A-Level Physics Mechanics: Core Formula Revision Guide | A-Level 物理力学：考纲核心公式梳理

📚 A-Level Physics Mechanics: Core Formula Revision Guide | A-Level 物理力学：考纲核心公式梳理

Mechanics is one of the most heavily weighted topics in the A-Level physics exam, spanning roughly 20-30% of all exam papers. A thorough command of the core equations – knowing not just what they are, but when and how to apply them – is essential for achieving top grades. This guide consolidates every essential formula from the standard Cambridge, Edexcel, and AQA mechanics syllabi into one systematic revision resource.

力学是 A-Level 物理考试中分值占比最高的板块之一，通常在全部试卷中占 20%-30%。熟练掌握核心公式——不仅知道公式本身，更清楚何时使用、如何应用——是冲击高分的必备条件。本指南将剑桥、爱德思和 AQA 三大考纲中力学部分的所有必备公式系统整合，供你一站式复习使用。

1. Kinematics – Uniform Acceleration | 运动学——匀变速直线运动

The five SUVAT equations form the foundation of all constant-acceleration problems. You must memorise these and identify which variables are given and which are sought before choosing the appropriate equation. s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.

五个 SUVAT 公式是所有匀变速直线运动问题的基础。你必须熟记这些公式，并在解题前先判断题目给出了哪些已知量、需要求哪个未知量，再选择相应的方程。s 表示位移，u 表示初速度，v 表示末速度，a 表示加速度，t 表示时间。

v = u + at

s = ut + ½at²

s = ½(u + v)t

v² = u² + 2as

s = vt − ½at²

For free fall under gravity, simply substitute a = g (9.81 m/s² on Earth) and choose a consistent positive direction. Remember that displacement, velocity, and acceleration are vectors – direction matters when assigning signs.

对于自由落体，只需将 a = g（地球表面取 9.81 m/s²）代入公式，并选定统一的正方向。切记位移、速度和加速度都是矢量——正负号的方向选择至关重要。

When an object is thrown upward, at its highest point v = 0 and a = −g.

物体竖直上抛时，最高点处 v = 0，且 a = −g。
The time of ascent equals the time of descent under uniform gravity.

在匀强重力场中，上升时间等于下降时间。

2. Projectile Motion | 抛体运动

Projectile problems are solved by resolving the initial velocity into horizontal and vertical components, then treating the two directions independently. The key insight: horizontal motion has constant velocity, vertical motion has constant acceleration g.

抛体运动问题需将初速度分解为水平和竖直两个分量，然后对两个方向分别独立处理。核心思路：水平方向为匀速运动，竖直方向为加速度为 g 的匀加速运动。

Horizontal: x = (u cos θ)t

Vertical: y = (u sin θ)t − ½gt²

v_y = u sin θ − gt

The range equation is derived by setting the vertical displacement to zero (landing at the same height):

射程公式可通过令竖直位移为零（落回同一高度）推导得出：

R = (u² sin 2θ) / g

Maximum range occurs at a launch angle of 45°.

当发射角为 45° 时射程最远。
For a fixed launch speed, angles θ and (90° − θ) give the same range.

在初速度大小固定时，θ 和 (90° − θ) 两个发射角对应相同射程。

3. Newton’s Laws of Motion | 牛顿运动定律

Newton’s three laws govern all classical dynamics. The second law is most commonly written in two equivalent forms:

牛顿三大定律支配着一切经典动力学问题。其中第二定律最常用的两种等价形式如下：

F = ma or equivalently F = Δp / Δt

F = ma 或等价形式 F = Δp / Δt

Law 1 \| 第一定律	A body remains at rest or in uniform motion unless acted upon by a net external force. 物体在不受合外力作用时，保持静止或匀速直线运动状态。
Law 2 \| 第二定律	F = ma – acceleration is proportional to net force and inversely proportional to mass. F = ma——加速度与合外力成正比，与质量成反比。
Law 3 \| 第三定律	Every action has an equal and opposite reaction. 作用力与反作用力大小相等、方向相反，且作用在同一条直线上。

Common exam trap: the action-reaction pair act on different bodies, so they never neutralize each other. Weight (W = mg) and the normal reaction are an action-reaction pair – the normal force is not always equal to mg (e.g., in an accelerating lift).

常见考试陷阱：作用力与反作用力作用在不同物体上，因此它们永远不会相互抵消。重力（W = mg）与支持力并非总是作用力与反作用力对——支持力也不总是等于 mg（例如在加速运动的电梯中）。

4. Friction and Inclined Planes | 摩擦力与斜面问题

When an object moves or tends to move on a rough surface, friction acts to oppose relative motion. Friction is calculated as:

当物体在粗糙表面上运动或趋向运动时，摩擦力总是阻碍相对运动。摩擦力的计算公式为：

f_max = μR

Here μ is the coefficient of friction (a unitless constant) and R is the normal reaction force. For a block on an inclined plane of angle θ, the components of weight parallel and perpendicular to the plane are:

其中 μ 为摩擦系数（无量纲常数），R 为法向支持力。对于斜面上倾角为 θ 的物体，重力沿斜面方向和垂直斜面方向的分量分别为：

Parallel: mg sin θ

Perpendicular: mg cos θ

沿斜面方向：mg sin θ

垂直斜面方向：mg cos θ

For a block at rest with friction, the friction force equals mg sin θ, not μR.

对于静止在斜面上且有摩擦的物体，摩擦力大小等于 mg sin θ，而非 μR。
Sliding begins when mg sin θ > μ mg cos θ, i.e., when tan θ > μ.

物体开始下滑的条件为 mg sin θ > μ mg cos θ，即 tan θ > μ。

5. Work, Energy, and Power | 功、能与功率

Work is done when a force causes displacement. Energy is the capacity to do work. These interlinked concepts appear in every mechanics exam paper. The three core definitions:

力使物体发生位移时即做功。能量是做功的本领。这三个相互关联的概念出现在每一份力学考卷中。以下三个核心定义：

W = Fd cos θ (work done by constant force)

KE = ½mv² (kinetic energy)

PE = mgh (gravitational potential energy)

When the force is parallel to displacement, θ = 0 and W = Fd. Power is the rate of doing work:

当力与位移方向平行时，θ = 0，此时 W = Fd。功率是做功的快慢：

P = W / t or P = Fv

The work-energy principle states that the net work done on a system equals its change in kinetic energy:

功能原理指出：合外力对系统所做的净功等于系统动能的变化量：

W_net = ΔKE = ½mv² − ½mu²

Gravitational PE depends only on height change, not the path taken.

重力势能只与高度变化有关，与路径无关。
In a closed system with no friction, mechanical energy (KE + PE) is conserved.

在无摩擦的封闭系统中，机械能（动能 + 势能）守恒。
Average power = total work ÷ total time; instantaneous power = F × instantaneous velocity.

平均功率 = 总功 ÷ 总时间；瞬时功率 = 力 × 瞬时速度。

6. Conservation of Energy | 能量守恒定律

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. For mechanics problems, this yields a powerful problem-solving strategy.

能量守恒定律指出：能量既不能凭空产生，也不能凭空消失，只能从一种形式转化为另一种形式。在力学问题中，这提供了一种强大的解题策略。

Initial total energy = Final total energy + Energy lost to surroundings

初始总能量 = 末态总能量 + 散失到周围环境的能量

For a frictionless system involving height changes and speed changes:

对于无摩擦且涉及高度变化和速度变化的系统：

mgh₁ + ½mv₁² = mgh₂ + ½mv₂²

Typical exam applications include a pendulum swinging, a ball rolling down a track, or a skier descending a slope. When friction is present, the energy lost equals the work done against friction, f × d.

典型考查场景包括：单摆摆动、小球沿轨道滚下、滑雪者沿斜坡下滑。当存在摩擦力时，损失的能量等于克服摩擦力所做的功，即 f × d。

7. Linear Momentum and Collisions | 动量与碰撞

Linear momentum is the product of mass and velocity. The principle of conservation of momentum states that the total momentum of an isolated system remains constant.

动量是质量与速度的乘积。动量守恒定律指出：孤立系统的总动量始终保持不变。

p = mv

Total momentum before = Total momentum after

碰撞前总动量 = 碰撞后总动量

Impulse is the change in momentum and equals the force multiplied by time:

冲量是动量的变化量，等于力乘以时间：

Impulse = Ft = Δp = mv − mu

Collisions are classified as elastic or inelastic. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not. In a completely inelastic collision, the two objects stick together.

碰撞分为弹性碰撞和非弹性碰撞。在弹性碰撞中，动量和动能均守恒。在非弹性碰撞中，动量守恒但动能不守恒。在完全非弹性碰撞中，两物体碰撞后粘合在一起运动。

Type 类型	Momentum 动量	Kinetic Energy 动能
Elastic 弹性碰撞	Conserved 守恒	Conserved 守恒
Inelastic 非弹性碰撞	Conserved 守恒	Not conserved (some lost as heat/sound) 不守恒（部分转化为热/声）
Completely inelastic 完全非弹性	Conserved 守恒	Maximum loss 损失最大

8. Circular Motion | 圆周运动

When an object moves in a circle of radius r at constant speed, its velocity direction continually changes, meaning it experiences centripetal acceleration directed toward the centre of the circle.

当物体以恒定速率沿半径为 r 的圆周运动时，其速度方向不断改变，因此它受到指向圆心的向心加速度。

a = v² / r or a = ω²r

ω = v / r = 2π / T

F_c = mv² / r = mω²r

Here ω is angular velocity (rad/s), T is the period, and F_c is the centripetal force. The centripetal force is not an independent force but the resultant of real forces – tension, gravity, friction, or the normal reaction – directed radially inward.

其中 ω 为角速度（rad/s），T 为周期，F_c 为向心力。向心力并非一种独立的力，而是指向圆心的真实合力——可以是张力、重力、摩擦力或支持力的合力。

For a car turning a flat corner, the centripetal force is provided by friction: F = μmg = mv²/r.

汽车在水平路面转弯时，向心力由摩擦力提供：F = μmg = mv²/r。
For a satellite in orbit, gravity provides the centripetal force: GMm/r² = mv²/r.

卫星在轨道上运行时，万有引力提供向心力：GMm/r² = mv²/r。
At the top of a loop-the-loop, the minimum speed requires that the normal reaction just reaches zero: v_min = √(gr).

在竖直圆环顶端，恰好不脱离轨道的最小速度为：v_min = √(gr)。

9. Gravitational Fields and Newton’s Law of Gravitation | 引力场与万有引力定律

Newton’s law of gravitation describes the attractive force between any two point masses. This is an inverse-square law, which means the force decreases rapidly with distance.

万有引力定律描述了两质点之间的相互吸引力。这是一条平方反比定律，意味着力随距离增大而迅速减小。

F = GMm / r²

The gravitational field strength g at a distance r from the centre of a mass M is:

距离质量 M 的中心 r 处的引力场强度 g 为：

g = GM / r²

For an object on the Earth’s surface, this reduces to g = 9.81 m/s². The gravitational potential energy near the Earth’s surface is PE = mgh, but the general formula for a body at distance r is:

对于地球表面的物体，上式即化为 g = 9.81 m/s²。地球表面附近的引力势能为 PE = mgh，但距地心距离为 r 处的通用引力势能公式为：

U = −GMm / r

Note the negative sign: gravitational potential energy is defined as zero at infinity and becomes more negative as the object approaches the mass. For circular orbits, equating gravitational force to centripetal force gives the orbital speed:

注意负号：引力势能以无穷远处为零点，物体越靠近质量源，势能越负。对于圆轨道，令万有引力等于向心力，可得轨道速度：

v = √(GM / r)

Kepler’s third law for orbital periods follows directly: T² ∝ r³.

开普勒第三定律可直接由此推出：T² ∝ r³。

10. Simple Harmonic Motion | 简谐运动

Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement from equilibrium and directed toward the equilibrium position. The defining equation:

当回复力与偏离平衡位置的位移成正比且指向平衡位置时，物体做简谐运动（SHM）。其定义方程为：

a = −ω²x

The displacement as a function of time takes a sinusoidal form:

位移随时间呈正弦（或余弦）规律变化：

x = A cos(ωt) or x = A sin(ωt)

The velocity and acceleration formulas are:

速度和加速度的公式为：

v = ±ω√(A² − x²)

v_max = ωA, a_max = ω²A

For a mass-spring system and a simple pendulum, the angular frequencies are:

对于弹簧振子和单摆，角频率分别为：

Mass-spring: ω = √(k / m)

Simple pendulum: ω = √(g / L)

Energy in SHM alternates between kinetic and potential forms. Total mechanical energy is constant and equals ½mω²A². At equilibrium x = 0, all energy is kinetic; at maximum displacement x = A, all energy is potential.

简谐运动中的能量在动能与势能之间交替转化。总机械能恒定，等于 ½mω²A²。在平衡位置 x = 0 处，能量全部为动能；在最大位移 x = A 处，能量全部为势能。

11. Damping and Resonance | 阻尼与共振

Damping is the process by which the amplitude of an oscillator gradually decreases due to energy dissipation. Light damping produces a gradual amplitude decay over many oscillations; heavy damping causes the system to decay without completing even one full oscillation.

阻尼是振荡器因能量耗散而振幅逐渐减小的过程。欠阻尼（轻阻尼）使振幅经过多个周期逐渐衰减；过阻尼（重阻尼）则使系统在完成一次完整振荡之前就衰减为零。

Critical damping returns the system to equilibrium in the shortest time without oscillation – essential in car suspension design.

临界阻尼使系统在不发生振荡的情况下以最短时间回到平衡位置——这在汽车悬挂设计中至关重要。
Resonance occurs when the driving frequency matches the natural frequency of the system, causing maximum amplitude.

当驱动频率与系统的固有频率相等时发生共振，振幅达到最大。
Real-world examples include opera singers shattering glass and suspension bridge oscillations.

现实案例包括歌剧演唱震碎玻璃杯、悬索桥振荡等。

12. Momentum, Torque, and Equilibrium in Static Systems | 力矩与静力平衡

A rigid body in equilibrium must satisfy two conditions: the net external force is zero, and the net external torque about any point is zero. Torque (moment of a force) is defined as:

刚体处于平衡状态必须满足两个条件：合外力为零，且对任意一点的合外力矩为零。力矩（力对点的矩）定义为：

τ = F × d

where d is the perpendicular distance from the pivot to the line of action of the force. The principle of moments states that for a body in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot.

其中 d 为从支点到力的作用线的垂直距离。力矩平衡原理指出：处于平衡状态的物体，对任意支点，顺时针力矩之和等于逆时针力矩之和。

The centre of gravity of an object is the point where the entire weight may be considered to act. For uniform regular shapes, it lies at the geometric centre. Equilibrium is classified into three types:

物体的重心是可将全部重力视为集中作用的点。对于均匀规则形状的物体，重心位于几何中心。平衡分为三类：

Type 类型	Behaviour 行为	Example 示例
Stable 稳定平衡	Returns to equilibrium after small displacement 受到微小扰动后回到原平衡位置	A ball in a bowl 碗中的小球
Unstable 不稳定平衡	Moves further away after small displacement 受到微小扰动后继续远离平衡位置	A ball on top of a dome 穹顶上的小球
Neutral 随遇平衡	Remains in new position after displacement 受到扰动后在新位置保持平衡	A ball on a flat table 水平桌面上的小球

Mastering these formulas is only half the battle. For each equation, always ask yourself: What are the units? What are the assumptions or limitations? Does the problem involve vectors (requiring direction breakdown)? Does energy or momentum conservation apply? Write down all known variables, draw a clear free-body diagram, and select the appropriate equation before substituting numbers. Practice past papers repeatedly – examiners award method marks, so always show your working clearly.

熟记这些公式只是成功的一半。每遇到一个公式，务必问自己：单位是什么？有哪些假设条件或适用范围？题目是否涉及矢量（需要分解方向）？能量守恒或动量守恒是否适用？先列出所有已知量，画出清晰的受力分析图，再选择正确的公式代入计算。反复练习历年真题——阅卷按照步骤给分，务必清晰展示完整的解题过程。

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A-Level Physics Mechanics: Core Formula Revision Guide | A-Level 物理力学：考纲核心公式梳理