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AP Course

AP Physics C: Mechanics

Updated for the AP Physics C: Mechanics framework effective fall 2024, with calculus-based derivations, labs, quizzes, and practice.

Build calculus-based mechanics with original derivations, experiments, quizzes, and practice sets.

Lessons
Scalars and VectorsDisplacement, Velocity, and AccelerationRepresenting MotionReference Frames and Relative MotionMotion in Two or Three DimensionsSystems and Center of MassForces and Free-Body DiagramsNewton’s Third LawNewton’s First LawNewton’s Second LawGravitational ForceKinetic and Static FrictionSpring ForcesResistive ForcesCircular MotionTranslational Kinetic EnergyWorkPotential EnergyConservation of EnergyPowerLinear MomentumChange in Momentum and ImpulseConservation of Linear MomentumElastic and Inelastic CollisionsRotational KinematicsConnecting Linear and Rotational MotionTorqueRotational InertiaRotational Equilibrium and Newton’s First Law in Rotational FormNewton’s Second Law in Rotational FormRotational Kinetic EnergyTorque and WorkAngular Momentum and Angular ImpulseConservation of Angular MomentumRollingMotion of Orbiting SatellitesDefining Simple Harmonic Motion (SHM)Frequency and Period of SHMRepresenting and Analyzing SHMEnergy of Simple Harmonic OscillatorsSimple and Physical Pendulums
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics C: Mechanics · Unit 2 · Reviewed representative lesson

Newton’s Second Law

Newton's second law states that the net external force equals the time rate of change of a system's momentum. The familiar constant-mass form \(\sum\vec F=m\vec a\) follows when the modeled mass does not change.

Definition and Key Formula

\[\sum\vec F_{\mathrm{external}}=\frac{d\vec p}{dt}\]

Momentum is \(\vec p=m\vec v\). For constant \(m\), \(d\vec p/dt=m\,d\vec v/dt=m\vec a\). The force and momentum components must be handled with a consistent coordinate system.

Integral Form

Integrating the net force over a time interval gives impulse and the corresponding momentum change.

\[\int_{t_i}^{t_f}\sum\vec F\,dt=\Delta\vec p\]

This form is the natural choice when force varies with time and a force–time function or graph is known.

Worked Example

Question: A \(2.0\,\mathrm{kg}\) cart starts from rest. The net force is \(F_x(t)=3.0t\,\mathrm N\) from \(t=0\) to \(4.0\,\mathrm s\). Find its final velocity.

\[\Delta p_x=\int_0^4 3.0t\,dt=\left.1.5t^2\right|_0^4=24\,\mathrm{N\,s}\]
\[v_f=\frac{\Delta p_x}{m}=\frac{24}{2.0}=12\,\mathrm{m\,s^{-1}}\]

Answer: The cart's final velocity is \(12\,\mathrm{m\,s^{-1}}\) in the positive x-direction.

Why This Method Was Chosen

The force is not constant, so multiplying one force value by the full time interval would be unjustified. Integration adds the continuously changing force contributions and gives the exact impulse for the model.

Common Mistakes

  • Using the final force as though it acted for the entire interval.
  • Confusing the area under a force–time graph with acceleration.
  • Dropping vector signs before calculating momentum change.
  • Using \(m\vec a\) without checking the constant-mass assumption.

Key Takeaways

  • The momentum form is the general starting point.
  • Derivative form connects instantaneous force and momentum rate.
  • Integral form connects impulse and finite momentum change.
Practice connection

Use the quiz and practice tabs to connect force functions, free-body diagrams, and differential equations of motion.

Official curriculum reference: College Board AP Physics C: Mechanics course page. The explanation and worked example are independently written for this study site.