📚 AP Physics C Formula Mind Map | AP物理C公式导图整理
Mastering AP Physics C demands a clear, interconnected understanding of formulas that span mechanics and electromagnetism. This mind map distills core equations into organized categories, emphasizing how differential and integral forms link kinematics, forces, energy, and fields. Use these tables to reinforce your revision, spot relationships between topics, and build the confidence to derive any result on exam day.
掌握AP物理C需要清晰、相互关联地理解跨越力学和电磁学的公式。这张导图将核心方程提炼成有条理的分类，强调微分和积分形式如何将运动学、力、能量和场联系在一起。利用这些表格巩固复习，发现主题之间的关联，并建立起在考试当天推导任何结果的信心。
1. Kinematics | 运动学
Kinematics describes motion without reference to its causes. In AP Physics C, you must be comfortable with calculus-based relationships between position, velocity, and acceleration.
运动学在不涉及原因的情况下描述运动。在AP物理C中，必须熟练掌握位置、速度和加速度之间基于微积分的关系。
| Formula | Explanation |
|---|---|
| v = dx/dt |
Instantaneous velocity is the time derivative of position. 瞬时速度是位置对时间的导数。 |
| a = dv/dt = d²x/dt² |
Acceleration is the first derivative of velocity or second derivative of position. 加速度是速度的一阶导数或位置的二阶导数。 |
| v = v₀ + ∫ a dt |
Velocity from acceleration when it may vary with time. 当加速度可能随时间变化时，速度由加速度积分得到。 |
| v = v₀ + a t |
Velocity under constant acceleration (uniform). 匀加速条件下的速度。 |
| x = x₀ + v₀ t + ½ a t² |
Position for constant acceleration motion. 匀加速运动的位移公式。 |
| v² = v₀² + 2 a Δx |
Time-independent kinematics equation. 不显含时间的运动学方程。 |
2. Newton’s Laws & Forces | 牛顿定律与力
Newton’s laws form the foundation of classical mechanics. Students must apply free-body diagrams and vector sums to solve for unknown forces and motion.
牛顿定律构成了经典力学的基础。学生必须运用受力图和向量求和来求解未知力和运动。
| Formula | Explanation |
|---|---|
| ΣF⃗ = m a⃗ |
Newton’s second law; net force causes acceleration. 牛顿第二定律；合力产生加速度。 |
| F_spring = -k x |
Hooke’s law: restoring force proportional to displacement. 胡克定律：回复力与位移成正比。 |
| f_s ≤ μ_s N, f_k = μ_k N |
Static friction (inequality) and kinetic friction (equality). 静摩擦力（不等式）和滑动摩擦力（等式）。 |
| F_drag = ½ ρ C_d A v² (approx) |
Air resistance magnitude; direction opposite velocity. 空气阻力大小；方向与速度相反。 |
3. Work, Energy & Power | 功、能与功率
Work–energy theorem and conservation of mechanical energy are powerful tools. Recognizing when non-conservative forces do work is essential.
功能定理和机械能守恒是强大的工具。识别非保守力何时做功至关重要。
| Formula | Explanation |
|---|---|
| W = ∫ F⃗ ⋅ d r⃗ |
Work done by a force along a path. 力沿路径所做的功。 |
| K = ½ m v² |
Translational kinetic energy. 平动动能。 |
| U_g = m g y |
Gravitational potential energy near Earth’s surface. 地表附近的重力势能。 |
| U_s = ½ k x² |
Elastic potential energy stored in a spring. 弹簧储存的弹性势能。 |
| E_mech = K + U |
Total mechanical energy; conserved if only conservative forces do work. 总机械能；若只有保守力做功则守恒。 |
| P = dW/dt = F⃗ ⋅ v⃗ |
Instantaneous power. 瞬时功率。 |
4. Momentum & Collisions | 动量与碰撞
Linear momentum and impulse introduce vector conservation laws that are especially useful for collision and explosion problems.
线动量和冲量引入了矢量守恒定律，对于碰撞和爆炸问题特别有用。
| Formula | Explanation |
|---|---|
| p⃗ = m v⃗ |
Linear momentum of a particle. 质点的线动量。 |
| J⃗ = ∫ F⃗ dt = Δp⃗ |
Impulse equals change in momentum. 冲量等于动量的变化。 |
| Σp_initial = Σp_final |
Conservation of linear momentum (no external forces). 线动量守恒（无外力）。 |
| e = (v₂’ – v₁’) / (v₁ – v₂) |
Coefficient of restitution; e=1 for elastic collisions. 恢复系数；弹性碰撞 e=1。 |
5. Rotational Motion | 转动
Rotational kinematics and dynamics mirror translational equations, with moment of inertia replacing mass and torque replacing force.
转动运动学和动力学与平动方程类似，转动惯量代替质量，力矩代替力。
| Formula | Explanation |
|---|---|
| ω = dθ/dt, α = dω/dt |
Angular velocity and angular acceleration. 角速度和角加速度。 |
| ω = ω₀ + α t, θ = θ₀ + ω₀ t + ½ α t² |
Constant angular acceleration equations. 匀角加速度运动方程。 |
| I = ∫ r² dm |
Moment of inertia definition. 转动惯量的定义。 |
| τ⃗ = r⃗ × F⃗, Στ = I α |
Torque and rotational second law. 力矩与转动的第二定律。 |
| K_rot = ½ I ω² |
Rotational kinetic energy. 转动动能。 |
| L⃗ = I ω⃗ ; τ⃗ = dL⃗/dt |
Angular momentum and its relationship to net torque. 角动量及其与合外力矩的关系。 |
6. Gravitation | 万有引力
Newton’s law of universal gravitation applies to point masses. For extended bodies, gravitational potential energy and field are central.
牛顿万有引力定律适用于质点。对于扩展物体，引力势能和引力场是核心。
| Formula | Explanation |
|---|---|
| F_g = G m₁ m₂ / r² |
Magnitude of gravitational force between two masses. 两质量之间的引力大小。 |
| U_g = – G m₁ m₂ / r |
Gravitational potential energy for point masses. 质点的引力势能。 |
| g = G M / r² |
Gravitational field strength at distance r from mass M. 距离质量M为r处的引力场强度。 |
| v_orb = √(G M / r) ; T² = (4π²/GM) a³ |
Orbital speed and Kepler’s third law (a = semi-major axis). 轨道速度和开普勒第三定律（a为半长轴）。 |
7. Simple Harmonic Motion | 简谐运动
SHM occurs when restoring force is proportional to displacement. The kinematics are sinusoidal, and energy continually interchanges between kinetic and potential.
当回复力与位移成正比时，发生简谐运动。运动学呈正弦变化，能量在动能和势能间持续转换。
| Formula | Explanation |
|---|---|
| a = – ω² x |
Condition for SHM; acceleration proportional to -x. SHM条件；加速度与-x成正比。 |
| ω = √(k/m) [spring]; ω = √(g/L) [pendulum] |
Angular frequency for mass-spring and simple pendulum (small angles). 弹簧振子和单摆（小角度）的角频率。 |
| x(t) = A cos(ω t + φ) |
Position function for SHM; A is amplitude. SHM的位移函数；A为振幅。 |
| v = -A ω sin(ωt+φ), a = -A ω² cos(ωt+φ) |
Velocity and acceleration as time derivatives. 速度和加速度作为时间导数。 |
| T = 2π/ω = 2π √(m/k) or 2π √(L/g) |
Period of oscillation. 振动周期。 |
| E_total = ½ k A² |
Total mechanical energy of an undamped spring oscillator. 无阻尼弹簧振子的总机械能。 |
8. Electric Fields & Potential | 电场与电势
Electrostatics describes forces, fields, and potential energy arising from stationary charges. Calculus helps move between field and potential.
静电学描述静止电荷产生的力、场和势能。微积分有助于在场和电势之间转换。
| Formula | Explanation |
|---|---|
| F⃗ = k q₁ q₂ / r² r̂ |
Coulomb’s law (k = 1/(4πε₀)). 库仑定律。 |
| E⃗ = F⃗ / q = k q / r² r̂ |
Electric field of a point charge. 点电荷的电场。 |
| ΔV = – ∫ E⃗ ⋅ d l⃗ |
Potential difference related to electric field. 电势差与电场的关系。 |
| U_e = q V = k q₁ q₂ / r |
Electric potential energy of a charge in a potential. 电荷在电势中的电势能。 |
| E_plate = σ / (2ε₀), V_capacitor = E d |
Field between parallel plates; relation to voltage. 平行板间的电场；与电压的关系。 |
9. Circuits & Capacitors | 电路与电容
Direct-current circuits combine Ohm’s law, Kirchhoff’s rules, and capacitor behavior. Time-varying circuits with RC elements are also tested.
直流电路结合了欧姆定律、基尔霍夫定律和电容器行为。含RC元件的时变电路也在考查范围。
| Formula | Explanation |
|---|---|
| V = I R |
Ohm’s law. 欧姆定律。 |
| P = I V = I² R = V² / R |
Electric power dissipated in a resistor. 电阻消耗的电功率。 |
| C = Q / V ; C_parallel-plate = ε₀ A / d |
Capacitance definition and parallel-plate formula. 电容定义及平行板电容公式。 |
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