📚 IB & CIE Physics: Mastering Mechanics – Motion, Forces, Energy and Momentum | IB与CIE物理：掌握力学——运动、力、能量和动量
Mechanics is the foundation of both IB Physics and CIE Physics courses. Whether you are preparing for IB Standard Level, Higher Level, or CIE AS/A Level exams, a strong grasp of motion, forces, energy and momentum is essential for solving problems and scoring well on Paper 1, Paper 2, or practical assessments. This article integrates the core syllabus points from both boards, highlights key equations, and gives you exam-focused strategies.
力学是 IB 物理和 CIE 物理课程的基础。无论你准备 IB 标准级别、高级别，还是 CIE AS/A Level 考试，扎实掌握运动、力、能量和动量对于解题和在试卷一、试卷二或实验评估中取得高分都至关重要。本文整合两大考试局的核心大纲要点，突出关键公式，并提供以考试为导向的备考策略。
Although IB and CIE write their papers differently, the underlying physics of mechanics is identical. The same equations, the same free-body diagrams and the same conservation laws appear in both specifications, so working through mechanics once - carefully - pays off across every paper you sit.
尽管 IB 与 CIE 的出题风格不同，但力学背后的物理原理完全一致。相同的方程、相同的受力分析图和相同的守恒定律出现在两份大纲中，因此把力学认真吃透一次，就能在每一份试卷上获得回报。
1. Scalars, Vectors and Resolving Components | 标量、矢量与分量分解
In both IB and CIE physics, physical quantities are classified as scalars or vectors. Scalars have magnitude only, while vectors have both magnitude and direction. A vector is only fully specified when you state its size, its unit and its direction.
在 IB 和 CIE 物理中，物理量分为标量和矢量。标量只有大小，而矢量既有大小又有方向。只有同时说明大小、单位和方向，一个矢量才算被完整描述。
Examples of scalar quantities include distance, speed, mass, energy and time. Vector quantities include displacement, velocity, acceleration, force and momentum. Note that distance and displacement are not interchangeable, and neither are speed and velocity: the scalar version ignores direction, the vector version does not.
标量的例子包括路程、速率、质量、能量和时间。矢量的例子包括位移、速度、加速度、力和动量。注意路程与位移不可互换，速率与速度同样不可互换：标量版本忽略方向，矢量版本则不然。
Vector components are found using F cos θ and F sin θ, where θ is the angle between the vector and a chosen axis. Always draw the angle first, then decide which component is adjacent to it - the adjacent component takes cos θ, the opposite component takes sin θ.
矢量分量可用 F cos θ 和 F sin θ 求得，其中 θ 是矢量与选定坐标轴之间的夹角。先画出角度，再判断哪个分量与它相邻——与角相邻的分量用 cos θ，对角所对的垂直分量用 sin θ。
Fₓ = F cos θ, Fᵧ = F sin θ
B cos θ, or by resolving each vector into components and adding the components separately.
当两个矢量成角度相加时，合矢量可由余弦定理求得，R² = A² + B² + 2AB cos θ，也可以把每个矢量分解为分量，再分别对分量求和。
2. Kinematic Equations and Motion Graphs | 运动学方程与运动图像
For uniformly accelerated motion, both boards use the same SUVAT equations. These equations only apply when acceleration is constant - if acceleration changes, you must work from the graph or from calculus.
对于匀加速运动，两大考试局使用相同的 SUVAT 方程。这些方程仅在加速度恒定时适用——若加速度变化，就必须借助图像或微积分处理。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
Each equation omits exactly one quantity. That is the practical trick for choosing the right one: list s, u, v, a and t, mark the value you are asked for and the value you are not given, and pick the equation missing that unwanted quantity.
每个方程恰好缺少一个物理量。这就是选方程最实用的技巧：列出 s、u、v、a、t，标出你要求的量和题目没给的量，然后选择缺少那个"不需要的量"的方程。
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. In both cases, upward or rightward slopes are positive, and a straight line means constant rate of change.
在位移-时间图像中，斜率表示速度。在速度-时间图像中，斜率表示加速度，图线下的面积表示位移。两种情况中，向上或向右的斜率为正，直线意味着变化率恒定。
Remember to convert units to SI before substituting values, especially km/h to m/s. Divide km/h by 3.6 to obtain m/s, and multiply m/s by 3.6 to return to km/h.
代入数值前记得将单位转换为国际单位制，特别是将 km/h 换算为 m/s。km/h 除以 3.6 得 m/s，m/s 乘以 3.6 则回到 km/h。
3. Newton's Laws and Free-Body Diagrams | 牛顿定律与受力分析图
Newton's first law states that a body remains at rest or moves with constant velocity unless acted on by a resultant force. This is why a rocket in deep space needs no engine to keep moving, only to change its motion.
牛顿第一定律指出，若不受合外力作用，物体保持静止或做匀速直线运动。这就是为什么深空中的火箭无需发动机就能持续运动，只有在改变运动状态时才需要。
Newton' F = ma, while IB Higher Level expects you to link force to the rate of change of momentum.
牛顿第二定律在质量恒定时写作 F = ma，但更普遍的形式是 F = Δp/Δt。CIE 大纲更强调 F = ma，而 IB 高级别要求你把力与动量变化率联系起来。
Newton's third law requires that the two forces act on different bodies, are of the same type, and are equal in magnitude but opposite in direction. "Equal and opposite" alone is not sufficient - a book resting on a table has weight and normal contact force, but these are not a third-law pair because they act on the same body.
牛顿第三定律要求两个力作用在不同物体上、属于同种性质的力、大小相等方向相反。仅说"大小相等方向相反"是不够的——放在桌上的书受到重力和支持力，但它们不是一对作用力与反作用力，因为二者作用在同一物体上。
Free-body diagrams are worth marks in both courses. Draw each force from the point where it acts, label it with its symbol, and never include a force the object exerts on something else. A missing arrow loses marks even when the final numerical answer is correct.
在两大课程中，受力分析图本身就有分数。从力的作用点出发画每个力，用符号标注，绝不画物体施加于其他物体的力。即使最终数值答案正确，漏画一个箭头也会失分。
4. Friction, Inclines and Connected Bodies | 摩擦力、斜面与连接体
he coefficient of friction and N is the normal contact force. Once the object is sliding, kinetic friction is usually treated as constant and, in most school problems, equal to the limiting value.
摩擦力阻碍相对运动。在即将滑动的临界状态，最大静摩擦力为 F = μN，其中 μ 是摩擦系数，N 是法向接触力。一旦物体开始滑动，滑动摩擦通常视为恒定，并且在大多数中学题目中与最大静摩擦力取相同数值。
On an incline of angle θ, the weight component down the slope is mg sin θ and the component into the slope is mg cos θ. This is the single most examined resolution in mechanics, so it is worth memorising which is which: the steeper the slope, the larger mg sin θ becomes.
在倾角为 θ 的斜面上，沿斜面向下的重力分量为 mg sin θ，垂直压向斜面的分量为 mg cos θ。这是力学中考得最多的分解方式，值得记牢哪个是哪个：斜面越陡，mg sin θ 越大。
For connected bodies - blocks joined by a string over a pulley, or a train of coupled trucks - treat the whole system as one body to find the acceleration, then isolate each body to find the tension. The acceleration is the same for every part of the system only if the string is inextensible and does not slip on the pulley.
对于连接体——用跨过滑轮的绳子相连的物块，或由多节车厢连成的列车——先把整个系统视为一个整体求加速度，再隔离其中任一部分求张力。只有在绳子不可伸长且不打滑时，系统各部分的加速度才相同。
Terminal velocity appears in both syllabuses: as a falling object speeds up, drag increases until the drag force equals the weight, the resultant force becomes zero, and acceleration falls to zero. Mark schemes want the argument in terms of resultant force, not just the statement that the forces balance.
终端速度在两份大纲中都有考查：下落物体速度增大时，阻力随之增大，直到阻力等于重力，合外力为零，加速度降为零。评分标准要求从合外力的角度论证，而不只是说"受力平衡"。
5. Circular Motion and Centripetal Force | 圆周运动与向心力
hat acceleration points towards the centre of the circle and has magnitude a = v²/r = ω²r.
做匀速圆周运动的物体仍在加速，因为速度方向不断改变。该加速度指向圆心，大小为 a = v²/r = ω²r。
a = v² / r = ω²r, F = mv² / r = mω²r
. A common exam error is mixing degrees and radians - angular velocity must be in rad s⁻¹.
角速度把周期和频率与线速度联系起来：ω = 2π/T = 2πf 且 v = ωr。常见失分点是角度与弧度混用——角速度必须以 rad s⁻¹ 为单位。
Centripetal force is not a new kind of force; it is a role played by a real force such as tension, gravity, friction or the normal contact force. Always name that real force in your answer, because descriptions such as "the centripetal force keeping it in a circle" earn no credit.
向心力不是一种新的力，而是张力、重力、摩擦力或法向接触力等真实力所扮演的角色。答题时务必点出这个真实力，因为"向心力使它做圆周运动"这类表述不得分。
6. Work, Energy and Power | 功、能量与功率
Work done by a force is the product of the force and the displacement in the direction of the force. When the force is at an angle θ to the displacement, only the component along the motion does work.
力做的功等于力与沿力方向位移的乘积。当力与位移成 θ 角时，只有沿运动方向的分量做功。
W = Fs cos θ, P = W / t = Fv
Kinetic energy is ½mv² and gravitational potential energy near the Earth's surface is mgh. Elastic potential energy stored in a stretched spring is ½kx² for a spring of stiffness k extended by x, and the work done in stretching it is also ½kx² - provided the limit of proportionality is not exceeded.
动能是 ½mv²，地球表面附近的重力势能是 mgh。劲度系数为 k 的弹簧伸长 x 时储存的弹性势能为 ½kx²，拉伸过程中所做的功同样是 ½kx²——前提是不超过比例极限。
e power, a larger resistive force must be balanced at a lower speed, which explains why a car climbs a hill more slowly in a high gear.
功率是做功或传递能量的速率。在车辆问题中，P = Fv 意味着发动机功率固定时，阻力越大，能达到的平衡速度越低，这也解释了为什么汽车上坡时用高档位反而更慢。
7. Conservation of Energy and Efficiency | 能量守恒与效率
Energy cannot be created or destroyed, only transferred from one store to another. In mechanics this is usually written as the total energy at one point equals the total energy at another, with any difference accounted for as work done against friction or drag.
能量既不能被创造也不能被消灭，只能从一种形式转移到另一种形式。在力学中通常表述为：某一位置的总能量等于另一位置的总能量，差额则记为克服摩擦或阻力所做的功。
Efficiency is the ratio of useful output energy to total input energy, or useful output power to total input power, and is often quoted as a percentage. Values above 100% are always a sign of an error in the calculation.
效率是有用输出能量与总输入能量之比，或有用输出功率与总输入功率之比，通常用百分数表示。超过 100% 的数值永远说明计算有误。
efficiency = (useful output ÷ total input) × 100%
ignificant, the shortfall is transferred to the thermal store of the surroundings, and the final speed is lower than the value predicted by v² = 2gh.
对于下落物体，只有空气阻力可忽略时，重力势能的减少量才等于动能的增加量。若阻力不可忽略，差额转移到了周围环境的热能中，最终速度会低于由 v² = 2gh 预测的数值。
8. Momentum, Impulse and Collisions | 动量、冲量与碰撞
r, so direction must be handled with a sign convention. Impulse is the product of force and the time for which it acts, and it equals the change in momentum.
线动量是质量与速度的乘积，p = mv，它是矢量，因此必须用正负号约定处理方向。冲量是力与作用时间的乘积，等于动量的变化量。
p = mv, J = FΔt = Δp, F = Δp / Δt
Total momentum is conserved in any collision or explosion, provided no external resultant force acts on the system. Kinetic energy, by contrast, is conserved only in perfectly elastic collisions; in inelastic collisions some kinetic energy is transferred to thermal energy, sound or deformation.
只要系统所受合外力为零，任何碰撞或爆炸中总动量守恒。相比之下，动能只在完全弹性碰撞中守恒；在非弹性碰撞中，一部分动能转化为热能、声能或形变能。
Explosions are the reverse of collisions: momentum before is zero, so the total momentum afterwards must also be zero. This is the key idea behind recoil problems and rocket propulsion.
爆炸是碰撞的逆过程：碰撞前总动量为零，所以碰撞后的总动量也必须为零。这是反冲问题和火箭推进问题的核心思想。
Adding an airbag or a crumple zone increases the collision time for the same change of momentum, so the average force on the occupant is reduced. Answers must mention Δp being fixed and Δt increasing - the words "cushions the impact" alone earn nothing.
加装安全气囊或溃缩区可以在动量变化量相同的情况下延长碰撞时间，从而减小乘员所受的平均力。答题时必须指出 Δp 固定、Δt 增大——只写"缓冲了冲击"是不得分的。
9. Moments, Couples and Equilibrium | 力矩、力偶与平衡
The perpendicular distance, not the distance along the rod, is what the mark scheme wants.
力对支点的力矩等于力与支点到力作用线的垂直距离的乘积，M = Fd。评分标准要求的是垂直距离，而不是沿杆的长度。
For a body in equilibrium, two conditions must both hold: the resultant force is zero and the resultant moment about any point is zero. The principle of moments states that the sum of the clockwise moments about a point equals the sum of the anticlockwise moments about that same point.
处于平衡的物体必须同时满足两个条件：合外力为零，且对任一点的合力矩为零。力矩平衡原理指出，对某一点顺时针力矩之和等于逆时针力矩之和。
A couple consists of two equal, opposite and parallel forces whose lines of action do not coincide, producing a pure rotational effect with moment Fd where d is the perpendicular distance between them. The centre of gravity is the point at which the whole weight of a body may be taken to act, and a body topples when its centre of gravity falls outside its base.
力偶由两个大小相等、方向相反、平行但不共线的力组成，产生纯转动效果，其力矩为 Fd，d 为两力之间的垂直距离。重心是物体全部重力可视为作用的点；当重心落到支撑面之外时，物体就会翻倒。
10. Practical Work and Graphical Analysis | 实验操作与图像分析
Practical skills carry a substantial share of marks in both courses - the IB internally assessed investigation and the CIE practical papers. In mechanics experiments, the most commonly assessed skills are measuring with correct precision, repeating readings, and quoting an uncertainty with a sensible number of significant figures.
实验技能在两门课程中都占相当比例的分数——IB 的内部评估探究和 CIE 的实验卷。在力学实验中，最常被考查的技能是用正确精度测量、重复读数，以及用合理有效数字给不确定度。
Percentage uncertainty is the absolute uncertainty divided by the measured value, multiplied by 100%. When quantities are multiplied or divided, percentage uncertainties add; when they are raised to a power, the percentage uncertainty is multiplied by that power.
百分比不确定度等于绝对不确定度除以测量值再乘以 100%。当量相乘或相除时，百分比不确定度相加；当量取幂时，百分比不确定度乘以该指数。
Graph work is heavily rewarded. Plot the manipulated variable on the x-axis, draw the line of best fit by eye, and use a large triangle from the line itself - not from data points - to find the gradient. A straight-line graph turns a proportional relationship into a testable claim: if the line passes through the origin within the uncertainty of the intercept, the two quantities are proportional.
作图题分值很高。把自变量画在 x 轴上，用肉眼画出最佳拟合直线，并用取自直线本身（而非数据点）的大三角形求斜率。直线图像把比例关系变成可检验的结论：若直线在截距不确定度范围内通过原点，则两量成正比。
For non-linear relationships such as T = 2π√l/g of gradient 4π²/g. The same log-plot technique handles relationships of the form y = kxⁿ.
对于 T = 2π√l/g 之类的非线性关系，应作 T² 对 l 的图像，得到斜率为 4π²/g 的直线。同样的对数作图技巧可处理 y = kxⁿ 形式的关系。
11. Command Words and Mark-Scheme Habits | 指令词与评分标准习惯
Command words decide how much you write. "State" needs a brief answer with no working; "describe" needs a sequence of observations or events; "explain" needs a physical reason, usually mentioning a force, an energy transfer or a conservation law; "determine" or "calculate" needs working and a unit.
指令词决定你该写多少。"State（写出）"只需简短答案，不需过程；"describe（描述）"需要一连串观察或事件的顺序；"explain（解释）"需要给出物理原因，通常要提到力、能量转移或守恒定律；"determine / calculate（求算）"需要写出过程和单位。
Mark schemes for mechanics are usually written as independent points, each worth one mark. That means a two-mark "explain" question is expecting two distinct physics ideas, not one idea written twice at greater length.
力学评分标准通常写成互相独立的得分点，每点一分。这意味着两分的"explain"题期待两个不同的物理观点，而不是把一个观点写两遍、写得更长。
Always give the equation first, then substitute, then state the answer with its unit and an appropriate number of significant figures. The final answer line should never be left without units, and intermediate rounding should be avoided until the very end.
永远先写方程，再代入数值，最后给出带单位和合适有效数字的答案。最后一行答案绝不能没有单位，并且在最后一步之前不要提前四舍五入。
12. Common Errors and a Revision Plan | 常见错误与复习计划
The most frequent mistakes in mechanics are treating a scalar as a vector, forgetting that SUVAT only applies at constant acceleration, omitting the direction of a force in a free-body diagram, and quoting a formula without naming the real force responsible for centripetal motion.
力学中最常见的错误是把标量当矢量、忘记 SUVAT 仅适用于匀加速、在受力分析图中漏掉力的方向，以及只写公式却不指明提供向心力的真实力。
Other recurring slips include mixing up mg sin θ and mg cos θ, adding kinetic energies in an inelastic collision, and forgetting to convert km/h to m/s before substitution. Each of these is worth a mark every time it appears, so a short personal error list is one of the highest-value revision tools you can keep.
其他反复出现的失误包括把 mg sin θ 和 mg cos θ 弄反、在非弹性碰撞中把动能相加，以及代入前忘记把 km/h 换算为 m/s。这些失误每出现一次就丢一分，因此保留一份属于自己的"错误清单"是性价比最高的复习工具之一。
A sensible eight-week plan is to spend two weeks on kinematics and graphs, two on forces and Newton's laws, two on energy and momentum, one on circular motion and moments, and the final week on past papers under timed conditions. Mark every paper against the official scheme, and rewrite any answer that lost marks.
一个合理的八周计划是：两周用于运动学与图像，两周用于力与牛顿定律，两周用于能量与动量，一周用于圆周运动与力矩，最后一周做限时真题。每份试卷都对照官方评分标准批改，并把任何失分的答案重写一遍。
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