A.1.4 Fluid resistance (qualitative only)

Drag Force: A Velocity-Dependent Force

Unlike friction, which is typically constant, drag force increases with velocity.

How Drag Force Works

Drag force arises from two main factors:

1. Viscous Drag: Caused by the friction between the fluid and the surface of the object.
2. Pressure Drag: Caused by the difference in pressure between the front and back of the object as it moves through the fluid.

Mathematical Models of Drag Force

The drag force, $F_d$, can be modeled using different equations depending on the velocity of the object:

Low Velocities

At low speeds, drag force is often proportional to velocity:

$${\overrightarrow{F}}_d=-k\overrightarrow{v}$$

where $k$ is a constant that depends on the shape and size of the object and the properties of the fluid.

High Velocities

At higher speeds, drag force is typically proportional to the square of the velocity:

$$F_d=k{v}^2$$

or in vector form:

$${\overrightarrow{F}}_d=-kv\overrightarrow{v}$$

Impact of Drag on Projectile Motion

1. In ideal projectile motion, we assume no air resistance.
2. However, in reality, drag significantly alters the trajectory of a projectile.

Changes to Trajectory

1. Reduced Range:
   1. Drag slows the projectile, causing it to travel a shorter horizontal distance.
2. Lower Maximum Height:
   1. The upward component of velocity decreases more quickly due to drag, reducing the height reached.
3. Asymmetrical Path:
   1. Unlike the symmetrical parabolic path in ideal conditions, the trajectory with drag is steeper on the descent.

Changes to Maximum Height and Time of Flight

1. Maximum Height:
   1. Drag reduces the vertical velocity more rapidly, leading to a lower maximum height.
2. Time of Flight:
   1. The time to reach the peak is shorter, but the descent may take longer due to reduced speed, often resulting in a slightly shorter overall time of flight.

Terminal Velocity: The Balance of Forces

At this point, the net force is zero, and the object stops accelerating.

How Terminal Velocity is Achieved

1. Initial Acceleration: When an object is dropped, gravity causes it to accelerate downward.
2. Increasing Drag: As the object speeds up, the drag force increases.
3. Equilibrium: Eventually, the drag force equals the gravitational force ($mg$), resulting in zero net force and constant velocity.

Calculating Terminal Velocity

1. Terminal velocity, $v_T$, can be found by setting the drag force equal to the gravitational force: $$mg=k{v}_T^2$$
2. Solving for $v_T$ gives: $$v_T=\sqrt{\frac{mg}{k}}$$

Appendix

Further, we present some more context on drag forces, which is not required by the syllabus, but should help with understanding.

Linear Drag Model

1. The linear drag model originates from Stokes' law, which describes the drag force on a small spherical object moving slowly through a viscous fluid (low Reynolds number): $$F_{\text{Stokes}}=6\pi \eta rv$$ where $\eta$ is the dynamic viscosity of the fluid and $r$ is the radius of the sphere.
2. Rigorously, it is derived from Navier-Stokes equations for a viscous, incompressible fluid around a small spherical object.

Quadratic Drag Model

1. The quadratic drag model assumes the drag force is proportional to the square of the object's speed: $${\overrightarrow{F}}_{\text{drag}}=-\frac{1}{2}{C}_d\rho A{v}^2\hat{v}$$ where: $C_d$ is the dimensionless drag coefficient, $\rho$ is the fluid density, $A$ is the cross-sectional area of the object perpendicular to the flow, $v$ is the speed of the object, $\hat{v}$ is the unit vector in the direction of velocity.
2. This model emerges from considerations of inertial forces in a fluid and could be derived from dimensional analysis and experimental observations for objects at higher Reynolds numbers.