AP Physics C Mechanics Calculus Versions Cheat Sheet

Key Facts

• Velocity and acceleration are derivatives of position: $v = \frac{dx}{dt}$ and $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$.
• Displacement and change in velocity can be found by integration: $\Delta x = \int_{t_1}^{t_2} v\,dt$ and $\Delta v = \int_{t_1}^{t_2} a\,dt$.
• Newton's second law in calculus form is $\sum \vec{F} = \frac{d\vec{p}}{dt}$, and for constant mass it becomes $\sum \vec{F} = m\vec{a}$.
• Work by a variable force is $W = \int_{x_1}^{x_2} F_x\,dx$, and the work-energy theorem is $W_{\text{net}} = \Delta K$.
• Linear momentum is $\vec{p} = m\vec{v}$, impulse is $\vec{J} = \int \vec{F}\,dt = \Delta \vec{p}$, and momentum is conserved when $\vec{F}_{\text{ext}} = 0$.
• Rotational motion follows $\tau_{\text{net}} = I\alpha$, $K_{\text{rot}} = \frac{1}{2}I\omega^2$, and $L = I\omega$ for rotation about a fixed axis.
• Newton's law of gravitation is $F_g = \frac{Gm_1m_2}{r^2}$, and gravitational potential energy for two masses is $U_g = -\frac{Gm_1m_2}{r}$.
• For simple harmonic motion, $a = -\omega^2x$, $x(t) = A\cos(\omega t + \phi)$, and for a spring-mass system $\omega = \sqrt{\frac{k}{m}}$.

Vocabulary

Derivative

A derivative gives the instantaneous rate of change of a quantity, such as $v = \frac{dx}{dt}$ for velocity.

Integral

An integral accumulates a changing quantity over an interval, such as $W = \int F_x\,dx$ for work.

Net Force

Net force is the vector sum of all forces on an object and determines its acceleration through $\sum \vec{F} = m\vec{a}$.

Conservative Force

A conservative force has path-independent work and can be related to potential energy by $F_x = -\frac{dU}{dx}$.

Torque

Torque measures the rotational effect of a force and is given by $\vec{\tau} = \vec{r} \times \vec{F}$.

Angular Momentum

Angular momentum describes rotational motion and is conserved when the net external torque is $\vec{\tau}_{\text{ext}} = 0$.