Drag Force Formula: F = ½ρv²C_D A with Worked Examples

The drag force formula is $F_D = \frac{1}{2}\rho v^2 C_D A$ : the resistance of air or water on a moving object depends on the fluid density ρ, the square of the speed v, the drag coefficient $C_D$ and the frontal area A. A cyclist at 10 m/s with $C_D A$ of about 0.45 m² meets roughly 27 N of air drag.

Because speed is squared, doubling the speed makes the drag four times larger. That one fact explains why fuel use climbs steeply on fast roads and why falling objects reach a terminal speed. Drag is a force like any other in the laws of motion, and it always points opposite to the motion through the fluid.

Quick facts

Formula
$F_D = \frac{1}{2}\rho v^2 C_D A$
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Speed dependence
Proportional to v² at ordinary speeds
Air density
About 1.2 kg/m³ near sea level
Slow, small spheres
Use Stokes' law, $F = 6\pi\eta r v$

Formula for drag force: each factor

Factors in the drag equation
Symbol	Meaning	SI unit
$F_D$	Drag force	N
$\rho$	Density of the fluid (air about 1.2, water about 1000)	kg/m³
$v$	Speed of the object relative to the fluid	m/s
$C_D$	Drag coefficient, set by shape and surface	none
$A$	Frontal area facing the flow	m²

The term $\frac{1}{2}\rho v^2$ is the dynamic pressure of the flow. Multiplying by an area turns pressure into force, and $C_D$ adjusts for shape. Water is about 800 times denser than air, which is why wading is so much harder than walking.

Typical drag coefficients

Approximate drag coefficients
Shape	Approximate C_D
Streamlined teardrop	about 0.04
Sphere	about 0.47
Modern car	about 0.25 to 0.35
Upright cyclist	about 0.9
Flat plate facing the flow	about 1.2 to 1.3
Skydiver spread flat	about 1.0

These values change with speed and surface details, so questions always state the coefficient to use. Streamlining lowers C_D; crouching lowers A. Racing cyclists attack both.

Worked examples on the drag force formula

Example 1: a cyclist

A cyclist rides at 10 m/s. Take ρ = 1.2 kg/m³, C_D = 0.90 and A = 0.50 m². Find the drag.

$F_D = 0.5 \times 1.2 \times 10^2 \times 0.90 \times 0.50 = 27\ \text{N}$

Example 2: motorway speed, with km/h

A car with C_D = 0.30 and frontal area 2.2 m² travels at 108 km/h. Find the drag. What happens at double the speed?

First, 108 km/h is 30 m/s.

$F_D = 0.5 \times 1.2 \times 30^2 \times 0.30 \times 2.2 = 356.4\ \text{N} \approx 360\ \text{N}$

At 60 m/s, v² is four times larger, so F_D is about 1430 N.

Example 3: terminal speed of a skydiver

An 80 kg skydiver falls belly down with C_D = 1.0 and A = 0.70 m². Find the terminal speed, with g = 9.8 m/s².

At terminal speed drag equals weight: $\frac{1}{2}\rho v^2 C_D A = mg$ .

$v = \sqrt{\frac{2mg}{\rho C_D A}} = \sqrt{\frac{2 \times 80 \times 9.8}{1.2 \times 1.0 \times 0.70}} = \sqrt{1867} \approx 43\ \text{m/s}$

That is about 155 km/h. Entering F = 784 N, rho = 1.2, Cd = 1.0, A = 0.70 in the calculator gives the same speed.

Quadratic drag and linear drag

The $v^2$ law holds for most everyday objects in air and water, where the flow behind them is turbulent. For tiny or very slow objects, such as dust grains, mist droplets or a ball bearing sinking in thick oil, the flow stays smooth and drag is proportional to v instead. That case follows the viscous force formula and Stokes' law. Which regime applies is set by the Reynolds number, $Re = \frac{\rho v L}{\eta}$ : small values mean smooth flow and linear drag, large values mean turbulence and quadratic drag.

Drag, power and terminal velocity

The power needed to push against drag is $P = F_D v = \frac{1}{2}\rho C_D A v^3$ . Doubling speed needs eight times the power. A falling object speeds up until drag balances its weight; see the terminal velocity formula. After that the net force is zero and the speed stays fixed, exactly as the net force formula predicts.

Drag force mistakes

Not squaring the speed. The drag on a car at 30 m/s uses 900, not 30.
Speed left in km/h. Divide by 3.6 first, or the answer is about 13 times too large.
Wrong area. A is the frontal area that faces the flow, not the total surface of the object.
Wrong fluid. A swimmer moves through water of density about 1000 kg/m³, not air.
Relative speed ignored. A headwind adds to the speed through the air; a tailwind subtracts.

Frequently asked questions

What is the formula for drag force?

F_D = ½ρv²C_D A, with ρ for fluid density, v for speed, C_D for the shape factor and A for the area facing the flow.

Why does drag depend on the square of velocity?

A faster object hits more fluid each second and gives each bit of fluid more momentum. Both effects grow with speed, so their product grows as v².

What is the unit of the drag coefficient?

None. C_D is a pure number that describes how the shape and surface affect drag.

What is the difference between drag and viscous force?

Viscous force is the friction between layers of fluid and dominates at low speeds, growing with v. Drag in everyday flows is mostly due to turbulence and pressure differences and grows with v².