Chapter 6 Lectures: Applying Newton's Laws: Drag Forces: Non Constant Acceleration

Fundamentals of Drag Forces and Fluid Resistance

• Definition of Drag Force:

  • A drag force is a frictional-type force that occurs when an object moves through a fluid medium.

  • Fluid mediums encompass both liquids (such as water) and gases (such as atmospheric air or any other gas material).

  • Drag force behaves mechanically like friction, opposing the relative motion of the moving body.

  • The vector direction of the drag force is strictly opposite to the instantaneous velocity vector of the object.

• Incorporation of Air Resistance in Mechanical Systems:

  • Elementary mechanics often ignores air resistance by assuming ideal vacuum conditions.

  • Incorporating air resistance introduces a drag force with velocity-dependent behavior, meaning the magnitude of the opposing force changes based on how fast the object travels.

Mathematical Formulation and the Drag Coefficient

• Functional Dependence of Drag:

  • The drag force depends directly on the speed (or velocity vector) of the object multiplied by a proportionality factor known as the drag coefficient, denoted as $b$.

  • In vector notation, the relationship is expressed as:     $\mathbf{D} = -b\,\mathbf{v}$

  • The negative sign signifies that the drag force vector $\mathbf{D}$ acts in the direct opposite direction to the velocity vector $\mathbf{v}$. For example, if an object has a positive velocity to the right, the drag force points to the left.

• Physical Determinants of the Drag Coefficient ($b$):

  • Type of Medium: The fluid or gaseous material through which the object moves (e.g., air vs. water).

  • Material Composition: The specific material out of which the object is fabricated.

  • Geometric Shape: The aerodynamic geometry of the object. Streamlined shapes yield significantly smaller drag coefficients compared to flat, blunt surfaces.

  • Surface Area: The total cross-sectional or surface area exposed to the fluid flow.

  • Standard Modeling Assumption: Although mechanical engineering and advanced fluid dynamics analyze the detailed components of $b$, introductory physics models treat $b$ as a constant parameter for a given object of fixed shape and material moving through a specific fluid.

  • Point Object Transition: Incorporating $b$ transitions physical analysis away from idealized dimensionless point masses toward realistic objects possessing physical geometry and surface area.

Dimensional Analysis of the Drag Coefficient

• Dimensional Derivation of the Units of $b$:

  • The drag force $D$ possesses dimensions of force, measured in newtons ($\text{N}$).

  • In fundamental dimensions of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$):     $\text{Force} = \frac{\text{Mass} \times \text{Length}}{\text{Time}^2}$

  • The velocity $v$ has dimensions of speed:     $\text{Velocity} = \frac{\text{Length}}{\text{Time}}$

  • Setting up the relationship based on the drag equation $D = b\,v$:     $b = \frac{D}{v}$

  • Substituting the fundamental dimensions yields:     $\text{Dimensions of } b = \frac{\frac{\text{Mass} \times \text{Length}}{\text{Time}^2}}{\frac{\text{Length}}{\text{Time}}}$

  • Simplifying the fractional expression:

  • The length dimension cancels out in the numerator and denominator.

  • One factor of time cancels out in the denominator.

  • The resulting required dimensions for $b$ are:     $\text{Dimensions of } b = \frac{\text{Mass}}{\text{Time}}$

  • The drag coefficient must represent a mass per unit time to yield a force when multiplied by velocity.

Dynamics of Freefall and Terminal Speed

• Freefall Example (Skydiver Dropped from High Altitude):

  • Scenario: A person jumps from an airplane at a substantial height and begins falling under the influence of gravity.

  • Initial State ($t = 0$):

  • The instant the skydiver drops from the plane, the initial velocity is zero ($v = 0$).

  • Since drag force is directly proportional to speed ($D = b\,v$), the drag force at this initial release point is zero ($D = 0$).

  • The sole force acting on the body at the moment of release is the downward gravitational force (weight, $w$).

  • Mid-Fall Evolution:

  • As gravity accelerates the body downward, the downward speed increases.

  • The gravitational force ($w = m\,g$) remains constant throughout the descent.

  • Because speed increases, the upward drag force ($D = b\,v$) increases proportionally in the direction opposite to motion.

  • As the opposing drag increases, the net downward force decreases, causing the downward acceleration to diminish over time.

• State of Terminal Speed:

  • As the object accelerates and speed continues to build, the drag force continues to grow until its magnitude equals the magnitude of the gravitational weight.

  • Once the drag force balances the weight force, the net force acting on the body becomes zero:     $\sum F = 0$

  • When the net force is zero, acceleration ceases ($a = 0$).

  • The object continues falling at a constant, steady speed called the terminal speed (or terminal velocity, $v_t$).

Mathematical Derivation of Terminal Speed

• Equilibrium Condition:

  • At terminal velocity, the sum of vertical forces is zero:     $\sum F = w - D = 0$     $w = D$

  • Substituting the expressions for gravitational weight and linear drag force:     $m\,g = b\,v_t$

• Determining Terminal Velocity and the Drag Coefficient:

  • Solving for terminal velocity ($v_t$):     $v_t = \frac{w}{b} = \frac{m\,g}{b}$

  • Solving conversely for the drag coefficient ($b$) using empirical terminal velocity measurements:     $b = \frac{w}{v_t} = \frac{m\,g}{v_t}$

Constant versus Variable Forces: Calculus in Non-Constant Motion

• Constant Forces vs. Variable Forces:

  • Constant Forces: When forces acting on an object are constant, Newton's second law yields a constant acceleration, allowing motion to be solved using standard algebraic kinematic equations.

  • Variable Forces: When forces vary with parameters such as velocity, Newton's second law produces a non-constant acceleration that varies with time ($a(t)$).

• Analytical Implications for Drag Force:

  • Because the drag force depends continuously on speed, it constitutes a variable force that changes as the object accelerates.

  • Consequently, algebraic kinematic formulas are insufficient to determine the instantaneous acceleration, velocity, or position during the transition to terminal speed.

  • Solving the equation of motion for a system experiencing a drag force requires formulating and solving differential equations using calculus:     $m\,\frac{dv}{dt} = m\,g - b\,v$