design parachutes, aerodynamic fairings on rockets, and safety equipment - knowing the drag force at a given speed lets them size structural members, predict heat generation, and estimate safe deployment speeds.
Typical drag coefficients for common objects
| Object | Cd | Notes |
|---|---|---|
| Skydiver (belly-down) | 1.0 | Standard freefall position |
| Skydiver (head-down) | 0.7 | Faster freefall position |
| Deployed parachute (round) | 1.5-1.75 | Typical sport canopy |
| Sphere | 0.47 | Smooth ball, turbulent-regime |
| Cylinder (long axis horizontal) | 0.82 | Broadside to flow |
| Cube | 1.05 | Flat face to flow |
| Streamlined airfoil | 0.04 | Minimal drag shape |
| Human (standing) | 1.0-1.3 | Highly variable |
| Car (sedan) | 0.25-0.35 | Modern aerodynamic design |
Cd values are approximate averages. Actual values depend on surface roughness, Reynolds number, and flow regime.
Frequently asked questions
What is terminal velocity?
Terminal velocity is the constant speed a falling object reaches when the upward drag force equals its weight. Once these forces balance, net acceleration is zero, so the object no longer speeds up. The value depends on the object's mass, size, shape, and the density of the fluid it is moving through. A skydiver belly-down reaches about 55 m/s (120 mph); a feather may reach only a fraction of a metre per second.
Does a heavier object fall faster with air resistance?
Yes, all else being equal. Terminal velocity is v_t = sqrt(m * g / k), so a heavier object (larger m) has a higher terminal velocity. This is why a cannonball falls much faster than a feather, even though they would fall identically in a vacuum. Mass appears only in the numerator of the fraction under the square root, so doubling mass increases terminal velocity by a factor of about 1.41.
What drag coefficient should I use?
It depends on the shape of your object. A sphere is about 0.47, a flat plate held perpendicular to the flow is about 1.0-1.2, a skydiver belly-down is roughly 1.0, and a streamlined aerodynamic shape can be as low as 0.04. The reference table on this page gives common values. If you do not know the drag coefficient experimentally, use a value from a similar shape in the table as a starting point.
How does altitude affect air resistance?
Air becomes less dense at higher altitudes, which reduces the air resistance coefficient k and therefore increases terminal velocity. A skydiver jumping from 4000 m experiences thinner air and accelerates to a somewhat higher speed before deploying a parachute than one jumping from 1000 m. This calculator assumes a constant air density throughout the fall; for very long falls (many kilometres) you would need to model the decreasing density as a function of altitude.
What is the difference between this and a free fall in vacuum calculator?
In a vacuum, the only force is gravity, so velocity grows linearly as v = g * t and the fall time is simply t = sqrt(2 * h / g). With air resistance, drag opposes motion, velocity follows a tanh curve instead of a straight line, a terminal velocity exists, and the fall takes longer than it would in vacuum. This calculator uses the full drag model; for vacuum conditions you can set the drag coefficient or cross-sectional area to an extremely small value.
Can I use this calculator for objects falling in water or other fluids?
Yes, to a good approximation. The physics is identical: change the fluid density to the density of the liquid (about 1000 kg/m^3 for water) and choose an appropriate drag coefficient for submerged motion. For high-speed motion in water, cavitation and other effects can make the simple drag model less accurate, but for moderate speeds the formula works well.