Introduction to Resistive Forces

In our previous chapters, we often ignored air resistance to keep things simple. However, in the real world, objects moving through fluids (like air or water) experience a "drag" that opposes their motion. This is a resistive force. Understanding these forces is a major step up in AP Physics C: Mechanics because it requires us to use calculus to describe how an object's velocity changes over time. Don't worry if the math seems intimidating at first—we will break down the steps together!

What is a Resistive Force?

A resistive force, often denoted as \( F_R \), is a force that acts in the opposite direction of an object's velocity. Unlike kinetic friction, which is generally constant regardless of speed, resistive forces depend on how fast the object is moving.

The two most common mathematical models you will see are:

  1. Linear Resistance: \( F_R = -bv \) (Usually for slow-moving objects or thick fluids).
  2. Quadratic Resistance: \( F_R = -cv^2 \) (Usually for fast-moving objects in air).

In these equations, \( b \) and \( c \) are constants that depend on the shape of the object and the properties of the fluid. The negative sign indicates the force opposes the motion.

Did you know? Parachutes work because they increase the surface area of a falling person, which significantly increases the constant \( c \) in the resistive force equation, allowing for a much slower and safer landing.

Newton's Second Law and Differential Equations

In Unit 2, we learned that \( \sum F = ma \). Since acceleration \( a \) is the derivative of velocity with respect to time (\( a = \frac{dv}{dt} \)), we can write a differential equation for an object falling under the influence of gravity and air resistance.

Imagine an object of mass \( m \) dropped from rest. The forces acting on it are gravity (downward) and the resistive force (upward). Using the linear model \( F_R = -bv \):

\( \sum F = mg - bv = m \frac{dv}{dt} \)

Important FBD Rule: On the AP Exam, when drawing a Free-Body Diagram (FBD) for this scenario, you must draw two separate arrows originating from a single dot: one pointing down for \( mg \) and one pointing up for \( F_R \). Never draw component arrows or net force arrows on an FBD!

Solving for Velocity: The Calculus Journey

To find the velocity as a function of time \( v(t) \), we must use a technique called separation of variables. Here is the step-by-step process for the linear case:

Step 1: Set up the equation

\( m \frac{dv}{dt} = mg - bv \)

Step 2: Separate the variables

Move all the \( v \) terms to one side and the \( t \) terms to the other:

\( \frac{dv}{mg - bv} = \frac{1}{m} dt \)

Step 3: Integrate both sides

We integrate from \( v = 0 \) to \( v \) and \( t = 0 \) to \( t \):

\( \int_{0}^{v} \frac{dv'}{mg - bv'} = \int_{0}^{t} \frac{1}{m} dt' \)

Step 4: Solve the integral

Using u-substitution (where \( u = mg - bv \)), the left side becomes a natural log:

\( -\frac{1}{b} \ln(mg - bv) \Big|_0^v = \frac{1}{m} t \)

\( \ln\left(\frac{mg - bv}{mg}\right) = -\frac{b}{m} t \)

Step 5: Exponentiate and Isolate \( v \)

\( \frac{mg - bv}{mg} = e^{-\frac{b}{m}t} \)

\( v(t) = \frac{mg}{b} (1 - e^{-\frac{b}{m}t}) \)

Key Takeaway: The term \( \frac{m}{b} \) is often called the time constant. It determines how quickly the object reaches its final speed. After a long time, the exponential term goes to zero, and the velocity levels off.

Terminal Velocity

Terminal velocity (\( v_T \)) is the constant speed reached by a falling object when the resistive force eventually becomes equal in magnitude to the gravitational force. At this point, the net force is zero and the acceleration is zero.

You don't need calculus to find the terminal velocity! Just set \( a = 0 \) in Newton's Second Law:

\( \sum F = 0 \implies mg - F_R = 0 \)

For the linear model (\( F_R = bv \)):
\( mg - bv_T = 0 \implies v_T = \frac{mg}{b} \)

For the quadratic model (\( F_R = cv^2 \)):
\( mg - cv_T^2 = 0 \implies v_T = \sqrt{\frac{mg}{c}} \)

Visualizing the Motion: Graphs

Understanding the qualitative behavior of these graphs is essential for Science Practice 1.C (creating qualitative sketches).

1. Velocity vs. Time Graph

The graph starts at the origin (\( 0,0 \)) and curves upward, gradually flattening out (approaching an asymptote) at the value of terminal velocity. The slope is always positive but decreasing.

2. Acceleration vs. Time Graph

The graph starts at its maximum value (\( g = 10 \, \text{m/s}^2 \)) and curves downward toward zero. This is because as the object speeds up, the resistive force increases, which reduces the net force and therefore reduces the acceleration.

Quick Tip: If the problem says "an object is thrown downward faster than terminal velocity," the resistive force will be greater than gravity. In that case, the object will actually slow down until it reaches terminal velocity!

Common Mistakes to Avoid

  • Confusing \( g \) and \( a \): Remember that \( g \) is the constant gravitational field strength (\( 10 \, \text{m/s}^2 \)), but the actual acceleration \( a \) of the object changes as the resistive force changes.
  • Forgetting the Chain Rule/U-Substitution: When integrating \( \frac{1}{mg-bv} \), many students forget the factor of \( -\frac{1}{b} \). This is a critical step!
  • Incorrect FBDs: Do not include "Net Force" on your diagram. Only draw the actual physical forces (\( F_g \) and \( F_R \)).
  • Sign Errors: Always define a coordinate system (e.g., downward is positive) and be consistent throughout your derivation.

Quick Review Summary

  • Resistive forces oppose motion and usually depend on velocity (\( v \) or \( v^2 \)).
  • Newton's Second Law leads to a separable differential equation because \( a = \frac{dv}{dt} \).
  • Terminal velocity occurs when \( F_{net} = 0 \), meaning the resistive force equals the driving force (like gravity).
  • Calculus is required to find the exact function of velocity over time, but the asymptote of the velocity graph is always the terminal velocity.