📚 AP Physics C: Formula Mind Map and Key Concepts Review | AP 物理C：公式导图与考点梳理
AP Physics C combines Mechanics and Electricity & Magnetism, demanding a firm grasp of calculus-based physics. A formula mind map links key equations, highlights common pitfalls, and reveals cross-topic connections. This guide organises essential formulas and concept checkpoints to strengthen your exam readiness.
AP 物理C融合力学与电磁学，需要扎实的微积分物理基础。公式导图能串联核心方程、揭示常见陷阱与跨专题联系。本文系统整理必备公式与考点，帮助你在考场上快速提取关键信息。
1. Kinematics | 运动学
Motion is described through position, velocity, and acceleration, with calculus providing the link. For constant acceleration, the kinematic equations below form the backbone of one-dimensional analysis. Projectile motion is treated as independent horizontal (constant velocity) and vertical (free fall) components.
运动由位置、速度、加速度描述，微积分是它们的纽带。匀加速运动的方程组构成一维分析的核心。抛体运动则分解为水平匀速和竖直自由落体两个独立分量。
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v = v₀ + at — velocity evolves linearly with time.
v = v₀ + at — 速度随时间线性变化。
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Δx = v₀t + ½ at² — displacement incorporates initial velocity and acceleration.
Δx = v₀t + ½ at² — 位移包含初速度和加速度的贡献。
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v² = v₀² + 2aΔx — eliminates time; handy for collision or stopping distance.
v² = v₀² + 2aΔx — 消去时间，常用于碰撞或刹车距离问题。
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Δx = ½(v + v₀)t — average velocity form.
Δx = ½(v + v₀)t — 平均速度形式。
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Calculus links: v = dx/dt, a = dv/dt = d²x/dt². Integral forms recover displacement from velocity.
微积分关联：v = dx/dt, a = dv/dt = d²x/dt²。积分形式由速度求位移。
2. Newton’s Laws and Forces | 牛顿定律与力
The three laws govern interactions. Free-body diagrams are essential to identify all forces. Common forces include weight, normal force, tension, friction, and spring force. Circular motion requires a net centripetal force pointing toward the centre.
三大定律主宰相互作用。受力图是识别所有力的关键。常见力有重力、法向力、张力、摩擦力与弹簧力。圆周运动需要指向圆心的净向心力。
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ΣF = ma — net force produces acceleration. Use components: ΣF_x = ma_x, ΣF_y = ma_y.
ΣF = ma — 合外力产生加速度。需分解：ΣF_x = ma_x, ΣF_y = ma_y。
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f_k = μ_k N (kinetic), f_s ≤ μ_s N (static maximum). Friction opposes relative motion.
f_k = μ_k N（动摩擦），f_s ≤ μ_s N（最大静摩擦）。摩擦力阻碍相对运动。
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F_spring = -kx — Hooke’s law, restoring force proportional to displacement.
F_spring = -kx — 胡克定律，回复力与位移成正比。
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F_c = mv²/r — centripetal force required for uniform circular motion; can be provided by tension, gravity, or normal force.
F_c = mv²/r — 匀速圆周运动的向心力；可由张力、重力或法向力提供。
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Atwood machines, inclined planes, and banked curves test force resolution. Remember to set up coordinates accordingly.
阿特伍德机、斜面与倾斜弯道经常考查力的分解，注意合理建立坐标系。
3. Work, Energy, and Power | 功、能与功率
The work-energy theorem links force to kinetic energy. Conservative forces allow potential energy functions, and mechanical energy is conserved when only conservative forces do work. Power is the rate of doing work.
功能定理将力与动能联系起来。保守力可定义势能函数；当只有保守力做功时机械能守恒。功率是做功的速率。
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W = ∫ F · dr — work by a force along a path. For constant force: W = Fd cosθ.
W = ∫ F · dr — 力沿路径做功。恒力时：W = Fd cosθ。
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W_net = ΔK = ½mv_f² – ½mv_i² — work-energy theorem.
W_net = ΔK = ½mv_f² – ½mv_i² — 功能定理。
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Potential energies: U_g = mgh (near surface), U_s = ½kx² (spring).
势能：U_g = mgh（近地表），U_s = ½kx²（弹簧）。
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Conservative force and potential: F = -dU/dx. Equilibrium occurs where dU/dx = 0.
保守力与势能：F = -dU/dx。平衡点满足 dU/dx = 0。
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P = dW/dt = F · v — instantaneous power. Average power P_avg = W/Δt.
P = dW/dt = F · v — 瞬时功率。平均功率 P_avg = W/Δt。
4. Momentum and Collisions | 动量与碰撞
Momentum is conserved in isolated systems, making it a powerful tool for collision problems. Impulse equals change in momentum. The centre of mass moves as if all mass were concentrated there.
动量在孤立系统中守恒，是处理碰撞问题的有力工具。冲量等于动量的变化。质心的运动如同所有质量集中于该点。
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p = mv — linear momentum vector. System momentum P = Σ m_i v_i.
p = mv — 线动量矢量。系统的总动量 P = Σ m_i v_i。
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J = ∫ F dt = Δp — impulse-momentum theorem. For average force: J = F_avg Δt.
J = ∫ F dt = Δp — 冲量-动量定理。平均力时：J = F_avg Δt。
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Conservation: Σ p_before = Σ p_after (no external net force).
动量守恒：Σ p_before = Σ p_after（无外净力）。
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Elastic collisions: kinetic energy conserved. For equal masses in 1D, velocities exchange. Inelastic: objects stick, K lost.
弹性碰撞：动能守恒。一维等质量碰撞时速度交换。完全非弹性：粘在一起，动能损失。
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Center of mass: r_cm = (Σ m_i r_i)/M. Velocity: v_cm = P/M. Newton’s second law for CM: ΣF_ext = M a_cm.
质心：r_cm = (Σ m_i r_i)/M。质心速度 v_cm = P/M。质心牛顿定律：ΣF_ext = M a_cm。
5. Rotational Kinematics and Dynamics | 转动运动学与动力学
Rotational motion mirrors linear motion with angular quantities. The key is to apply analogous equations while distinguishing tangential and centripetal accelerations. Moment of inertia depends on mass distribution.
转动与平动在数学上高度对称。关键是用角量建立相应方程，同时区分切向与向心加速度。转动惯量取决于质量分布。
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Angular definitions: ω = dθ/dt, α = dω/dt. Constant α formulas resemble linear ones.
角量定义：ω = dθ/dt, α = dω/dt。匀角加速度公式与直线类似。
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Linear-angular connection: s = rθ, v = rω, a_t = rα, a_c = v²/r = ω²r.
线量与角量关系：s = rθ, v = rω, a_t = rα, a_c = v²/r = ω²r。
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I = Σ m_i r_i² (discrete) or I = ∫ r² dm (continuous). Parallel axis: I = I_cm + Md².
转动惯量：I = Σ m_i r_i²（分立）或 I = ∫ r² dm（连续）。平行轴定理：I = I_cm + Md²。
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τ = r × F (magnitude rF sinθ). Newton’s second law for rotation: Στ = Iα.
力矩：τ = r × F（大小 rF sinθ）。转动牛顿第二定律：Στ = Iα。
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Angular momentum: L = Iω (rigid body), L = r × p (particle). Conservation: L_i = L_f when net torque is zero.
角动量：L = Iω（刚体），L = r × p（质点）。合外力矩为零时角动量守恒：L_i = L_f。
6. Gravitation and Orbits | 引力与轨道
Newton’s law of gravitation explains planetary motion and satellite orbits. The gravitational force is conservative, with potential energy negative at large distances. Kepler’s laws summarise orbital geometry.
牛顿引力定律解释行星运动与卫星轨道。引力是保守力，在遥远距离势能为负。开普勒定律概括轨道几何特征。
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F_g = G Mm / r² (attractive, along the line joining centres). Vector form: F = – (GMm/r²) r̂.
F_g = G Mm / r²（吸引，沿中心连线）。矢量形式：F = – (GMm/r²) r̂.
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U_g = -GMm/r — zero at infinite separation. Total mechanical energy E = -GMm/(2a) for elliptical orbits.
U_g = -GMm/r — 无穷远处为零势能。椭圆轨道总机械能 E = -GMm/(2a)。
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Orbital speed (circular): v = √(GM/r). Period: T² = (4π²/GM) r³ — Kepler’s third law.
圆轨道速度：v = √(GM/r)。周期：T² = (4π²/GM) r³ — 开普勒第三定律。
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