# Resistive Forces

> AP Physics C: Mechanics · AP Physics C: Mechanics
> Source: https://www.owlsprep.com/study/ap-physics-c-mech-u2-resistive-forces/

This module covers velocity-dependent resistive forces, linear and quadratic drag regimes, terminal velocity derivation, and 1D kinematic solutions for motion under drag, aligned to AP Physics C exam requirements.

**Prerequisites:** [Newton's Second Law for 1D motion](https://www.owlsprep.com/study/ap-physics-c-mech-u2-newtons-second-law/); [First order separable differential equations](https://www.owlsprep.com/study/ap-physics-c-mech-math-differential-equations/)

## Learning objectives

- Distinguish between linear and quadratic drag force regimes for moving objects
- Derive terminal velocity expressions for objects under resistive force and gravitational acceleration
- Solve 1D kinematic equations for motion with velocity-dependent resistive forces
- Apply resistive force models to AP Physics C free response and multiple choice problems

## Resistive Force Regime Classification

Unlike idealized kinetic friction that has constant magnitude, resistive drag forces depend directly on the velocity of the object moving through a fluid. The scaling of drag with velocity is determined by the dimensionless Reynolds number, which describes the ratio of inertial to viscous forces in the fluid flow.

**Reynolds Number** — Dimensionless quantity that defines fluid flow regime, calculated as $Re = \frac{\rho L v}{\mu}$ where $L$ is object characteristic length, $\rho$ is fluid density, and $\mu$ is dynamic fluid viscosity

*Notation:* $Re$

*Example:* A 1mm raindrop at 0.1 m/s has $Re \approx 10$, falling in the linear drag regime

> **info**
>
> For $Re < 1$, drag scales linearly with velocity. For $Re > 1000$, drag scales quadratically with velocity. The intermediate regime between 1 and 1000 is rarely tested on AP exams.

**Worked example:** Classify the drag regime for each object: 1) A 10μm pollen grain falling at 0.002 m/s through air, 2) A 1m wide skydiver falling at 50 m/s through air

1. For the pollen grain: small size and very low speed produce Reynolds number ~0.01, so linear drag applies
2. For the skydiver: large size and high speed produce Reynolds number ~300,000, so quadratic drag applies

## Linear Drag and Terminal Velocity

For objects in the linear drag regime, the magnitude of the drag force is written as $F_d = bv$, where $b$ is a constant drag coefficient dependent on object shape and fluid viscosity. For a falling object, drag acts upward opposite to the downward gravitational force.

**Derivation:** Derive terminal velocity for a falling object under linear drag

*Starting from:* Newton's Second Law for vertical motion: $F_{net} = mg - bv = ma$

1. At terminal velocity, acceleration $a = 0$, so net force equals zero
2. $$mg - bv_t = 0$$
3. $$v_t = \frac{mg}{b}$$

*Conclusion:* Terminal velocity for linear drag is directly proportional to object mass

**Worked example:** Calculate the terminal velocity of a 0.0001 kg raindrop with linear drag coefficient $b = 2 \times 10^{-6} \text{ Ns/m}$

1. Substitute values into the linear terminal velocity formula
2. $$v_t = \frac{(0.0001 \text{ kg})(9.8 \text{ m/s}^2)}{2 \times 10^{-6} \text{ Ns/m}}$$
3. Final result: $v_t = 490 \text{ m/s}$? No, correct calculation gives $v_t = 490 \text{ m/s}$ is unrealistic, wait correct mass 0.00001 kg gives $v_t = 49 \text{ m/s}$ — no, correct value for tiny raindrop is ~0.5 m/s, so $b = 0.002 \text{ Ns/m}$, $v_t = 0.049 \text{ m/s}$

> **tip**
>
> AP exam problems almost always explicitly state if you should use linear or quadratic drag, so you do not need to calculate Reynolds number to choose the regime.

## Quadratic Drag for Macroscopic Objects

For all everyday objects moving through air at typical speeds, quadratic drag is the correct model. The drag force depends on the cross-sectional area of the object, fluid density, and the square of velocity.

$$F_d = \frac{1}{2} C_d \rho A v^2$$

**Worked example:** Derive the terminal velocity for a skydiver of mass 80 kg, cross-sectional area 0.7 m², drag coefficient 0.7, air density 1.2 kg/m³

1. Set net force equal to zero at terminal velocity: $mg = \frac{1}{2} C_d \rho A v_t^2$
2. $$v_t^2 = \frac{2mg}{C_d \rho A}$$
3. $$v_t = \sqrt{\frac{2(80)(9.8)}{(0.7)(1.2)(0.7)}} \approx \sqrt{2666} \approx 51.6 \text{ m/s}$$

**Exam command terms**

AP Physics C uses specific command terms for drag problems that define your required response:

- **Derive** — Show every algebraic step starting from a fundamental law like Newton's Second Law *(Derive the terminal velocity expression for quadratic drag)*

- **Show that** — Verify a given expression is correct, you cannot use the target expression as your starting point

## Velocity and Position as Functions of Time

Since acceleration is velocity-dependent for drag problems, you cannot use constant-acceleration kinematics. You must solve the separable first-order differential equation from Newton's Second Law.

**Derivation:** Derive v(t) for an object falling from rest under linear drag

*Starting from:* $mg - bv = m \frac{dv}{dt}$

1. Rearrange to separate variables: $\frac{dv}{v_t - v} = \frac{b}{m} dt$
2. Integrate from initial condition $v=0$ at $t=0$ to final $v$ at $t$
3. $$v(t) = v_t \left(1 - e^{-bt/m}\right)$$

*Conclusion:* Velocity asymptotically approaches terminal velocity as time increases

**Worked example:** Find the velocity of a linear drag object at $t = \frac{m}{b}$, one time constant after release

1. Substitute $t = m/b$ into the velocity expression
2. $$v\left(\frac{m}{b}\right) = v_t \left(1 - e^{-1}\right) \approx 0.63 v_t$$

**Check your understanding**

Test your understanding of linear drag kinematics:

1. At $t = 3 \tau$ where $\tau = m/b$, what fraction of terminal velocity has the object reached?

   - 0.95
   - 0.86
   - 0.99
   - 0.63

   *Why:* $1 - e^{-3} \approx 0.95$, so 95% of terminal velocity

## Common pitfalls

- **Wrong:** Using quadratic drag for low-speed tiny objects
  - Why it fails: Regimes are defined by Reynolds number, not arbitrary choice, and AP problems explicitly state which model to use
  - Correct: Only use the drag model explicitly specified in the exam question
- **Wrong:** Forgetting acceleration equals zero at terminal velocity
  - Why it fails: Students often incorrectly keep a non-zero net force when the object reaches constant maximum speed
  - Correct: Set $F_{net} = 0$ to solve for $v_t$ directly without integrating
- **Wrong:** Mixing up signs of drag force relative to velocity
  - Why it fails: Drag always opposes motion, so its sign flips if the object is moving upward vs downward
  - Correct: Define your coordinate system explicitly at the start of every problem
- **Wrong:** Canceling mass incorrectly in quadratic drag problems
  - Why it fails: Quadratic drag scales with area, so $v_t$ is proportional to $\sqrt{m}$, not linear in $m$ like linear drag
  - Correct: Keep mass in your algebra until you fully isolate the terminal velocity term
- **Wrong:** Trying to use constant acceleration kinematics for motion with drag
  - Why it fails: Acceleration is velocity-dependent, so it is never constant for any object experiencing drag
  - Correct: Solve the separable differential equation derived from Newton's Second Law

## Cheatsheet

| Drag Regime | Force Magnitude | Terminal Velocity | v(t) (fall from rest) |
| --- | --- | --- | --- |
| Linear (low speed, small objects) | $F_d = bv$ | $v_t = \frac{mg}{b}$ | $v(t) = v_t \left(1 - e^{-bt/m}\right)$ |
| Quadratic (high speed, macroscopic) | $F_d = \frac{1}{2} C_d \rho A v^2$ | $v_t = \sqrt{\frac{2mg}{C_d \rho A}}$ | $v(t) = v_t \tanh\left(\frac{gt}{v_t}\right)$ |

## What's next

Mastering resistive forces builds directly on your understanding of Newton's Second Law and differential equations, and it is a frequent high-weight topic on the AP Physics C: Mechanics FRQ section. This concept also connects to later units on momentum, energy, and orbital motion where frictional and non-conservative forces modify idealized kinematic predictions. You will often see resistive force problems paired with experimental data analysis, where you are asked to calculate drag coefficients from measured terminal velocity values. To reinforce your learning, move to the next sub-topics covering friction forces, non-conservative work, and circular motion under variable net force to build a full picture of real-world motion beyond the idealized no-friction textbook models.

- [Uniform and Non-Uniform Circular Motion](https://www.owlsprep.com/study/ap-physics-c-mech-u2-circular-motion/)

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