Drag Force

When an object moves through a fluid—either air or liquid—it experiences a drag force that resists its motion. You feel this when:

• Moving your hand quickly through water

• Extending your arm out a car window at high speed

• Walking against a strong wind

Drag increases dramatically with speed and depends on several factors:

• Shape of the object

• Size (frontal area)

• Velocity

• Fluid density (e.g., air vs. water)

Like friction, drag always opposes motion. But unlike simple friction, drag increases nonlinearly with speed. For most human-scale activities in air, the drag force [latex]F_D[/latex] is proportional to the square of velocity:

[latex]F_D \propto v^2[/latex]

This becomes more precise in the following form:

[latex]F_D = \frac{1}{2} C \rho A v^2[/latex]

Where:

• [latex]C[/latex] is the drag coefficient (dimensionless)

• [latex]\rho[/latex] is the fluid density (kg/m³)

• [latex]A[/latex] is the frontal area of the object (m²)

• [latex]v[/latex] is the velocity (m/s)

Figure 31.1 shows how aerodynamic shaping improves performance in sports and vehicles by minimizing drag.

Reducing Drag in Health and Sports

In athletics and medical biomechanics:

• Swimmers and cyclists wear smooth suits to reduce drag.

• Prosthetics and wheelchairs are shaped to improve aerodynamics and ease movement.

• In medicine, drag affects how air or liquid moves through tubes, airways, or blood vessels.

The drag coefficient [latex]C[/latex] is often determined experimentally using wind tunnels or water channels.

The value of the drag coefficient, [latex]C[/latex] , is determined empirically, usually with the use of a wind tunnel. (See Figure 31.2).

The drag coefficient can depend upon velocity, but we will assume that it is a constant here. Table 31.1 lists some typical drag coefficients for a variety of objects. Notice that the drag coefficient is a dimensionless quantity. At highway speeds, over 50% of the power of a car is used to overcome air drag. The most fuel-efficient cruising speed is about 70–80 km/h (about 45–50 mi/h). For this reason, during the 1970s oil crisis in the United States, maximum speeds on highways were set at about 90 km/h (55 mi/h).

The smoother and narrower the object, the lower the drag coefficient. Athletes often shave body hair or wear body suits to decrease drag and increase performance (see Figure 31.3).

Terminal Velocity

When a falling object accelerates, drag increases until it balances the gravitational force:

[latex]F_{\text{net}} = mg - F_D = 0[/latex]
[latex]mg = \frac{1}{2} C \rho A v^2[/latex]

Solving for terminal velocity [latex]v_t[/latex]:

[latex]v_t = \sqrt{\frac{2mg}{\rho C A}}[/latex]

This is the maximum velocity an object can reach in a fluid, such as a person falling through air. More massive or streamlined objects reach higher terminal velocities.

Example: Skydiver

• Mass: 75 kg

• Area: [latex]A = 0.18,\text{m}^2[/latex]

• Drag coefficient: [latex]C = 0.70[/latex]

• Air density: [latex]\rho = 1.21,\text{kg/m}^3[/latex]

[latex]v = \sqrt{\frac{2 \cdot 75 \cdot 9.8}{1.21 \cdot 0.70 \cdot 0.18}} = 98,\text{m/s} = 350,\text{km/h}[/latex]

A skydiver in a head-first position reaches up to 350 km/h. In a spread-eagle pose, the terminal velocity drops to about 200 km/h due to greater area and drag.

Why Small Animals and Microorganisms Survive Falls

Size plays a critical role in how organisms experience air resistance and terminal velocity. A fall from a 5-meter tree could injure a human—but a squirrel does it unharmed. Why?

Large animals like humans do not reach terminal velocity in a short fall, so they continue accelerating under gravity until they hit the ground. But small animals like squirrels or mice quickly reach terminal velocity, where drag balances gravity and their speed becomes constant and non-lethal.

As J.B.S. Haldane famously wrote in On Being the Right Size (1928):

“You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away… For the resistance presented to movement by the air is proportional to the surface… the small animal has ten times greater resistance per unit weight than the large one.”

Stokes’ Law: Drag at Small Scales

At very small scales or low speeds, the drag force behaves differently. Instead of being proportional to velocity squared, it is directly proportional to velocity. This is known as Stokes’ Law:

[latex]F_s = 6\pi r\eta v[/latex]

Where:

• [latex]r[/latex] is the radius of the particle

• [latex]\eta[/latex] is the fluid’s viscosity

• [latex]v[/latex] is the velocity

This law explains how dust particles, pollen, microorganisms, and cells move slowly through air or liquid at terminal velocity.

Examples:

• A bacterium (~1 μm size) moves at a terminal velocity of ~2 μm/s.

• Sediment in lakes (~5 μm/s) takes days to settle.

• Sperm cells are streamlined to efficiently swim through viscous fluids using flagella.

Evolution and Drag

Drag has influenced the evolution of animal form:

• Fish, whales, and dolphins are sleek and tapered to reduce drag in water.

• Birds have narrow profiles and long wings for efficient migration.

• Geese fly in a V-formation to reduce drag and conserve energy (see Figure 31.4.

In humans, the streamlined shape of sperm is a biological adaptation that maximizes propulsion while minimizing resistance in fluid environments.