Physics C: Mechanics · 16 min read · Updated 2026-05-11
Newton's Laws (Calculus-Based) — AP Physics C: Mechanics
AP Physics C: Mechanics · College Board AP Physics C: Mechanics CED · 16 min read
1. Calculus Formulation of Newton's Second Law★★☆☆☆⏱ 3 min
The algebra-level form $\vec{F}_{net} = m\vec{a}$ is a special case of the general calculus-based formulation of Newton's Second Law, which explicitly defines acceleration as the time derivative of velocity. This form works for all non-relativistic motion, even when acceleration is not constant.
2. Variable Force Problem Solving★★★☆☆⏱ 5 min
Most real-world forces are non-constant: they can vary with time, position, or velocity. For variable forces, you cannot use constant-acceleration kinematics, and must instead rearrange Newton's Second Law and integrate to find velocity or position.
**Force as a function of time ($F(t)$)**: Rearrange to $\int_{v_0}^{v_f} dv = \frac{1}{m}\int_{t_0}^{t_f} F(t) dt$
**Force as a function of position ($F(x)$)**: Use the chain rule to rewrite $\frac{dv}{dt} = v\frac{dv}{dx}$, then rearrange to $\int_{v_0}^{v_f} v dv = \frac{1}{m}\int_{x_0}^{x_f} F(x) dx$
3. Drag Force and Terminal Velocity★★★☆☆⏱ 4 min
Drag is a resistive force exerted by a fluid on a moving object, always acting opposite to the object's velocity relative to the fluid. AP Physics C almost exclusively tests linear drag (proportional to velocity) for terminal velocity problems.
Terminal velocity occurs when net force on a falling object equals zero, so upward drag balances downward gravity, and acceleration becomes zero. For linear drag, this gives $v_t = \frac{mg}{b}$. Solving the differential equation for motion gives the time-dependent velocity for a dropped object: $v(t) = v_t\left(1 - e^{-bt/m}\right)$.
4. Free-Body Diagrams in Non-Inertial Frames★★★★☆⏱ 4 min
Non-inertial reference frames are frames that accelerate relative to an inertial (constant-velocity) frame like the ground. Newton's First Law does not hold in these frames unless you add fictitious forces: apparent forces that are not real interactions, but allow you to use $F=ma$ in the accelerating frame.
Draw all real forces first: gravity, normal force, friction, tension, drag, etc.
Add the fictitious force: $\vec{F}_{fict} = -m\vec{a}_{frame}$, where $\vec{a}_{frame}$ is the acceleration of the non-inertial frame relative to an inertial frame. The fictitious force always points opposite to the frame's acceleration.
In the non-inertial frame, net force equals $m$ times the object's acceleration *relative to the frame*.
5. Concept Check (AP-Style Practice)★★★★☆⏱ 4 min
Common Pitfalls
Why: Students default to memorized algebra-level equations from AP Physics 1, even when force is non-constant
Why: Students treat all force and velocity values as scalars, ignoring coordinate system direction
Why: Students confuse low-speed linear drag and high-speed quadratic drag cases
Why: Students do not adjust the integration variable for position-dependent forces
Why: Students do not confirm the reference frame type before drawing the FBD