Use this calculator to find terminal velocity in air or water, or solve for mass, area, or drag coefficient from a known speed.
Advanced options
How to use our Terminal Velocity Calculator
- Choose an option in Solve for. Leave it on terminal velocity if you want the simplest setup.
- Enter the visible core inputs: Mass of object (kg), Cross-sectional area (m^2), Drag coefficient (unitless), Fluid, Fluid density rho (kg/m^3), and Gravitational acceleration g (m/s^2). If you picked a solve-for mode other than terminal velocity, also enter Known terminal velocity v_t (m/s).
- If you want a standard skydiving-style estimate, keep Fluid as Air, use sea-level Fluid density rho (kg/m^3) of 1.225, and leave buoyancy off.
- Open Advanced options only if needed. Turn on Include buoyancy? and enter Displaced volume (m^3) if the object pushes aside enough fluid for buoyancy to matter.
- For high-altitude air, turn on Use altitude-based air density? and enter Altitude above sea level (m). The calculator will then use the computed density instead of the rho input.
- Pick Number display if you want plain numbers, scientific notation, or automatic formatting for very large or very small results.
- Click Calculate and read the main result plus the support outputs like Fluid density used, Dynamic pressure at terminal speed q, and Net downward force used.
- Sanity-check the result: lower Fluid density rho (kg/m^3), smaller Cross-sectional area (m^2), or smaller Drag coefficient (unitless) should usually make terminal velocity go up, while stronger buoyancy should make it go down.
Definitions
Terminal velocity v_t: The steady speed where upward drag balances the net downward force, so the speed stops increasing [1][3].
Drag coefficient (unitless): A shape-dependent number that tells how strongly an object resists motion through a fluid in the drag equation.
Cross-sectional area (m^2): The effective front-facing area exposed to the flow. Bigger area usually means more drag.
Fluid density rho (kg/m^3): How much mass is packed into the fluid. Denser fluid usually creates more drag at the same speed.
Net downward force used: The force balanced by drag at terminal speed. With buoyancy off it is just weight, and with buoyancy on it is weight minus buoyant force.
Dynamic pressure q: A speed-related pressure defined here as half of fluid density times speed squared. Higher q means stronger airflow loading.
Common mistakes and quick fixes
Mistake: Entering 70 in Cross-sectional area (m^2) when you meant 0.70.
Fix: Check the unit in the label. Area must be in square meters, so human-sized values are usually well below 2 m^2.
Mistake: Leaving Known terminal velocity v_t (m/s) blank when Solve for is set to mass, area, or drag coefficient.
Fix: Enter the speed in m/s, or switch Solve for back to terminal velocity.
Mistake: Turning on Use altitude-based air density? and still expecting the typed Fluid density rho (kg/m^3) value to be used.
Fix: Read Fluid density used . That output shows the actual density used in the math.
Mistake: Using a zero or negative value for Drag coefficient (unitless) , Cross-sectional area (m^2) , Fluid density rho (kg/m^3) , or Gravitational acceleration g (m/s^2) .
Fix: Each of those inputs must be greater than 0 for this model. Replace the invalid entry with a positive value.
Mistake: Turning on Include buoyancy? but forgetting Displaced volume (m^3) or entering a volume so large that the object would float.
Fix: Check Net downward force used . If it is negative, the object does not fall downward in this simple model.
Mistake: Reading Solved value (selected mode) without checking whether the answer is physically reasonable.
Fix: Also review Notes and warnings and compare the result to your shape, fluid, and speed assumptions.
Limitations & Key Assumptions / Boundary Conditions
This calculator uses the quadratic-drag model, which is most useful when drag is roughly proportional to speed squared. That is a common approximation for faster motion through air or water, but it is not the best model for every size, speed, or fluid condition.
The calculator treats Drag coefficient (unitless) as constant. In real life, drag coefficient can change with speed, body position, surface roughness, and flow regime.
The optional altitude model is a simplified troposphere-style atmosphere calculation. It assumes the inputs keep temperature positive and that the standard lapse-rate approach is appropriate for the altitude entered.
Buoyancy is included only through displaced-fluid weight. The model does not include lift, spin, tumbling, changing posture, gusts, or side forces.
If Net downward force used is zero or negative, the calculator reports that downward terminal speed is not defined in this simple falling model. That means the object is neutrally buoyant or would rise instead of fall.
Results can differ from field measurements because real objects do not always keep the same shape, area, or orientation while moving.
Methodology
Model used
This calculator balances downward force against quadratic drag. The drag equation used is the standard form for force proportional to speed squared [1][2]. Terminal speed is reached when drag matches the net downward force [1][3].
F_d = (1/2) * rho * C_d * A * v^2
W_net = m*g - rho*V*g
v_t = sqrt((2 * W_net) / (rho * C_d * A))
Here, rho is fluid density, C_d is drag coefficient, A is cross-sectional area, m is mass, g is gravitational acceleration, and V is displaced volume. If buoyancy is off, the calculator uses V = 0, so W_net = m*g.
Solve-for modes
When terminal velocity is known, the same force balance can be rearranged to solve for another unknown.
m = ((1/2) * rho * C_d * A * v_t^2) / g + rho*V
A = (2 * W_net) / (rho * C_d * v_t^2)
C_d = (2 * W_net) / (rho * A * v_t^2)
If the selected denominator would be zero, such as v_t = 0 when solving for area or drag coefficient, the calculator returns N/A instead of forcing an impossible result.
Extra outputs
The calculator also reports dynamic pressure and the drag force at terminal speed.
q = (1/2) * rho * v_t^2
F_d at v_t = (1/2) * rho * C_d * A * v_t^2
At terminal speed, that drag force equals the net downward force used by the model.
Altitude-based air density
If you enable altitude-based density for air, the calculator uses a simplified ISA-like troposphere relation to estimate pressure, temperature, and density before doing the drag calculation.
T = T0 - L*h
p = p0 * (T/T0)^(g0/(R*L))
rho = p / (R*T)
The defaults are T0 = 288.15 K, p0 = 101325 Pa, L = 0.0065 K/m, R = 287.05 J/(kg*K), and g0 = 9.80665 m/s^2. If the computed temperature is not positive, the input is rejected.
Mini example
Using m = 80 kg, A = 0.70 m^2, C_d = 1.0, rho = 1.225 kg/m^3, and g = 9.80665 m/s^2 with buoyancy off:
W_net = 80 * 9.80665 = 784.532 N
v_t = sqrt((2 * 784.532) / (1.225 * 1.0 * 0.70)) = 42.747 m/s
q = (1/2) * 1.225 * 42.747^2 = about 1119 Pa
That speed is about 95.6 mph or 153.9 km/h.
Assumptions behind the math
The result assumes constant mass, constant area, constant drag coefficient, and straight motion through a fluid with one density value. Real results can differ if the object changes orientation, compresses air strongly, or moves in conditions where drag is not well described by the quadratic model.