Terminal Velocity
Terminal velocity is the constant maximum speed attained by an object falling through a fluid when the downward gravitational force equals the sum of the upward drag force and buoyancy force, resulting in zero net acceleration. At terminal velocity the object no longer accelerates; its speed depends on its mass, size, shape, and the fluid's density and viscosity. Terminal velocity is encountered in skydiving, raindrop formation, sedimentation in liquids, and the design of parachutes and falling bodies in engineering.
Key Formula
v_t = sqrt(2mg / (ρ_f × C_D × A))
LaTeX: v_t = \sqrt{\frac{2mg}{\rho_f C_D A}}
| Symbol | Meaning | Unit |
|---|---|---|
| v_t | Terminal velocity | m/s |
| m | Mass of the falling object | kg |
| g | Acceleration due to gravity | m/s² |
| ρ_f | Density of the fluid | kg/m³ |
| C_D | Drag coefficient | dimensionless |
| A | Reference (cross-sectional) area | m² |
Worked Example
Problem
A skydiver with mass m = 80 kg (including equipment) has a frontal area A = 0.70 m² and drag coefficient C_D = 1.0 in a spread-eagle position. Air density ρ = 1.2 kg/m³. Calculate the terminal velocity. (g = 9.81 m/s²)
Solution
Step 1 — Use v_t = sqrt(2mg / (ρ C_D A)). Step 2 — Numerator inside sqrt: 2 × 80 × 9.81 = 1569.6. Step 3 — Denominator: 1.2 × 1.0 × 0.70 = 0.84. Step 4 — Ratio: 1569.6 / 0.84 = 1868.6 m²/s². Step 5 — v_t = sqrt(1868.6) ≈ 43.2 m/s.
Answer
v_t ≈ 43.2 m/s ≈ 155 km/h
Terminal Velocities of Objects Falling Through Air at Sea Level (ρ = 1.2 kg/m³)
| Object | Mass (kg) | C_D | Area (m²) | Terminal Velocity (m/s) |
|---|---|---|---|---|
| Raindrop (2 mm dia.) | 4.2 × 10⁻⁶ | 0.47 | 3.1 × 10⁻⁶ | ~9 |
| Badminton shuttlecock | 0.005 | 0.60 | 0.003 | ~20 |
| Skydiver (spread eagle) | 80 | 1.00 | 0.70 | ~43 |
| Skydiver (headfirst) | 80 | 0.70 | 0.18 | ~85 |
| Deployed parachute | 80 | 1.75 | 44 | ~5.5 |
| Hailstone (2 cm dia.) | 0.004 | 0.47 | 3.1 × 10⁻⁴ | ~25 |
Interactive Tools
PhET Forces and Motion Basics
Simulate falling objects and observe force balance at terminal velocity
Open ToolWolfram Alpha — Terminal Velocity
Compute terminal velocity for any object given mass, area, and drag coefficient
Open ToolKhan Academy — Terminal Velocity
Step-by-step explanation with force diagrams and real-world examples
Open Tool
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Related Terms
Drag Force
Drag force is the resistive force exerted by a fluid on a body moving through it, acting opposite to the direction of relative motion and composed of pressure drag (form drag) and skin-friction drag. For objects moving at moderate to high speeds, drag is proportional to the square of velocity, the fluid density, the frontal area, and a dimensionless drag coefficient that depends on shape and flow regime. Understanding and minimising drag is critical in vehicle and aircraft design, sports engineering, and offshore structure analysis.
Turbulent Flow
Turbulent flow is a chaotic, irregular fluid motion characterised by rapid fluctuations in velocity and pressure, eddies, vortices, and vigorous lateral mixing between fluid layers. It occurs when inertial forces overcome viscous forces, typically at Reynolds numbers above 4000 in pipe flow, and is the dominant regime in most industrial, atmospheric, and oceanic flows. Despite its complexity, turbulent flow enhances heat and mass transfer, making it beneficial in heat exchangers and combustion systems.
Fluid Pressure
Fluid pressure is the force exerted per unit area by a fluid on any surface in contact with it, arising from the continuous collisions of fluid molecules. In a static fluid, pressure at a given depth depends on the fluid's density, gravitational acceleration, and the depth below the free surface. It is fundamental to hydraulics, hydrostatics, and the design of dams, pipelines, and pressure vessels.
From Latin "terminalis" (pertaining to a boundary or end), from "terminus" (boundary, limit) and "velocitas" (speed, swiftness), from "velox" (swift). The concept was implicit in Galileo's studies on falling bodies (early 17th century) and formalised once Newton's second law and fluid drag models were combined in the 18th century.