Drag Equation - Force, Terminal Velocity, Regime

The drag equation is the standard physics model for the resistive force a moving body experiences as it pushes through a fluid such as air or water, and this calculator turns velocity, density, drag coefficient, and area into a single drag force plus a terminal velocity for falling objects.

• • Skydivers and parachutists: Estimate the steady falling speed for a person and rig so you can plan a jump profile or rate of descent.
• • Cyclists and vehicle designers: Quantify how much horsepower is needed to overcome air resistance at a given speed.
• • Physics and engineering coursework: Cross-check homework and lab results against the canonical formula without re-deriving it each time.
• • What is the drag equation, in one line?: If you only remember one line, the drag equation is F_d = 0.5 * rho * v^2 * C_d * A: fluid density times velocity squared, halved, times a shape factor, times area.

The drag equation is taught in introductory physics and revisited in fluid-dynamics, aerospace, and mechanical-engineering courses because it captures the dominant resistive force for almost any object moving slower than the speed of sound, and that is the same reason most introductory problems simplify the flow to a single drag coefficient times the dynamic pressure.

By keeping the four variables visible, the calculator doubles as a quick sensitivity check: doubling speed quadruples drag, while doubling area only doubles it.

To see how drag interacts with the other forces on a body, Forces & Newton's Laws Calculator is a useful next stop.

The formula starts with the dynamic pressure of the moving fluid and scales that pressure by a dimensionless shape factor and the area presented to the flow, which is why drag depends so strongly on speed, area, and body shape.

• F_d: Aerodynamic drag force, measured in newtons (N).
• rho: Fluid density in kilograms per cubic meter; air at sea level is 1.225 kg/m^3 and water is 1000 kg/m^3.
• v: Speed of the body relative to the undisturbed fluid in meters per second.
• C_d: Drag coefficient, a dimensionless number that captures shape and surface roughness.
• A: Reference cross-sectional area in square meters, taken perpendicular to the flow.

The factor of 0.5 comes from the kinetic energy per unit volume of the moving fluid, which is the dynamic pressure q. The drag force is then q times the drag coefficient times the area, which is the form NASA uses in its beginner aerodynamics guide.

According to Wikipedia Drag (physics), this expression is F_D = 0.5 * rho * v^2 * C_D * A, where rho is the density of the fluid, v is the speed of the object relative to the fluid, C_D is the dimensionless drag coefficient, and A is the cross-sectional area.

To extend the analysis once you know the drag force, Projectile Motion Calculator lets you follow the same body along a curved trajectory.

Four ideas come up every time this equation appears: dynamic pressure, the drag coefficient itself, the velocity-squared dependence, and the link to terminal velocity for falling objects.

When drag is small enough to ignore, Free Fall Time Calculator gives the simpler fall-time result for vacuum motion.

Enter the four drag-equation variables, then the mass and gravity you want to use for the terminal-velocity branch, and read the drag force and terminal velocity straight from the results panel.

1. 1 Enter the velocity: Type the speed of the body relative to the fluid, in meters per second. Use 0 to see the no-motion baseline.
2. 2 Enter the fluid density: Type the fluid density in kilograms per cubic meter. The default 1.225 kg/m^3 is sea-level air.
3. 3 Enter the drag coefficient and area: Type the body's C_d (about 0.47 for a sphere, 0.7 for a cyclist, 0.3 for a car) and its frontal area in square meters.
4. 4 Enter the mass and gravity: Type the body mass in kilograms and the local gravity in meters per second squared; the default 9.80665 m/s^2 is standard Earth gravity.
5. 5 Read drag force and terminal velocity: The results panel shows the drag force in newtons, the terminal velocity in meters per second, the dynamic pressure, and a Reynolds-regime hint.

Try velocity 55 m/s, C_d 1.0, area 0.7 m^2, mass 80 kg on standard Earth gravity to see a skydiver-class terminal velocity in the 40-50 m/s range.

If you need to track how the velocity itself changes with time, Kinematics Motion Calculator is the right next step.

The formula is simple to write down but easy to mistype by a factor of two; the calculator removes the arithmetic and surfaces the most useful derived numbers in one place.

• • Saves rework: It eliminates the need to rearrange F_d = 0.5 rho v^2 C_d A every time you need a drag force or terminal velocity, which matters most when C_d has to be solved for indirectly.
• • Two outputs in one tool: You get both the drag force at the current speed and the terminal velocity for a falling object, so the same inputs cover two of the most common physics homework and design questions.
• • Reasonable defaults: Sea-level air density and standard Earth gravity are preloaded, so a first estimate is one form load away without reaching for a reference table.
• • Built-in regime check: The Reynolds-regime hint tells you when the velocity-squared drag law is the right model, and flags low-Reynolds cases where the assumption weakens.
• • Editable inputs: Every variable is editable, so the same form handles a skydiver, a cyclist, a raindrop, or a custom lab shape with the same workflow.

Drag force and terminal velocity are sensitive to four real-world factors, and two of the largest uncertainties are worth flagging before you trust a single number.

• • The formula assumes a steady flow, so it is most accurate for moderate Reynolds numbers well above the laminar regime and well below the transonic range.
• • It treats the drag coefficient as a single number, so it cannot capture strong compressibility effects near the speed of sound or strong viscosity changes at very low Reynolds numbers.

According to Engineering Toolbox, a smooth sphere has a drag coefficient near 0.47 in the turbulent Reynolds-number range and streamlined bodies sit near 0.04.

According to Wikipedia Terminal velocity, terminal velocity is v_t = sqrt(2 m g / (rho A C_d)) and a belly-to-earth skydiver reaches a terminal speed of about 55 m/s.

For a worked example of how time-of-flight depends on launch conditions, Time of Flight Projectile Motion Calculator is a good companion read.