Resistive Forces - AP Physics C: Mechanics Study Guide

Getting Started

Nearly every object moving through a fluid, such as air or water, experiences a force that opposes its motion. This resistive force, often called drag, is fundamentally different from the simple friction forces you have studied, as its magnitude depends on the object's speed. We will investigate the motion of an object when subject to both a constant force, like gravity, and a velocity-dependent resistive force, asking the core question: How does an object's velocity change over time when the force opposing its motion grows stronger as it moves faster?

What You Should Be Able to Do

After working through this section, you should be able to perform the following tasks:

• Set up a first-order differential equation for an object's velocity using Newton's second law when a linear resistive force is present.

• Solve this differential equation for velocity as a function of time, $v(t)$, using the method of separation of variables and initial conditions.

• Determine the terminal velocity of an object by analyzing the net force when acceleration is zero or by finding the asymptotic limit of the velocity function.

• Integrate the velocity function, $v(t)$, to derive the position of the object as a function of time, $x(t)$.

Key Concepts & Mechanisms

System & Preconditions

Our system is a single object of mass $m$ (in kilograms, kg), treated as a point particle. It moves through a fluid that exerts a resistive force. We make the following idealizations:

• The resistive force, ${\vec{F}}_r$, is linearly proportional to the object's velocity, $\vec{v}$ (in meters per second, m/s), and acts in the opposite direction. This is a valid model for objects moving at low speeds in a viscous fluid (laminar flow).

• The constant of proportionality, $k$, known as the drag coefficient (in kg/s), is constant and depends on the fluid's properties and the object's shape.

• Other forces, such as buoyancy, are considered negligible.

• If the object is falling, the gravitational field is assumed to be uniform, providing a constant downward force ${\vec{F}}_g=m\vec{g}$.

Key Steps / Relations

We can determine the object's motion by applying Newton's second law and solving the resulting equation. Let's consider the case of an object falling from rest under the influence of gravity and a linear resistive force. We define the downward direction as positive.

1. Represent the Forces: A free-body diagram shows two forces: the gravitational force, $F_g=mg$, acting downward (positive), and the resistive force, $F_r=kv$, acting upward (negative), since it opposes the downward velocity.

2. Apply Newton's Second Law: The net force on the object is the sum of the forces acting on it.

   $$\sum F=ma$$

   $$mg-kv=ma$$

3. Formulate the Differential Equation: Acceleration, $a$, is the first derivative of velocity with respect to time, $a=\frac{dv}{dt}$. Substituting this into Newton's second law gives the governing differential equation for velocity.

   $$mg-kv=m\frac{dv}{dt}$$

   This equation shows that the rate of change of velocity depends on the velocity itself.

4. Solve for Velocity: We solve this first-order differential equation using the method of separation of variables, with the initial condition that the object starts from rest, $v(0)=0$.

   $$m\frac{dv}{dt}=mg-kv$$

   $$\frac{dv}{mg-kv}=\frac{1}{m}dt$$

   Integrate both sides. Let the velocity go from $0$ at $t=0$ to $v$ at time $t$.

   $${\int }_0^v\frac{d{v}^{\prime }}{mg-k{v}^{\prime }}={\int }_0^t\frac{1}{m}d{t}^{\prime }$$

   The left integral is of the form $\int \frac{dx}{a+bx}=\frac{1}{b}\ln |a+bx|$. Here, $a=mg$ and $b=-k$.

   $${\left[-\frac{1}{k}\ln (mg-k{v}^{\prime })\right]}_0^v={\left[\frac{t^{\prime }}{m}\right]}_0^t$$

   $$-\frac{1}{k}\left[\ln (mg-kv)-\ln (mg)\right]=\frac{t}{m}$$

   $$\ln \left(\frac{mg-kv}{mg}\right)=-\frac{k}{m}t$$

   Exponentiate both sides to solve for $v$.

   $$\frac{mg-kv}{mg}=e^{-\frac{k}{m}t}$$

   $$1-\frac{k}{mg}v=e^{-\frac{k}{m}t}$$

   $$v(t)=\frac{mg}{k}\left(1-e^{-\frac{k}{m}t}\right)$$

Outputs & Effects

The solution reveals the key features of the motion.

• Terminal Velocity ($v_T$): As time $t\to \infty$, the exponential term $e^{-\frac{k}{m}t}$ approaches zero. The velocity thus approaches a maximum, constant value called the terminal velocity.

  $$v_T=\underset{t\to \infty }{\lim }v(t)=\frac{mg}{k}$$

  Terminal velocity is reached when the upward resistive force ($k{v}_T$) perfectly balances the downward gravitational force ($mg$), resulting in zero net force and zero acceleration.

• Time Constant ($\tau$): The quantity $\tau =m/k$ has units of time and is called the characteristic time or time constant. It governs how quickly the object approaches terminal velocity. The velocity function can be written as:

  $$v(t)=v_T(1-e^{-t/\tau })$$

  After one time constant ($t=\tau$), the object has reached $(1-e^{-1})\approx 0.63$, or 63% of its terminal velocity.

• Acceleration: The acceleration as a function of time is found by differentiating $v(t)$:

  $$a(t)=\frac{dv}{dt}=v_T\left(-e^{-t/\tau }\cdot \left(-\frac{1}{\tau }\right)\right)=\frac{v_T}{\tau }{e}^{-t/\tau }$$

  Since $v_T=mg/k$ and $\tau =m/k$, we have $v_T/\tau =(mg/k)/(m/k)=g$.

  $$a(t)=g{e}^{-t/\tau }$$

  At $t=0$, $a(0)=g$. As $t\to \infty$, $a(t)\to 0$. The acceleration starts at its maximum value and exponentially decays to zero.

Regulation & Limits

• Validity Domain: The linear drag model (${\vec{F}}_r=-k\vec{v}$) is an approximation. For many real-world objects moving at higher speeds (turbulent flow), the resistive force is better modeled as being proportional to the square of the velocity, ${\vec{F}}_r\propto -v^2\hat{v}$. This quadratic model leads to a different, more complex differential equation.

• Graphical Interpretation: A plot of $v(t)$ starts at the origin, rises, and asymptotically approaches the horizontal line $v=v_T$. A plot of $a(t)$ starts at $(0,g)$, and exponentially decays, asymptotically approaching the horizontal axis $a=0$.

Key Models & Diagrams

The process of analyzing motion with resistive forces can be mapped with the following flowchart, connecting the physical setup to the predicted motion.

Physical System (e.g., Object falling in a fluid)

↓

Representation (Free-Body Diagram showing ${\vec{F}}_g$ and ${\vec{F}}_r$)

↓

Governing Law (Newton's Second Law: $\sum \vec{F}=m\vec{a}$)

↓

Differential Equation ($m\frac{dv}{dt}=mg-kv$)

↓

Mathematical Solution (Separation of Variables & Integration)

↓

Predicted Observables

• Velocity function: $v(t)=v_T(1-e^{-t/\tau })$

• Acceleration function: $a(t)=g{e}^{-t/\tau }$

• Terminal Velocity: $v_T=mg/k$

• Time Constant: $\tau =m/k$

Key Components & Evidence

• Resistive Force (${\vec{F}}_r$): A force exerted by a fluid that opposes an object's motion through it. In the linear model, ${\vec{F}}_r=-k\vec{v}$. Its SI unit is the newton (N).

• Drag Coefficient ($k$): A constant of proportionality that depends on the fluid's viscosity and the object's size and shape. Its SI unit is kilograms per second (kg/s).

• Newton's Second Law ($\sum \vec{F}=m\vec{a}$): The fundamental principle relating the net force on an object to its mass and acceleration. It is the basis for deriving the equation of motion.

• Differential Equation: An equation that relates a function with its derivatives. For resistive motion, it connects velocity $v$ with its own rate of change, $\frac{dv}{dt}$.

• Separation of Variables: A technique for solving certain types of differential equations by isolating variables on opposite sides of the equation and integrating.

• Terminal Velocity ($v_T$): The constant, maximum velocity reached by an object when the resistive force on it is equal in magnitude and opposite in direction to the driving force (e.g., gravity).

• Time Constant ($\tau$): A parameter, $\tau =m/k$, that characterizes the time it takes for the system to approach its terminal state. A larger $\tau$ means a slower approach to $v_T$.

Skill Snapshots

Causation

• Driver: A net force ($mg-kv$) that is initially large but decreases as velocity increases.

• Change: Causes an acceleration ($a=g-\frac{k}{m}v$) that is initially equal to $g$ but decreases over time.

• Driver: An increasing velocity.

• Change: Causes the magnitude of the resistive force ($F_r=kv$) to increase, which in turn reduces the net force.

• Driver: The net force approaching zero.

• Change: Causes the acceleration to approach zero, resulting in the velocity approaching a constant terminal value.

Comparison

• Motion with Drag vs. Free Fall: An object in free fall has a constant acceleration $g$ and its velocity increases linearly with time. An object with drag has a decreasing acceleration and its velocity asymptotically approaches a finite terminal velocity.

• Linear Drag Model ($F_r\propto v$) vs. Quadratic Drag Model ($F_r\propto v^2$): The linear model is mathematically simpler and applies to low-speed, viscous fluid motion. The quadratic model is more accurate for higher-speed, turbulent motion (like a skydiver) but results in a more complex differential equation.

• High Mass vs. Low Mass (same $k$): An object with a higher mass has a larger time constant ($\tau =m/k$) and a higher terminal velocity ($v_T=mg/k$). It will take longer to approach its higher terminal speed.

Change and Continuity Over Time

• Baseline (t=0): The object is at rest ($v=0$), so the resistive force is zero. The net force is maximal ($F_{net}=mg$) and the initial acceleration is $a=g$.

• Change 1: As time progresses, the velocity increases, but at a continuously decreasing rate.

• Change 2: As velocity increases, the resistive force grows, causing the net force and acceleration to decrease exponentially.

• Continuity: The object's mass $m$, the drag coefficient $k$, and the gravitational acceleration $g$ are assumed to be constant throughout the motion.

Common Misconceptions & Clarifications

1. Misconception: An object with zero acceleration must be at rest.

   • Clarification: Zero acceleration implies zero net force, not necessarily zero velocity. An object moving at its constant terminal velocity has zero acceleration because the resistive force perfectly balances the gravitational force.

2. Misconception: The resistive force on a falling object is a constant value.

   • Clarification: The resistive force is a function of velocity ($F_r=kv$). It is zero when the object is at rest and increases as the object speeds up, only becoming constant when the object reaches terminal velocity.

3. Misconception: Heavier objects always fall faster than lighter objects.

   • Clarification: While a more massive object will have a higher terminal velocity (assuming the same shape and size, i.e., same $k$), its initial acceleration in the presence of air resistance is still $g$ (just like a lighter object). "Falling faster" is ambiguous; the heavier object accelerates for a longer duration and to a higher final speed, but it does not have a greater acceleration at all times.

4. Misconception: An object reaches terminal velocity at a specific moment in time.

   • Clarification: The velocity function $v(t)=v_T(1-e^{-t/\tau })$ is asymptotic. This means the velocity approaches $v_T$ as time goes to infinity but, in the mathematical model, never strictly reaches it in a finite amount of time. In practice, we consider the object to have reached terminal velocity when its speed is negligibly different from $v_T$.

One-Paragraph Summary

Resistive forces are velocity-dependent forces that oppose motion, commonly modeled for low speeds by the linear relation ${\vec{F}}_r=-k\vec{v}$. Applying Newton's second law to an object moving under a constant force (like gravity) and this resistive force yields a first-order linear differential equation. The solution to this equation reveals that the object's velocity does not increase indefinitely but instead approaches a constant maximum speed, the terminal velocity, where the driving and resistive forces balance. This behavior is characterized by a time constant, $\tau =m/k$, which dictates how quickly the terminal velocity is approached. This model provides a powerful, calculus-based framework for understanding realistic motion in fluids, explaining why falling objects have a maximum speed and how factors like mass and shape influence that speed.