📚 Core Mechanics Formulas and Common Applications | 力学公式梳理与常见应用
Mechanics is the foundation of A-Level physics. A clear command of its core formulas — and how to apply them under the correct conditions — is essential for solving exam problems quickly and accurately. This article organizes the most frequently tested equations by topic, with paired explanations in English and Chinese.
力学是 A-Level 物理的基础。熟练掌握核心公式，并清楚知道它们在什么条件下适用，是快速而准确解题的关键。本文按专题梳理考试中最常考的力学公式，并给出对应的中英文说明与应用提示。
1. Physical Quantities and Units | 物理量与单位
Every mechanics formula must be used with consistent SI units. Displacement is measured in metres (m), time in seconds (s), mass in kilograms (kg), and force in newtons (N), where 1 N = 1 kg·m/s².
所有力学公式都必须使用统一的 SI 单位：位移用米（m）、时间用秒（s）、质量用千克（kg）、力用牛顿（N），且 1 N = 1 kg·m/s²。
When a formula gives you a value, always check the units of every quantity before substitution. For example, speed must be in m/s, not km/h, unless the question explicitly requires otherwise.
代入计算前，务必检查各物理量的单位。例如速度应换算成 m/s 而非 km/h，除非题目另有明确要求。
F = ma, 1 N = 1 kg·m/s²
2. Uniform Acceleration Formulae (SUVAT) | 匀变速直线运动公式
For motion with constant acceleration, five variables are connected: displacement s, initial velocity u, final velocity v, acceleration a, and time t. The following three equations are used most frequently.
在匀变速直线运动中，五个关键物理量相互关联：位移 s、初速度 u、末速度 v、加速度 a 和时间 t。最常用的是以下三个公式。
v = u + at
s = ut + ½at²
v² = u² + 2as
Choose the equation that contains the three known quantities and the one unknown you need. Draw a diagram and label the positive direction before solving.
选择公式时，应优先挑选包含三个已知量和待求量的那一个。解题前先画图，并标出正方向。
3. Motion Graphs | 运动图像
Graphs are a compact way to represent motion. In a displacement–time graph, the gradient gives velocity; in a velocity–time graph, the gradient gives acceleration and the area under the graph gives displacement.
图像是描述运动最直观的工具。在位移–时间图中，斜率表示速度；在速度–时间图中，斜率表示加速度，图线与横轴围成的面积表示位移。
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s–t 图：斜率为速度 v，瞬时速度由切线斜率得到。
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v–t 图：斜率为加速度 a，面积为位移 s。
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a–t 图：面积为速度变化量 Δv。
For an object falling under gravity near Earth’s surface, acceleration is approximately 9.81 m/s² downwards, so its v–t graph is a straight line with a constant negative or positive slope depending on the chosen direction.
在地球表面附近自由下落的物体，加速度约为 9.81 m/s²，方向竖直向下，因此其 v–t 图是一条斜率恒定的直线，具体斜率的正负取决于所选正方向。
4. Newton’s Laws of Motion | 牛顿三大运动定律
Newton’s first law states that an object remains at rest or in uniform motion unless acted on by a resultant external force. Newton’s second law gives the quantitative relationship: the resultant force equals the rate of change of momentum.
牛顿第一定律指出：物体在不受合外力作用时，将保持静止或匀速直线运动状态。牛顿第二定律给出了定量关系：合外力等于动量对时间的变化率。
F = ma (for constant mass)
Newton’s third law states that forces always occur in pairs: if object A exerts a force on object B, then B exerts an equal and opposite force on A.
牛顿第三定律说明力总是成对出现：如果物体 A 对物体 B 施力，则 B 必定同时对 A 施加大小相等、方向相反的力。
Remember to distinguish action–reaction pairs from balanced forces. Action–reaction forces act on different bodies and never cancel each other.
注意区分作用力与反作用力、以及平衡力。作用力与反作用力作用在不同物体上，因此不能互相抵消。
5. Force Decomposition and Equilibrium | 力的分解与平衡条件
When a force F acts at an angle θ to the horizontal, its components are Fcosθ horizontally and Fsinθ vertically. This decomposition is essential for analysing objects on slopes.
当力 F 与水平方向成夹角 θ 时，其水平分量为 Fcosθ，竖直分量为 Fsinθ。力的分解在处理斜面上的物体时尤其重要。
Fₓ = Fcosθ, Fᵧ = Fsinθ
For an object in equilibrium, the vector sum of all forces is zero. Students should resolve forces in two perpendicular directions and set each sum equal to zero.
物体处于平衡状态时，所有力的矢量和为零。解题时应将力沿两个垂直方向分解，并分别令合力为零。
6. Friction | 摩擦力
Friction is a contact force that opposes relative motion or attempted motion. Limiting static friction reaches a maximum value before motion starts, while kinetic friction acts during sliding.
摩擦力是阻碍相对运动或相对运动趋势的接触力。物体开始运动前，静摩擦力可达到最大值（最大静摩擦力）；而在滑动过程中，作用的是滑动摩擦力。
f ≤ μₛN (static), f = μₖN (kinetic)
Here μₛ is the coefficient of static friction and μₖ is the coefficient of kinetic friction. Friction acts parallel to the contact surface and always opposes relative motion.
其中 μₛ 为静摩擦因数，μₖ 为动摩擦因数。摩擦力沿接触面的切线方向，且总是阻碍相对运动。
7. Work, Energy and Power | 功、能量与功率
Work is done when a force moves an object through a displacement. Energy is the capacity to do work. The work–energy principle states that the net work done on an object equals its change in kinetic energy.
力使物体发生位移时，力就对物体做了功。能量是做功的本领。动能定理指出：合外力对物体做的总功等于物体动能的变化量。
W = Fs cosθ, Eₖ = ½mv², Eₚ = mgh
P = W/t = Fv
When friction and air resistance are negligible, mechanical energy is conserved: the sum of kinetic and potential energy remains constant.
当摩擦力和空气阻力可忽略不计时，机械能守恒：动能与势能之和保持不变。
8. Momentum and Impulse | 动量与冲量
Momentum is the product of mass and velocity. Impulse is the product of the average force and the time interval over which it acts. The impulse–momentum theorem states that impulse equals the change in momentum.
动量是质量与速度的乘积，冲量是平均力与其作用时间的乘积。动量定理指出：合外力的冲量等于物体动量的变化量。
p = mv, J = FΔt = Δp
In any collision or explosion with no external resultant force, total momentum is conserved. For perfectly elastic collisions, kinetic energy is also conserved; for inelastic collisions, it is not.
在没有合外力作用的情况下，任何碰撞或爆炸过程中总动量守恒。完全弹性碰撞动能也守恒；非弹性碰撞动能不守恒。
9. Circular Motion | 圆周运动
Uniform circular motion requires a resultant force directed toward the centre of the circle, called the centripetal force. This force changes the direction of the velocity, not its speed.
匀速圆周运动需要指向圆心的合外力来提供向心力。向心力只改变速度方向，不改变速度大小。
a = v²/r = ω²r, F = mv²/r
Be careful: centrifugal force is not a real force acting on the object. When analysing circular motion, identify the real forces (tension, gravity, normal reaction) that together provide the required centripetal force.
注意：离心力并不是真实作用在物体上的力。分析圆周运动时，应找出真实的力（如拉力、重力、支持力），它们共同提供所需的向心力。
10. Simple Harmonic Motion (SHM) | 简谐运动
Simple harmonic motion is a special oscillating motion in which acceleration is proportional to displacement from equilibrium and always directed toward it. The defining equation is a = −ω²x, where ω is the angular frequency.
简谐运动是一种特殊的往复运动：加速度与相对平衡位置的位移成正比，且方向始终指向平衡位置。其定义式为 a = −ω²x，其中 ω 为角频率。
a = −ω²x, T = 2π√(m/k) (mass–spring), T = 2π√(L/g) (simple pendulum)
Energy in SHM oscillates continually between kinetic energy and potential energy, but the total mechanical energy remains constant for ideal systems.
在简谐运动中，动能与势能不断相互转化，但在理想系统中总机械能保持不变。
11. Problem-Solving Strategy: Energy Method vs Momentum Method | 解题策略：能量法与动量法
Many mechanics problems can be approached by more than one method. Choosing the right tool saves time and reduces mistakes.
许多力学问题并不只有一种解法。选对方法能节省时间、减少错误。
| Approach | Best used when | Typical example |
| Energy method | Only speeds and heights are involved | Object sliding down a frictionless slope |
| Momentum method | Collisions, explosions, or impulses | Two carts colliding on a track |
When a collision occurs, use momentum conservation to find velocities. When the question involves distance along a curved path or changing height, consider energy conservation instead of detailed force analysis.
遇到碰撞问题时，应使用动量守恒求解速度；遇到涉及曲面路径或高度变化的问题，则优先考虑机械能守恒，而不必逐点分析受力。
12. Formula Summary Table | 力学公式总览表
The table below summarizes the most important formulas covered in this article. Keep it nearby during revision.
下表归纳了本文涉及的核心公式，复习时可作为速查手册。
| Topic | Formula |
| Kinematics | v = u + at; s = ut + ½at²; v² = u² + 2as |
| Dynamics | F = ma |
| Friction | f ≤ μₛN; f = μₖN |
| Work, Energy, Power | W = Fs cosθ; Eₖ = ½mv²; Eₚ = mgh; P = Fv |
| Momentum | p = mv; FΔt = Δp |
| Circular motion | a = v²/r; F = mv²/r |
| SHM | a = −ω²x; T = 2π√(m/k) |
Mastering this formula sheet is only the first step. True understanding comes when you can derive, compare, and apply these equations in unfamiliar exam scenarios.
掌握这张公式表只是第一步。真正理解意味着你能够在陌生的题目情境中推导、比较并灵活运用这些方程。
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