Calculus-Based Motion

• Velocity: \( v = \frac{dx}{dt} \)
• Acceleration: \( a = \frac{dv}{dt} = \frac{d^2x}{dt^2} \)
• Average velocity: \( v_{\text{avg}} = \frac{\Delta x}{\Delta t} \)

Constant Acceleration Equations

• \( v = v_0 + at \)
• \( x = x_0 + v_0 t + \frac{1}{2}at^2 \)
• \( v^2 = v_0^2 + 2a(x – x_0) \)

• Newton’s 2nd Law: \( \Sigma F = ma \)
• Momentum form: \( F = \frac{dp}{dt} \)
• Weight: \( F_g = mg \)
• Friction: \( f = \mu N \)
• Hooke’s Law: \( F_s = -kx \)

• Work (calculus form): \( W = \int F \cdot dx \)
• Kinetic energy: \( KE = \frac{1}{2}mv^2 \)
• Gravitational PE: \( U_g = mgy \)
• Spring PE: \( U_s = \frac{1}{2}kx^2 \)
• Power: \( P = \frac{dW}{dt} = F \cdot v \)
• Work-Energy Theorem: \( W_{\text{net}} = \Delta KE \)

• Momentum: \( p = mv \)
• Impulse: \( J = \int F \, dt = \Delta p \)
• Conservation: \( \Sigma p_{\text{initial}} = \Sigma p_{\text{final}} \)
• Center of mass: \( r_{\text{cm}} = \frac{\Sigma m_i r_i}{\Sigma m_i} \)

• Torque: \( \tau = r \times F = rF\sin\theta \)
• Net torque: \( \Sigma\tau = I\alpha \)
• Moment of inertia: \( I = \Sigma m_i r_i^2 \)
• Angular momentum: \( L = I\omega \)
• Torque relation: \( \tau = \frac{dL}{dt} \)
• Rolling without slipping: \( v = \omega r \)

• Rotational KE: \( KE_{\text{rot}} = \frac{1}{2}I\omega^2 \)
• Total KE (rolling): \( KE_{\text{total}} = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2 \)
• Angular momentum (particle): \( L = r \times p \)
• Angular momentum (rigid body): \( L = I\omega \)
• Conservation: \( L_{\text{initial}} = L_{\text{final}} \)

• Hooke’s Law: \( F = -kx \)
• Period of spring: \( T = 2\pi\sqrt{\frac{m}{k}} \)
• Period of pendulum (small angle): \( T = 2\pi\sqrt{\frac{L}{g}} \)
• SHM general solution: \( x(t) = A\cos(\omega t + \phi) \)
• Angular frequency (spring): \( \omega = \sqrt{\frac{k}{m}} \)
• Angular frequency (pendulum): \( \omega = \sqrt{\frac{g}{L}} \)

• Acceleration due to gravity: \( g = 9.8 \, \text{m/s}^2 \)
• Pi: \( \pi \approx 3.1416 \)
• Gravitational constant: \( G = 6.67 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2 \)
• Mass of Earth: \( M_E = 5.97 \times 10^{24} \, \text{kg} \)
• Radius of Earth: \( R_E = 6.37 \times 10^6 \, \text{m} \)

Q1: Do I need to memorize all these formulas?

Not all of them. The College Board provides an equation sheet during the exam, but you should memorize the most common formulas (Units 1-4) to save time. Focus on understanding when to use each formula rather than rote memorization.

Q2: What’s the difference between the calculus forms and regular forms?

Calculus forms (like \( W = \int F \cdot dx \)) handle variable forces and non-uniform motion precisely. Regular forms (like \( W = Fd\cos\theta \)) only work for constant forces. Physics C requires you to know when to use each.

Q3: How are rotational formulas related to linear formulas?

They’re direct analogs! Replace: \( x \rightarrow \theta \), \( v \rightarrow \omega \), \( a \rightarrow \alpha \), \( m \rightarrow I \), \( F \rightarrow \tau \), \( p \rightarrow L \). Understanding this pattern helps you remember twice as many formulas with half the effort.

Q4: Which units are tested most heavily on the AP exam?

Units 3-6 (Energy, Momentum, and Rotation) make up about 60-70% of the exam. Kinematics and oscillations are important but appear less frequently. Focus your study time accordingly.

Q5: What if I forget a formula during the exam?

Derive it! Most formulas can be derived from fundamental principles. For example, if you forget the period of a pendulum, you can derive it from \( F = -mg\sin\theta \approx -mg\theta \) and SHM equations. Show your work for partial credit.

✍️ Written by Dr. Sarah Mitchell

Ph.D. in Molecular Biology, Stanford University | 15+ years teaching AP Sciences

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Dr. Sarah Mitchell holds a Ph.D. in Molecular Biology from Stanford University and has been teaching AP Sciences for over 15 years. As a former AP Biology exam reader for the College Board, she brings insider knowledge of scoring rubrics and exam expectations. Her students consistently achieve a 95% pass rate with over 70% scoring 4s and 5s.