6.5 — Air drag and terminal velocity

An object moving through air experiences a force opposite its motion. Unlike surface friction (nearly constant), air drag is speed-dependent — and this generates interesting behavior.

Two regimes of drag

Viscous regime (low speed, small object):

\[ F_\text{drag} = b v \]

• b depends on shape, size, air viscosity
• Applies: raindrops, dust, slow motion in oil

Inertial regime (high speed, large object):

\[ F_\text{drag} = \frac{1}{2}\rho C_d A v^2 \]

• ρ air density (1.2 kg/m³)
• C_d drag coefficient (dimensionless, 0.05-2 depending on shape)
• A cross-sectional area
• Applies: skydivers, high-speed cars, heavy projectiles

Equation of motion with drag

Vertical fall with linear drag:

\[ m\frac{dv}{dt} = mg - bv \]

Terminal velocity (when dv/dt = 0):

\[ v_t = \frac{mg}{b} \]

Time evolution (initial condition v(0) = 0):

\[ v(t) = v_t\left(1 - e^{-bt/m}\right) \]

Note: the object never truly reaches v_t — approaches exponentially. After ~3-5 \tau (with \tau = m/b time constant), essentially at terminal.

Equation of motion — inertial regime

Fall with v² drag:

\[ m\frac{dv}{dt} = mg - kv^2 \]

Terminal velocity:

\[ v_t = \sqrt{\frac{2mg}{\rho C_d A}} = \sqrt{\frac{mg}{k}} \]

Skydiver: m = 80 kg, \rho = 1.2, C_d \approx 0.6, A \approx 0.8 m²:

\[ v_t \approx 52\ m/s \approx 188\ km/h \]

With parachute open (larger A, higher C_d): v_t \approx 6\ m/s.

Example — raindrop

Small raindrop, viscous regime. For a 1 mm droplet:

• Mass m \approx 4 \times 10^{-6} kg
• b \approx 3.4 \times 10^{-5} (Stokes)
• v_t \approx 1.2\ m/s

Larger drops (heavy rain), inertial regime, terminal velocity can be ~10 m/s.

Why a skydiver falls faster than a feather

Feather:

• Low mass, large area → small v_t (~1 m/s)
• Quickly reaches terminal

Skydiver:

• High mass, relatively small area → large v_t
• Takes several seconds to reach terminal

In vacuum (no drag): both fall at g. Apollo 15.

Terminal-velocity "paradox"

If a skydiver falls at terminal, acceleration is zero — so net force is zero. Is that "free fall"? No — she's still moving. Is there force? Yes — two that cancel (gravity and drag).

Distinction: "acceleration" means changing velocity, not "moving". You can move fast with zero acceleration.

A few notes and common mistakes

1. Drag depends on speed. Unlike surface friction (nearly constant).

2. Terminal velocity depends on shape and size. Mass matters, but so do drag coefficient and area.

3. Fall with drag is not §2.4 kinematics. Acceleration is variable — must integrate carefully.

4. At high altitude, ρ is smaller. Skydiver at 30 km altitude has much larger v_t (up to ~370 m/s — Redbull Stratos).

What you should be able to do

• Distinguish viscous (F \propto v) vs inertial (F \propto v²) regimes
• Compute terminal velocity in both
• Solve the differential equation for linear drag
• Apply to skydiving and raindrops
• Understand exponential approach to terminal

Preview of §6.6

The fundamental forces of nature — gravity, electromagnetism, strong and weak nuclear. §6.6.

📚 See also: Halliday Vol 1, Ch 6, §6.6.

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