Drag Force and Terminal Speed

Learning Objectives

• Express the drag force mathematically.
• Describe applications of the drag force.
• Define terminal velocity.
• Determine an object's terminal velocity given its mass.

Introduction to Drag Forces

• Drag is the force experienced by an object moving through a fluid (gas or liquid).
• Similar to friction, drag opposes motion but is proportional to some function of the object's velocity.
• Key factors influencing drag force:
  • Shape of the object
  • Size of the object
  • Velocity of the object
  • Fluid's density

Mathematical Model of Drag Force

• For large objects, drag is proportional to the square of speed: [ F_D = \frac{1}{2} C \rho A v^2 ]
  • C is the drag coefficient, A is the area, and ( \rho ) is the fluid density.
• Generalized form: ( F_D = b v^n ) where ( n = 2 ) for high speed.
• Drag coefficient (C) is determined empirically, often using wind tunnels.
• Table of typical drag coefficients for various objects provided.

Application of Drag Forces

• Aerodynamic design reduces drag (e.g., cars, sports equipment).
• Innovations like body suits in sports reduce drag to improve performance.

Terminal Velocity

• Defined as the constant speed reached when drag force equals the gravitational force.
• At terminal velocity:
  • Net force = 0
  • ( mg = F_D )
  • Solving gives: ( v_T = \sqrt{\frac{2mg}{C \rho A}} )
• Example: A skydiver reaching terminal velocity.

Example Calculations

• Example 6.17: Calculate terminal velocity for a skydiver in a spread-eagle position.
• Example 6.18: Motorboat velocity and position functions with resistive force.

Stokes Law

• For small particles or low-speed objects, drag is proportional to velocity.
• Drag force for spherical objects: ( F_R = 6\pi \eta r v )
  • r is radius, ( \eta ) is viscosity.

Implications and Applications

• Biological implications: evolution of streamlined shapes in animals to reduce drag.
• Environmental factors: small animals like squirrels are less susceptible to fall damage due to higher relative drag.

Calculus of Velocity-Dependent Frictional Forces

• For liquids: ( f_R = bv )
• Differential equations and integration used to determine velocity and position.
• Limiting values (terminal velocity) reached over time.

Additional Understanding

• Calculation exercises for understanding terminal velocity and resistive forces.
• Discussion on the effects of air resistance on various objects and animals.