📚 AP Physics 1&2: Formula Mind Map & Exam Content Review | AP 物理1&2：公式导图与考点梳理
Welcome to the ultimate formula mind map for AP Physics 1 & 2. This article connects every key equation to its core concept, helping you visualize the big picture and nail those free-response questions. You will find essential formulas organized by topic, along with concise exam tips that highlight common pitfalls and shortcuts.
欢迎查看 AP 物理1&2 的终极公式导图。本文将每一个关键方程与其核心概念联系起来，帮助你构建整体框架，轻松应对自由作答题。我们按专题整理了必备公式，并提供简洁的考试提示，让你避开常见陷阱并掌握解题捷径。
1. Kinematics | 运动学
Kinematics describes motion without regard to its causes. The three fundamental equations for constant acceleration form the backbone of motion analysis.
运动学描述运动本身，不涉及引起运动的原因。三个匀变速直线运动基本方程是分析运动的核心。
v = v₀ + at
Δx = v₀t + ½at²
v² = v₀² + 2aΔx
- Use the first equation when time and acceleration are known, but displacement is not required.
- 当已知时间和加速度而不需要位移时，使用第一式。
- The second equation gives displacement directly; remember the ½ factor for constant acceleration from rest.
- 第二式直接给出位移；从静止开始的匀加速运动务必保留系数½。
- The third equation avoids time; it is ideal for problems that give velocities and displacement.
- 第三式避开时间变量，最适合已知初、末速度和位移的问题。
- Free fall uses a = g = 9.8 m/s² downward. Replace a with –g if upward is positive.
- 自由落体使用 a = g = 9.8 m/s² 向下。若取向上为正方向，则 a = –g。
| Equation | Missing Variable |
|---|---|
| v = v₀ + at | Δx |
| Δx = v₀t + ½at² | v |
| v² = v₀² + 2aΔx | t |
For projectile motion, split the problem into horizontal (constant velocity) and vertical (constant acceleration) components. The time of flight is determined by the vertical motion.
对于抛体运动，将问题分解为水平（匀速）和竖直（匀变速）两个分量。飞行时间由竖直运动决定。
2. Dynamics & Newton’s Laws | 动力学与牛顿定律
Newton’s laws relate force, mass, and acceleration. The net force determines the motion of a system.
牛顿定律将力、质量和加速度联系起来。合外力决定系统的运动。
ΣF = ma
Always draw a free-body diagram before applying ΣF = ma. Sum forces along each axis independently; if the object is in equilibrium, ΣF = 0.
应用 ΣF = ma 之前一定要画受力分析图。沿各轴分别求合力；如果物体处于平衡状态，则 ΣF = 0。
Common forces include weight (Fg = mg), normal force (N), tension (T), friction (f ≤ μN), and spring force (Fs = –kx). Friction direction opposes relative motion or attempted motion.
常见的力有重力 (Fg = mg)、支持力 (N)、绳拉力 (T)、摩擦力 (f ≤ μN) 和弹簧力 (Fs = –kx)。摩擦力方向与相对运动或运动趋势相反。
fsₘₐₓ = μₛN, fₖ = μₖN
An object on an inclined plane experiences a component of gravity down the slope equal to mg sin θ and a normal force of mg cos θ.
斜面上的物体受到沿斜面向下的重力分量为 mg sin θ，支持力为 mg cos θ。
3. Circular Motion & Gravitation | 圆周运动与万有引力
Uniform circular motion requires a net centripetal force pointing toward the center. The centripetal acceleration arises from a change in direction, not speed.
匀速圆周运动需要一个指向圆心的净向心力。向心加速度源于速度方向的变化，而非大小。
a = v²/r, Fc = mv²/r
Gravity provides the centripetal force for orbiting bodies. Newton’s law of universal gravitation states:
引力为环绕天体提供了向心力。万有引力定律为：
F = Gm₁m₂/r²
At the surface of a planet, the gravitational field strength g is derived from GM/R². The acceleration due to gravity decreases with altitude.
行星表面的引力场强度 g 由 GM/R² 导出。重力加速度随高度增加而减小。
For objects moving in vertical circles, the speed is not constant; energy conservation often combines with centripetal force at the top and bottom.
物体在竖直面内做圆周运动时速率不恒定；通常需要结合能量守恒分析最高点和最低点的向心力。
4. Work, Energy & Power | 功、能量与功率
Work done by a constant force is the product of the force component along displacement and the displacement itself. The work–energy theorem connects net work to the change in kinetic energy.
恒力做功等于力沿位移方向的分量与位移的乘积。功能定理将合力做功与动能变化联系起来。
W = Fd cos θ, Wₙₑₜ = ΔK = ½mv² − ½mv₀²
Conservative forces (gravity, spring force, electrostatic) have associated potential energies. Mechanical energy is conserved when only conservative forces do work.
保守力（重力、弹簧力、静电力）具有相应的势能。当只有保守力做功时，机械能守恒。
Ug = mgh, Us = ½kx²
Non-conservative forces (friction, air resistance) change mechanical energy into thermal energy. The work done by non-conservative forces equals the change in mechanical energy.
非保守力（摩擦力、空气阻力）将机械能转化为内能。非保守力做功等于机械能的变化量。
Power is the rate of doing work, also expressed as force times velocity when the force is constant and parallel to velocity.
功率是做功的快慢，当力恒定且与速度平行时也可表示为力乘速度。
P = W/t = Fv
5. Momentum & Impulse | 动量与冲量
Linear momentum is a vector quantity defined as mass times velocity. Impulse delivered by a net force changes momentum.
线动量是矢量，定义为质量与速度的乘积。合力提供的冲量改变动量。
p = mv, J = FΔt = Δp
The area under a force–time graph gives the impulse. In collisions, momentum of an isolated system is conserved, but kinetic energy may not be.
力–时间图像下的面积表示冲量。碰撞过程中，孤立系统的动量守恒，但动能可能不守恒。
m₁v₁ + m₂v₂ = m₁v₁’ + m₂v₂’
Perfectly elastic collisions conserve both momentum and kinetic energy. Perfectly inelastic collisions involve objects sticking together, maximizing energy loss.
完全弹性碰撞同时守恒动量和动能。完全非弹性碰撞中物体粘合在一起，能量损失最大。
For two-dimensional collisions, break momentum into perpendicular components and apply conservation along each axis.
对于二维碰撞，将动量分解为相互垂直的分量，分别沿各轴应用守恒定律。
6. Simple Harmonic Motion | 简谐运动
SHM occurs when the restoring force is proportional to displacement and points toward equilibrium. The mass–spring system and simple pendulum are canonical examples.
当回复力与位移成正比且指向平衡位置时，发生简谐运动。弹簧–振子和单摆是经典实例。
F = –kx, Tₛ = 2π√(m/k), Tₚ = 2π√(L/g)
Amplitude does not affect the period for ideal springs and pendulums (small angle approximation). The angular frequency ω = 2π/T = √(k/m) for a spring system.
对于理想弹簧和单摆（小角度近似），振幅不影响周期。弹簧系统的角频率 ω = 2π/T = √(k/m)。
Energy in SHM continuously transforms between kinetic energy and potential energy. The total energy is constant and equals ½kA².
简谐运动中的能量不断在动能和势能之间转化。总能量恒定，等于 ½kA²。
Eₜₒₜ = ½kA² = ½mvₘₐₓ²
The position, velocity, and acceleration as functions of time: x = A cos(ωt + φ), v = –ωA sin(ωt + φ), a = –ω²A cos(ωt + φ).
位移、速度和加速度随时间变化：x = A cos(ωt + φ)，v = –ωA sin(ωt + φ)，a = –ω²A cos(ωt + φ)。
7. Torque & Rotational Motion | 扭矩与旋转运动
Torque produces rotational acceleration. Its magnitude depends on force, lever arm, and the angle between them. Use the right-hand rule for direction.
力矩产生角加速度，其大小取决于力、力臂及其夹角。方向由右手定则确定。
τ = rF sin θ = r⊥F
Rotational kinematics mirrors linear kinematics with analogous variables: θ ↔ x, ω ↔ v, α ↔ a.
转动运动学与线运动学对应：θ 对应 x，ω 对应 v，α 对应 a。
ω = ω₀ + αt, Δθ = ω₀t + ½αt², ω² = ω₀² + 2αΔθ
Newton’s second law for rotation: net torque equals moment of inertia times angular acceleration.
转动情况下的牛顿第二定律：合外力矩等于转动惯量乘以角加速度。
Στ = Iα
Moment of inertia depends on mass distribution. Common formulas: I = mr² for a point mass, I = ½MR² for a solid cylinder, I = ⅖MR² for a solid sphere.
转动惯量取决于质量分布。常见公式：点质量 I = mr²，实心圆柱 I = ½MR²，实心球 I = ⅖MR²。
Angular momentum L = Iω is conserved when no external net torque acts. This explains the spinning skater pulling arms in.
角动量 L = Iω 在没有外力矩时守恒，这解释了花样滑冰运动员收臂时转速加快的现象。
8. Mechanical Waves & Sound | 机械波与声
Waves transfer energy without transferring matter. The relationship among speed, frequency, and wavelength holds for all periodic waves.
波传递能量而不传递物质。波速、频率和波长的关系适用于所有周期性波动。
v = fλ
Transverse waves have oscillations perpendicular to propagation (string waves), while longitudinal waves have oscillations parallel (sound).
横波的振动方向与传播方向垂直（绳波），纵波的振动方向与传播方向平行（声波）。
For a standing wave on a string fixed at both ends, the wavelength of the nth harmonic is λₙ = 2L/n, and frequency fₙ = nv/(2L).
对于两端固定的绳上的驻波，第 n 次谐波的波长为 λₙ = 2L/n，频率 fₙ = nv/(2L)。
Sound intensity in decibels is logarithmic: β = 10 log(I/I₀), where I₀ = 10⁻¹² W/m². Beat frequency equals the absolute difference between two close frequencies.
声强级用分贝表示，为对数尺度：β = 10 log(I/I₀)，其中 I₀ = 10⁻¹² W/m²。拍频等于两个相近频率之差的绝对值。
The Doppler effect shifts observed frequency when source and observer move relative to the medium. Approaching increases the frequency.
当波源和观察者相对于介质运动时，多普勒效应使接收频率发生偏移。相互靠近则频率升高。
9. Electrostatics & DC Circuits (AP1) | 静电学与直流电路 (AP物理1)
Electric charge is conserved and quantized. Coulomb’s law gives the force between two point charges.
电荷守恒且量子化。库仑定律给出两点电荷之间的作用力。
F = k|q₁q₂|/r²
Electric field E = F/q points away from positive charges. The potential difference (voltage) V = ΔU/q, and for a uniform field E = ΔV/d.
电场 E = F/q 的方向背离正电荷。电势差（电压）V = ΔU/q，在匀强电场中 E = ΔV/d。
Ohm’s law governs many ohmic materials: V = IR. Power dissipated in a resistor: P = IV = I²R = V²/R.
欧姆定律适用于许多欧姆材料：V = IR。电阻消耗的功率：P = IV = I²R = V²/R。
In a series circuit, current is constant and voltages add; equivalent resistance Rₛ = R₁ + R₂ + … In parallel, voltage is constant and currents add; 1/Rₚ = 1/R₁ + 1/R₂ + …
串联电路中电流处处相等，电压相加；等效电阻 Rₛ = R₁ + R₂ + …… 并联电路中电压相同，电流相加；1/Rₚ = 1/R₁ + 1/R₂ + ……
10. Electricity & Magnetism (AP2): Capacitance, Magnetic Force, Induction | 电与磁 (AP物理2)：电容、磁力、电磁感应
A capacitor stores charge and energy; capacitance C = Q/V. The energy stored is U = ½CV² = ½QV = ½Q²/C.
电容器储存电荷和能量；电容 C = Q/V。储存的能量为 U = ½CV² = ½QV = ½Q²/C。
For a parallel-plate capacitor, C = ε₀A/d, where ε₀ = 8.85×10⁻¹² F/m. A dielectric increases capacitance by a factor κ.
平行板电容器的电容 C = ε₀A/d，其中 ε₀ = 8.85×10⁻¹² F/m。介质使电容增大 κ 倍。
Magnetic force on a moving charge: F = qvB sin θ. Right-hand rule determines the direction. The force on a current-carrying wire: F = ILB sin θ.
运动电荷受到的磁力：F = qvB sin θ。右手定则确定方向。通电导线受到的力：F = ILB sin θ。
Faraday’s law: an induced emf equals the rate of change of magnetic flux, ε = –ΔΦ/Δt. Lenz’s law gives the direction of the induced current.
法拉第电磁感应定律：感应电动势等于磁通量的变化率，ε = –ΔΦ/Δt。楞次定律确定感应电流的方向。
11. Fluids & Thermodynamics | 流体与热力学
Density ρ = m/V; pressure P = F/A. In a static fluid, pressure increases with depth: P = P₀ + ρgh.
密度 ρ = m/V；压强 P = F/A。在静止流体中，压强随深度增加：P = P₀ + ρgh。
Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished. The buoyant force equals the weight of displaced fluid (Archimedes).
帕斯卡原理：施加在封闭流体上的压强会大小不变地传递。浮力等于排开流体的重量（阿基米德原理）。
Fb = ρfVg
The continuity equation for ideal fluids: A₁v₁ = A₂v₂. Bernoulli’s equation describes pressure, speed, and height along a streamline.
理想流体的连续性方程：A₁v₁ = A₂v₂。伯努利方程描述流线上压强、流速和高度的关系。
P + ½ρv² + ρgh = constant
Ideal gases obey PV = nRT. The kinetic theory links temperature to average translational kinetic energy: Kavg = ³/₂ kT.
理想气体遵从 PV = nRT。分子动理论将温度与平均平动动能联系起来：Kavg = ³/₂ kT。
First law of thermodynamics: ΔU = Q − W, where work done BY the system is positive. For an isothermal process, ΔU = 0; for adiabatic, Q = 0.
热力学第一定律：ΔU = Q − W，系统对外做功取正。等温过程 ΔU = 0；绝热过程 Q = 0。
Heat engine efficiency e = W/Qₕ. The maximum (Carnot) efficiency depends on reservoir temperatures: eₘₐₓ = 1 − T/Tₕ.
热机效率 e = W/Qₕ。最大（卡诺）效率取决于热源温度：eₘₐₓ = 1 − T/Tₕ。
12. Modern Physics: Optics, Quantum & Nuclear | 现代物理：光学、量子与核物理
Reflection and refraction follow Snell’s law: n₁ sin θ₁ = n₂ sin θ₂. Total internal reflection occurs when light goes from higher to lower n at an angle exceeding the critical angle.
反射和折射遵循斯涅尔定律：n₁ sin θ₁ = n₂ sin θ₂。光从光密介质射向光疏介质且入射角大于临界角时发生全反射。
Thin-film interference and double-slit interference yield constructive maxima when d sin θ = mλ. Single-slit diffraction minima: a sin θ = mλ.
薄膜干涉和双缝干涉中，当 d sin θ = mλ 时出现亮纹。单缝衍射暗纹条件：a sin θ = mλ。
Photon energy E = hf = hc/λ. The photoelectric effect gives maximum kinetic energy: Kₘₐₓ = hf − Φ, where Φ is the work function.
光子能量 E = hf = hc/λ。光电效应中最大动能为：Kₘₐₓ = hf − Φ，其中 Φ 为逸出功。
Electron energy levels in hydrogen: Eₙ = –13.6 eV/n². Emitted/absorbed photon energy equals the difference between levels.
氢原子能级：Eₙ = –13.6 eV/n²。发射或吸收的光子能量等于
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