6.4 Drag Force and Terminal Speed

When objects move through fluids, they experience drag force. This force opposes motion and depends on factors like fluid density, object velocity, and shape. Understanding drag is crucial for designing vehicles, parachutes, and buildings to optimize performance and safety.

Terminal speed occurs when drag force equals gravity, causing an object to fall at constant velocity. Factors like mass, fluid density, and object shape affect terminal speed. This concept is important in various applications, from skydiving to designing efficient transportation.

Drag Force

Calculation of drag force

• Drag force ($F_D$) exerted by a fluid (liquid or gas) on an object moving through it
• Opposes the motion of the object
• Drag force equation: $F_D = \frac{1}{2} \rho v^2 C_D A$
  • $\rho$ represents the density of the fluid (air, water)
  • $v$ is the velocity of the object relative to the fluid
  • $C_D$ is the drag coefficient determined by the shape of the object (streamlined, blunt) and the properties of the fluid (viscosity, compressibility)
  • $A$ is the cross-sectional area of the object perpendicular to the flow (frontal area)

Real-world applications of drag

• Aerodynamics in vehicle design
  • Streamlined shapes (teardrop, bullet) reduce drag force leading to improved fuel efficiency and performance (cars, airplanes, bicycles)
  • Spoilers and air dams on race cars increase downforce and stability at high speeds
• Parachutes
  • Large surface area increases drag force slowing descent for safe landings (skydivers, cargo drops)
  • Parachute material and shape affect drag coefficient and stability
• Swimming and aquatic animals
  • Streamlined body shapes (dolphins, sharks) minimize drag force allowing for efficient movement through water
  • Swimsuits designed to reduce drag and improve performance (Olympic swimmers)
• Wind resistance on buildings and structures
  • Drag force considered in design to ensure stability and safety (skyscrapers, bridges)
  • Wind tunnels used to test scale models and optimize designs

Fluid Dynamics and Forces

• Aerodynamics: The study of air flow and its interaction with solid objects, crucial for aircraft and vehicle design
• Hydrodynamics: The study of fluid motion in liquids, important for ship design and underwater vehicles
• Lift force: An upward force perpendicular to the direction of fluid flow, essential for aircraft wings and hydrofoils
• Buoyancy: An upward force exerted by a fluid on an immersed object, affecting the apparent weight and flotation of objects in liquids

Terminal Speed

Definition of terminal speed

• Terminal speed (terminal velocity) is the constant speed reached by an object when the drag force equals the force of gravity
• Acceleration becomes zero at terminal speed as the net force acting on the object is zero
• Conditions for reaching terminal speed:
  1. Object must be falling through a fluid (air, liquid)
  2. Sufficient time and distance for the object to accelerate until drag force balances gravitational force

Factors affecting terminal velocity

• Mass of the object
  • Heavier objects have a higher terminal velocity due to greater gravitational force ($F_g = mg$)
  • Example: a bowling ball reaches a higher terminal velocity than a tennis ball
• Fluid density
  • Denser fluids (water) exert greater drag force resulting in a lower terminal velocity compared to less dense fluids (air)
  • Example: a skydiver reaches a lower terminal velocity in water than in air
• Cross-sectional area of the object
  • Larger cross-sectional area perpendicular to the flow results in greater drag force and lower terminal velocity
  • Example: a flat piece of paper falls slower than a crumpled piece of paper
• Drag coefficient
  • Depends on the shape of the object (streamlined, blunt) and fluid properties (viscosity, turbulence)
  • Streamlined shapes (bullet) have lower drag coefficients resulting in higher terminal velocities compared to blunt shapes (flat plate)