Resistive Forces: Air Drag and Terminal Velocity

Introduction

Key Concepts

Understanding Resistive Forces

Resistive forces are forces that oppose the motion of an object through a medium, such as air or water. These forces are essential in analyzing real-world scenarios where ideal conditions (like frictionless surfaces) do not exist. In the realm of Physics C: Mechanics, resistive forces help explain the behavior of objects in motion, particularly when external forces are balanced by opposing forces.

Air Drag: Definition and Characteristics

Air drag, a type of resistive force, occurs when an object moves through air, experiencing a force opposite to its direction of motion. This force depends on several factors:

• Velocity ($v$): The speed of the object directly influences air drag. As velocity increases, air drag increases.
• Cross-Sectional Area ($A$): A larger area facing the airflow results in greater air drag.
• Air Density ($\rho$): Denser air amplifies the resistive force experienced by the object.
• Drag Coefficient ($C_d$): A dimensionless number that characterizes the object's shape and its aerodynamic properties.

The equation representing air drag is: $$ F_d = \frac{1}{2} \rho v^2 C_d A $$

Terminal Velocity: Concept and Calculation

Terminal velocity is the constant speed attained by an object when the net force acting upon it becomes zero. At this point, the downward gravitational force equals the upward resistive force, resulting in no further acceleration.

Mathematically, terminal velocity ($v_t$) can be derived by setting the gravitational force equal to the air drag: $$ mg = \frac{1}{2} \rho v_t^2 C_d A $$ Solving for $v_t$, we get: $$ v_t = \sqrt{\frac{2mg}{\rho C_d A}} $$

Where:

• $m$: Mass of the object
• $g$: Acceleration due to gravity
• $\rho$: Air density
• $C_d$: Drag coefficient
• $A$: Cross-sectional area

Factors Affecting Terminal Velocity

Several factors influence terminal velocity, including:

• Mass ($m$): Heavier objects generally achieve higher terminal velocities because more gravitational force is required to balance air drag.
• Shape and Surface Area ($A$): Streamlined shapes with smaller cross-sectional areas encounter less air resistance, leading to higher terminal velocities.
• Air Density ($\rho$): Higher air density increases air drag, reducing terminal velocity.
• Drag Coefficient ($C_d$): Objects with lower drag coefficients experience less resistive force, enabling higher terminal velocities.

Applications of Terminal Velocity

Understanding terminal velocity has practical applications in various fields:

• Skydiving: Terminal velocity determines the maximum speed a skydiver can achieve during freefall.
• Engineering: Designing vehicles and structures that minimize air resistance improves performance and efficiency.
• Meteorology: Terminal velocity helps in understanding the behavior of particles in the atmosphere, such as raindrops and dust.

Deriving Terminal Velocity from Newton's Second Law

Newton's Second Law states that the net force ($F_{net}$) acting on an object is equal to the mass ($m$) of the object multiplied by its acceleration ($a$): $$ F_{net} = ma $$ At terminal velocity, acceleration is zero ($a = 0$), so: $$ F_{net} = 0 $$ This implies that the gravitational force is balanced by the air drag: $$ mg = \frac{1}{2} \rho v_t^2 C_d A $$ Solving for terminal velocity ($v_t$): $$ v_t = \sqrt{\frac{2mg}{\rho C_d A}} $$

Graphical Representation of Velocity vs. Time

A typical velocity vs. time graph for an object falling under gravity with air resistance shows that velocity increases initially but approaches a constant value—terminal velocity—as time progresses. The graph illustrates how the forces balance out over time, leading to steady motion.


Energy Considerations at Terminal Velocity

At terminal velocity, the kinetic energy input from gravity is balanced by the energy dissipated due to air drag. This balance ensures that the object's speed remains constant, as there's no net work being done to accelerate it further.

Real-World Examples

• Skydivers: When a skydiver jumps from a plane, they accelerate until the air drag equals the gravitational force, reaching terminal velocity.
• Falling Objects: Objects like feather and hammer in the same environment eventually reach terminal velocity, although their specific terminal speeds differ based on shape and mass.
• Parachutes: Parachutes increase the cross-sectional area ($A$), thereby increasing air drag and reducing terminal velocity to ensure a safe landing.

Mathematical Example: Calculating Terminal Velocity

Consider a skydiver with a mass of $80\,kg$, a drag coefficient ($C_d$) of $1.0$, and a cross-sectional area ($A$) of $0.7\,m^2$. Assuming the air density ($\rho$) is $1.225\,kg/m^3$ and acceleration due to gravity ($g$) is $9.81\,m/s^2$, the terminal velocity ($v_t$) can be calculated as follows:

$$ v_t = \sqrt{\frac{2 \times 80\,kg \times 9.81\,m/s^2}{1.225\,kg/m^3 \times 1.0 \times 0.7\,m^2}} $$

$$ v_t = \sqrt{\frac{1569.6\,kg \cdot m/s^2}{0.8575\,kg/m \cdot s^2}} $$

$$ v_t = \sqrt{1826.05\,m^2/s^2} $$

$$ v_t \approx 42.73\,m/s $$

Limitations of the Terminal Velocity Model

While the terminal velocity model provides valuable insights, it has certain limitations:

• Assumption of Constant Air Density: In reality, air density decreases with altitude, affecting terminal velocity.
• Neglecting Wind and Turbulence: External factors like wind can alter the resistive forces experienced by the object.
• Rigid Body Assumption: The model assumes the object's shape and size remain constant, which may not hold true for flexible or deformable objects.

Advanced Topics: Reynolds Number and Flow Regimes

The Reynolds number ($Re$) is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It helps determine whether the flow around an object is laminar or turbulent, which in turn affects the drag coefficient ($C_d$). $$ Re = \frac{\rho v D}{\mu} $$ Where:

• $\rho$: Fluid density
• $v$: Velocity of the object relative to the fluid
• $D$: Diameter of the object
• $\mu$: Dynamic viscosity of the fluid

Different flow regimes influence terminal velocity by altering the drag coefficient. For instance, at low Reynolds numbers, flow is laminar, and $C_d$ tends to be higher, whereas at high Reynolds numbers, flow becomes turbulent, often resulting in lower $C_d$ values.

Impact of Altitude on Terminal Velocity

As altitude increases, air density ($\rho$) decreases, leading to reduced air drag. Consequently, an object's terminal velocity increases with altitude. This phenomenon is evident in high-altitude skydives, where jumpers can achieve higher terminal velocities compared to lower altitudes.

Effect of Temperature on Air Drag

Temperature influences air density and viscosity, thereby affecting air drag. Warmer air is less dense, resulting in lower air drag and higher terminal velocity. Conversely, colder air increases air drag, reducing terminal velocity.

Comparison Table

Summary and Key Takeaways