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AP Physics C Mechanics Calculus Versions cheat sheet - grade 11-12

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Physics Grade 11-12

AP Physics C Mechanics Calculus Versions Cheat Sheet

A printable reference covering calculus-based kinematics, Newton's laws, work-energy, momentum, rotation, gravitation, and oscillations for grades 11-12.

TopicsAP PhysicsNewton's LawsForcesMotionEnergyMomentumGravityRotation
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This cheat sheet covers the calculus-based tools used in AP Physics C Mechanics, including motion, forces, energy, momentum, rotation, gravitation, and simple harmonic motion. Students need it because the course often asks for both physical reasoning and fast recognition of differential and integral relationships. A clear reference helps connect graphs, equations, and free-body diagrams to the correct method.

Key Facts

  • Velocity and acceleration are derivatives of position: v=dxdtv = \frac{dx}{dt} and a=dvdt=d2xdt2a = \frac{dv}{dt} = \frac{d^2x}{dt^2}.
  • Displacement and change in velocity can be found by integration: Δx=∫t1t2v dt\Delta x = \int_{t_1}^{t_2} v\,dt and Δv=∫t1t2a dt\Delta v = \int_{t_1}^{t_2} a\,dt.
  • Newton's second law in calculus form is ∑F⃗=dp⃗dt\sum \vec{F} = \frac{d\vec{p}}{dt}, and for constant mass it becomes ∑F⃗=ma⃗\sum \vec{F} = m\vec{a}.
  • Work by a variable force is W=∫x1x2Fx dxW = \int_{x_1}^{x_2} F_x\,dx, and the work-energy theorem is Wnet=ΔKW_{\text{net}} = \Delta K.
  • Linear momentum is p⃗=mv⃗\vec{p} = m\vec{v}, impulse is J⃗=∫F⃗ dt=Δp⃗\vec{J} = \int \vec{F}\,dt = \Delta \vec{p}, and momentum is conserved when F⃗ext=0\vec{F}_{\text{ext}} = 0.
  • Rotational motion follows τnet=Iα\tau_{\text{net}} = I\alpha, Krot=12Iω2K_{\text{rot}} = \frac{1}{2}I\omega^2, and L=IωL = I\omega for rotation about a fixed axis.
  • Newton's law of gravitation is Fg=Gm1m2r2F_g = \frac{Gm_1m_2}{r^2}, and gravitational potential energy for two masses is Ug=−Gm1m2rU_g = -\frac{Gm_1m_2}{r}.
  • For simple harmonic motion, a=−ω2xa = -\omega^2x, x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi), and for a spring-mass system ω=km\omega = \sqrt{\frac{k}{m}}.

Vocabulary

Derivative
A derivative gives the instantaneous rate of change of a quantity, such as v=dxdtv = \frac{dx}{dt} for velocity.
Integral
An integral accumulates a changing quantity over an interval, such as W=∫Fx dxW = \int F_x\,dx for work.
Net Force
Net force is the vector sum of all forces on an object and determines its acceleration through ∑F⃗=ma⃗\sum \vec{F} = m\vec{a}.
Conservative Force
A conservative force has path-independent work and can be related to potential energy by Fx=−dUdxF_x = -\frac{dU}{dx}.
Torque
Torque measures the rotational effect of a force and is given by τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}.
Angular Momentum
Angular momentum describes rotational motion and is conserved when the net external torque is τ⃗ext=0\vec{\tau}_{\text{ext}} = 0.

Common Mistakes to Avoid

  • Using constant-acceleration equations when aa is not constant is wrong because formulas like x=x0+v0t+12at2x = x_0 + v_0t + \frac{1}{2}at^2 only apply when acceleration is constant.
  • Forgetting vector directions in Newton's laws is wrong because ∑F⃗=ma⃗\sum \vec{F} = m\vec{a} must be applied separately in each coordinate direction.
  • Treating work as W=FdW = Fd for a variable force is wrong because changing forces require W=∫x1x2Fx dxW = \int_{x_1}^{x_2} F_x\,dx.
  • Mixing up torque and force is wrong because torque depends on both force and lever arm through τ=rFsin⁡θ\tau = rF\sin\theta.
  • Assuming mechanical energy is always conserved is wrong because nonconservative work changes mechanical energy according to Wnc=ΔK+ΔUW_{\text{nc}} = \Delta K + \Delta U.

Practice Questions

  1. 1 A particle moves along the xx-axis with velocity v(t)=3t2−4tv(t) = 3t^2 - 4t in m/s\text{m/s}. Find its displacement from t=0t = 0 to t=2 st = 2\,\text{s}.
  2. 2 A force on a block is given by Fx=5x2F_x = 5x^2 in newtons, where xx is in meters. Find the work done as the block moves from x=0x = 0 to x=3 mx = 3\,\text{m}.
  3. 3 A solid disk of mass M=2.0 kgM = 2.0\,\text{kg} and radius R=0.30 mR = 0.30\,\text{m} rotates about its center with angular speed ω=10 rad/s\omega = 10\,\text{rad/s}. Using I=12MR2I = \frac{1}{2}MR^2, find KrotK_{\text{rot}}.
  4. 4 A cart attached to a spring passes through equilibrium with maximum speed. Explain why its acceleration is 00 at that instant even though its speed is greatest.

Understanding AP Physics C Mechanics Calculus Versions

Calculus earns its place when a quantity changes continuously. A car can speed up at different rates during one trip. A spring pulls with a different strength at each position.

Gravity changes as a spacecraft moves farther from Earth. In these cases, an average value can hide important behavior. A derivative describes what is happening at one instant.

An integral combines many tiny changes over an interval. On graphs, this gives a useful visual rule. Slope tells how one quantity responds to another, while signed area tells accumulated change.

Students should always inspect graph axes before using either idea. The same shape can represent very different physics when the axes change.

Free body diagrams remain the starting point for most force problems, even when calculus is required later. Draw the object by itself. Include only forces acting on that object.

Then choose positive directions that make the motion easier to describe. Friction opposes relative slipping or the tendency to slip, not necessarily the direction an object is moving. Normal force is not automatically equal to weight.

On a curved track, the inward direction can change from point to point. A force equation in that direction often determines whether contact is maintained.

This matters for roller coasters, turning cars, elevators, and objects moving in vertical circles. The acceleration direction follows the net force, not the velocity.

Energy methods are especially useful when the path is complicated but the beginning and ending states are clear. They can avoid solving for time completely. Still, energy conservation requires careful system boundaries.

If a chosen system includes a spring and a block, spring energy can change within that system. If friction acts, mechanical energy may decrease because energy becomes internal thermal energy. That energy has not vanished.

Variable-force work is best understood as many tiny pushes over tiny distances. The area under a force versus position graph can be positive or negative.

A negative result means the force removes kinetic energy from the object. This is why braking distance depends strongly on speed.

Momentum is often the cleanest tool during a short collision or explosion. Internal forces can be large, but they occur in equal and opposite pairs within the selected system. External impulse is what can change the total momentum of that system.

This distinction explains recoil, airbags, catching a ball, and rocket motion. In rotation, students need the same discipline with direction and reference point. A force has greater turning effect when its line of action is farther from the axis.

Mass distribution matters too. Two objects with equal mass can be very different to spin if one has more mass far from the axis. For oscillations, the restoring effect must point toward equilibrium.

The negative direction in the acceleration relationship expresses that return tendency. At the endpoints, speed is zero but the restoring effect is largest.

At equilibrium, speed is largest while the restoring effect is zero. These checkpoints catch many sign errors.