Drag Force Formula
Reference for drag force F_D = 0.5 × C_D × ρ × A × v².
Covers drag coefficients for cars, spheres, and airfoils with Reynolds number and vehicle examples.
The Formula
F_D = ½ × ρ × v² × C_D × A
Drag force opposes the motion of any object moving through a fluid (air, water, etc.). It increases with the square of velocity — doubling speed quadruples drag.
Variables
| Symbol | Meaning |
|---|---|
| F_D | Drag force (Newtons) |
| ρ | Fluid density (kg/m³) — air ≈ 1.225, water ≈ 1000 |
| v | Velocity of the object relative to the fluid (m/s) |
| C_D | Drag coefficient (unitless, depends on shape) |
| A | Reference area — typically frontal cross-section (m²) |
Example 1
Find the drag on a car (C_D = 0.3, A = 2.2 m²) at 100 km/h in air
v = 100 km/h = 27.78 m/s, ρ = 1.225 kg/m³
F_D = 0.5 × 1.225 × (27.78)² × 0.3 × 2.2
F_D = 0.5 × 1.225 × 771.7 × 0.3 × 2.2
F_D ≈ 312 N
Example 2
Find the drag on a sphere (C_D = 0.47, diameter 0.1 m) at 5 m/s in water
A = π × (0.05)² = 0.00785 m², ρ = 1000 kg/m³
F_D = 0.5 × 1000 × 25 × 0.47 × 0.00785
F_D ≈ 46.1 N
When to Use It
Use the drag force formula when:
- Designing vehicles, aircraft, or boats for efficiency
- Calculating fuel consumption at different speeds
- Estimating terminal velocity of falling objects
- Optimizing the aerodynamic shape of structures
Key Notes
- Formula: F_D = ½ρv²C_D A: ρ is fluid density, v is velocity, C_D is the dimensionless drag coefficient, and A is the reference area (usually frontal area). The drag coefficient is shape-dependent: a sphere ≈ 0.47, a streamlined car ≈ 0.3, a flat plate ≈ 1.2.
- Quadratic dependence on velocity: Doubling speed quadruples drag force. This is why aerodynamic efficiency matters most at high speeds — a car at 100 mph faces 4× the drag of one at 50 mph, requiring 8× the power to overcome it.
- Terminal velocity: When drag equals weight (F_D = mg), acceleration stops and the object falls at constant speed. Skydivers reach ~55 m/s (120 mph) belly-down; headfirst they can exceed 90 m/s.
- Reynolds number governs the regime: At low Re (viscous flow), drag scales with velocity (Stokes' law: F = 6πηrv). At high Re (inertial flow), the quadratic drag formula applies. The transition affects which formula is valid.
- Engineering applications: Drag force is used to design vehicles, aircraft, and wind turbines. Streamlining reduces C_D; spoilers and diffusers generate downforce by manipulating pressure differences around the body.