Drag and air resistance Knowledge-Based Study

Study notes

1. What is Drag?

   When an object moves through a fluid such as air or water, it experiences a resistive force opposing its motion. This contact force exerted by the fluid is called drag, or specifically air resistance when the fluid is air.

   Key Properties of Drag

   • Direction: Drag always acts parallel to the relative flow of the fluid and strictly in the direction opposite to the velocity vector of the object relative to the fluid.
   • Velocity Dependence: Unlike dry kinetic friction, which is approximately independent of relative speed, drag increases systematically with the speed of the moving body.
   • Surface and Shape Effects: Drag depends on the effective cross-sectional area of the object, its aerodynamic or hydrodynamic profile, and the density of the surrounding fluid.

   Mathematical Representations

   In classical mechanics, the dependence of drag on speed $v$ is generally modeled in two regimes depending on flow characteristics:

   1. Linear Drag (Low Reynolds Number)

   For very small particles or objects moving at very low speeds in viscous fluids, drag is directly proportional to speed:

   $$F_d = b v$$

   where $b$ is a constant determined by the fluid's viscosity and the geometry of the body.

   2. Quadratic Drag (High Reynolds Number)

   For macroscopic objects moving at standard speeds through air (such as falling balls, skydivers, or vehicles), the drag force is proportional to the square of the speed:

   $$F_d = \frac{1}{2} C_d \rho A v^2$$

   In this equation:

   • $\rho$ is the density of the fluid ($\text{kg/m}^3$).
   • $A$ is the effective cross-sectional area perpendicular to the velocity ($\text{m}^2$).
   • $C_d$ is the dimensionless drag coefficient, which accounts for the object's shape and surface roughness.
   • $v$ is the speed of the object relative to the fluid ($\text{m/s}$).

   Dynamic Analysis of Free Fall with Drag

   Consider an object of mass $m$ dropped from rest and falling vertically downward under gravity with quadratic air resistance. Taking the downward direction as positive:

   Equation of Motion

   By applying Newton's second law ($\Sigma F_y = m a_y$):

   $$m g - F_d = m a$$$$m g - \frac{1}{2} C_d \rho A v^2 = m a$$

   Solving for the downward acceleration $a$:

   $$a(v) = g - \frac{C_d \rho A}{2m} v^2$$

   Evolution of Motion Over Time

   • At release ($t = 0$, $v = 0$): Drag is zero ($F_d = 0$), so the initial acceleration equals the gravitational acceleration: $a = g$.
   • During descent ($v > 0$): As the object gains speed, drag increases. Consequently, the net downward force decreases, causing acceleration to diminish ($0 < a < g$).
   • As speed increases further: The acceleration continuously decreases toward zero as drag approaches the weight of the object.

   Reaching Dynamic Equilibrium

   When the upward drag force grows large enough to balance the downward force of gravity, the net force on the falling body becomes zero.

   Condition for Terminal Velocity

   Setting the acceleration to zero ($a = 0$):

   $$\Sigma F = m g - F_d = 0 \implies F_d = m g$$

   Substituting the quadratic drag expression:

   $$\frac{1}{2} C_d \rho A v_t^2 = m g$$

   Solving for the terminal velocity $v_t$:

   $$v_t = \sqrt{\frac{2 m g}{C_d \rho A}}$$

   Physical Implications

   • Mass Dependency: Heavier objects of identical size, shape, and orientation have a higher terminal velocity because a larger drag force is needed to balance their greater weight.
   • Area and Orientation Dependency: Increasing the projected area $A$ (e.g., deploying a parachute) significantly reduces terminal velocity.
   • Subsequent Motion: Once terminal velocity is reached, the object moves downward with constant velocity, exhibiting zero acceleration in accordance with Newton's first law.