📚 AQA Maths Mechanics Revision Notes | AQA 数学：力学 考点精讲
This revision guide covers the essential mechanics topics for the AQA A-Level Mathematics specification, including kinematics, forces, Newton’s laws, momentum, moments and energy. Each section presents key concepts and equations, with clear explanations in both English and Chinese to support bilingual learning and exam preparation.
本复习指南涵盖AQA A-Level数学力学部分的核心考点，包括运动学、力、牛顿定律、动量、力矩和能量。每个小节以中英双语呈现关键概念与公式，帮助同学们深入理解并为考试做好充分准备。
1. Kinematics and SUVAT Equations | 运动学与SUVAT方程
Kinematics describes the motion of objects without considering the forces that cause the motion. The fundamental quantities are displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).
运动学描述物体的运动，而不涉及引起运动的力。基本量包括位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)。
For motion in a straight line with constant acceleration, we use the SUVAT equations. There are four key equations linking these five variables; each equation omits one variable.
对于匀加速直线运动，我们使用SUVAT方程。共有四个关键方程联系这五个变量，每个方程省略一个变量。
v = u + at
This equation connects velocity, initial velocity, acceleration and time; displacement is absent.
该方程将末速度、初速度、加速度和时间联系起来，不含位移。
s = ut + ½at²
This gives displacement when initial velocity, acceleration and time are known; final velocity is not needed.
已知初速度、加速度和时间时求位移，不需要末速度。
s = (u+v)t / 2
This uses the average velocity over the time interval; acceleration is omitted.
利用时间间隔内的平均速度求位移，省略加速度。
v² = u² + 2as
This equation relates velocities, acceleration and displacement; time is the missing variable.
该方程联系速度、加速度和位移，不含时间。
You must choose the appropriate equation depending on which variables are known and which is required. Always define a positive direction and ensure signs are consistent.
解题时必须根据已知量和待求量选择适当的方程。记得规定正方向并保持符号一致。
2. Displacement-Time and Velocity-Time Graphs | 位移—时间图与速度—时间图
Motion can be represented graphically. A displacement-time graph shows how displacement (or distance from a fixed point) changes with time. The gradient of the graph gives the velocity.
运动可用图像表示。位移—时间图显示位移随时间的变化，图线的斜率表示速度。
A straight line on a displacement-time graph indicates constant velocity. A curved line represents acceleration; the instantaneous velocity is the gradient of the tangent.
位移—时间图中的直线表示匀速运动，曲线表示加速运动；瞬时速度等于切线的斜率。
A velocity-time graph shows velocity against time. The gradient of this graph gives acceleration, and the area under the graph gives the displacement.
速度—时间图表示速度随时间的变化。其斜率表示加速度，图线下方的面积表示位移。
For constant acceleration, the velocity-time graph is a straight line with non-zero gradient. The area under the graph can often be found as a trapezium, linking back to the SUVAT equations.
对于匀加速运动，速度—时间图是一条倾斜直线。图线下方面积通常是梯形，这直接联系到SUVAT方程。
Always check the units on the axes and pay attention to where the graph crosses the time axis, which indicates a change in direction.
务必注意坐标轴的单位，并留意图线与时间轴的交点，该处表明运动方向改变。
3. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law: An object remains at rest or moves with constant velocity unless acted on by a resultant external force.
牛顿第一定律：物体将保持静止或匀速直线运动状态，除非受到合外力的作用。
Newton’s Second Law: The net force acting on a particle is equal to the product of its mass and acceleration: F = ma, where F is the resultant force in the direction of motion.
牛顿第二定律：作用在质点上的净力等于质量乘以加速度：F=ma，其中F是沿运动方向的合力。
Newton’s Third Law: If body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These forces act on different bodies.
牛顿第三定律：若物体A对物体B施加一个力，则物体B同时对A施加大小相等、方向相反的力。这两个力作用在不同物体上。
When applying Newton’s Second Law, resolve forces parallel and perpendicular to the direction of motion. Draw a clear force diagram to avoid sign errors.
应用牛顿第二定律时，沿运动方向和垂直方向分解力。画出清晰的受力图以避免符号错误。
4. Forces: Weight, Tension, Normal Reaction and Friction | 力：重力、张力、法向反力与摩擦力
Weight is the gravitational force acting on a mass: W = mg, where g = 9.8 m/s² (unless stated otherwise). Its direction is always vertically downwards.
重力是作用在物体上的万有引力：W=mg，g通常取9.8 m/s²。方向总是竖直向下。
Tension is the pulling force transmitted through a string or cable. In light inextensible strings, the tension is the same throughout the string and acts along its length.
张力是通过细绳或缆索传递的拉力。对于轻质且不可伸长的绳子，张力处处相等且沿绳子方向。
The normal reaction is the perpendicular contact force exerted by a surface on an object. It balances the component of weight perpendicular to the surface if no other vertical forces act.
法向反力是接触面对物体的垂直支持力。若无其他垂直力，它平衡重力的垂直分量。
Friction opposes relative motion or the tendency for motion. The maximum static friction is Fmax = μR, where μ is the coefficient of friction and R is the normal reaction. Kinetic friction is given by F = μR.
摩擦力阻碍相对运动或运动趋势。最大静摩擦力为Fmax=μR，μ为摩擦系数，R为法向反力。滑动摩擦力同样满足F=μR。
Always consider whether friction is limiting. In limiting equilibrium, the friction takes its maximum value; otherwise it adjusts to maintain equilibrium.
需判断摩擦是否达到极限。在极限平衡时，摩擦力取最大值；否则摩擦力会自动调节以维持平衡。
5. Connected Particles and Pulleys | 连接体与滑轮
When two particles are connected by a light inextensible string passing over a smooth pulley, the magnitudes of their accelerations are equal and the tension is uniform throughout the string.
当两物体通过轻质不可伸长的绳子跨过光滑滑轮相连时，它们的加速度大小相等，且绳中张力处处相同。
To solve such problems, treat each particle separately. Draw a force diagram for each mass, write the equation of motion F = ma for each, and solve the simultaneous equations.
解此类问题时，单独分析每个质点。分别画出受力图，对每个物体列出运动方程F=ma，然后联立求解。
For a mass hanging freely, the resultant force is weight minus tension. For a mass on a horizontal or inclined table, remember to include friction and normal reaction where appropriate.
对于自由悬挂的质量，合力为重力和张力的差值。对于水平面或斜面上的物体，需考虑摩擦力和法向反力。
If the string is inextensible, the speed of both particles is the same at any instant, and the distance moved is equal.
绳子不可伸长意味着任意时刻两物体速率相同，移动距离也相等。
6. Momentum and Impulse | 动量和冲量
Momentum is defined as the product of mass and velocity: p = mv. It is a vector quantity with direction given by the velocity.
动量定义为质量与速度的乘积：p=mv，是矢量，方向与速度方向相同。
Impulse is the change in momentum caused by a force acting over a time interval. Impulse = Force × time = mv – mu, where u and v are the initial and final velocities.
冲量是力在一段时间内作用引起的动量改变。冲量=力×时间=mv-mu，u和v分别为初、末速度。
The principle of conservation of momentum states that when no external forces act, the total momentum of a system remains constant. In collisions and explosions, total momentum before equals total momentum after.
动量守恒定律：当无外力作用时，系统的总动量保持不变。碰撞和爆炸中，作用前总动量等于作用后总动量。
For one-dimensional collisions: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Choose a positive direction and keep track of signs.
一维碰撞满足：m₁u₁+m₂u₂=m₁v₁+m₂v₂。规定正方向并注意速度的符号。
Always check whether a collision is perfectly elastic (kinetic energy conserved) or inelastic. The coefficient of restitution e = (v₂ – v₁)/(u₁ – u₂) is used for direct impacts.
需判断碰撞是完全弹性（动能守恒）还是非弹性。恢复系数e=(v₂-v₁)/(u₁-u₂)用于研究对心碰撞。
7. Moments and Equilibrium | 力矩与平衡
The moment of a force about a point is a measure of its turning effect. Moment = Force × perpendicular distance from the pivot to the line of action of the force. The unit is newton-metre (Nm).
力矩是力对某点的转动效应的量度。力矩=力×转轴到力作用线的垂直距离。单位是牛顿·米(Nm)。
For a body in equilibrium, the sum of the forces in any direction is zero and the sum of the moments about any point is zero. These conditions allow us to find unknown forces and distances.
处于平衡的物体，任何方向的合力为零，对任意点的合力矩也为零。利用这些条件可求解未知力和距离。
A uniform rod has its weight acting at its centre. When a rod is hinged, the reaction at the hinge can have both horizontal and vertical components.
匀质杆的重力作用于其中心。当杆用铰链连接时，铰链处的反力可分解为水平和竖直分量。
When taking moments, choose a point that eliminates as many unknown forces as possible, such as a hinge or the point where multiple forces intersect.
选取力矩中心时，应尽量使尽可能多的未知力通过该点（力矩为零），如铰链或多力交点。
Tilting: A body is on the point of tilting about a pivot when the normal reaction at all other supports becomes zero. This condition is used to determine maximum overhangs or sliding limits.
倾覆：当除支点外所有支撑处的法向反力变为零时，物体将绕支点翻倒。该条件用于求最大悬伸长度或稳定极限。
8. Projectile Motion | 抛体运动
A projectile moves under the influence of gravity alone, with constant downward acceleration g. The horizontal component of velocity remains constant, while the vertical component changes linearly with time.
抛体仅在重力作用下运动，具有恒定的向下加速度g。速度的水平分量保持不变，竖直分量随时间线性变化。
Using standard notation: horizontal displacement x = uₓ t, vertical displacement y = uᵧ t – ½gt², where uₓ = u cosθ and uᵧ = u sinθ.
采用标准符号：水平位移x=uₓt，竖直位移y=uᵧ t – ½gt²，其中uₓ=u cosθ，uᵧ=u sinθ。
The time of flight is found by setting y = 0 (assuming launch and landing at same vertical level): T = 2u sinθ / g. Maximum height occurs when vertical velocity becomes zero: H = u² sin²θ / (2g).
飞行时间由y=0求出（假设发射和落地同高）：T=2u sinθ/g。最大高度出现在竖直速度为零时：H=u² sin²θ/(2g)。
Range R = (u² sin 2θ) / g. Note that the maximum range for a given initial speed occurs when θ = 45°.
射程R=(u² sin 2θ)/g。给定初速度下，最大射程出现在θ=45°时。
Always resolve initial velocity into horizontal and vertical components. The kinematic equations are applied separately in each direction.
务必先将初速度分解为水平和竖直分量，然后分别对各方向应用运动学方程。
9. Work, Energy and Power | 功、能和功率
Work done by a constant force is given by W = Fs cosθ, where s is the displacement and θ is the angle between the force and the direction of motion. Work is measured in joules (J).
恒力做功定义为W=Fs cosθ，s为位移，θ为力与运动方向的夹角。功的单位是焦耳(J)。
Kinetic energy is the energy due to motion: KE = ½mv². Gravitational potential energy is given by GPE = mgh, where h is the vertical height above an arbitrary reference level.
动能是因运动而拥有的能量：KE=½mv²。重力势能GPE=mgh，h是相对于任意参考水平的竖直高度。
The work-energy principle states that the total work done by all forces (including gravity and friction) equals the change in kinetic energy between two points.
功能原理：所有力（包括重力和摩擦力）所做的总功等于物体动能的变化量。
Power is the rate of doing work. Average power = work done / time taken = Fv, where v is the speed of the object when the force F is applied in the direction of motion. The unit is the watt (W).
功率是做功的快慢。平均功率=做功/时间=Fv，其中v是物体在力F沿运动方向作用下的速度。单位是瓦特(W)。
In problems involving engines or resistive forces, use the relationship P = Fv to find the tractive force or maximum speed when power is constant.
涉及发动机或阻力的题目中，利用P=Fv可求牵引力或恒定功率下的最大速度。
10. Vectors in Mechanics | 力学中的向量
Many quantities in mechanics, such as displacement, velocity, acceleration and force, are vectors. They can be expressed in component form using unit vectors i and j (and k in 3D).
力学中许多物理量是矢量，如位移、速度、加速度和力。它们可以用单位向量i和j（三维时还有k）的分量形式表示。
To add vectors, add the i-components and j-components separately. The magnitude of a vector r = ai + bj is √(a² + b²), and its direction is given by an angle measured from the i direction.
向量相加时，分别对i分量和j分量求和。向量r=ai+bj的大小为√(a²+b²)，方向用与i方向的夹角表示。
For motion with variable velocity, the velocity vector is the derivative of the displacement vector with respect to time, and acceleration is the derivative of velocity.
对于变速运动，速度向量是位移向量对时间的导数，加速度是速度对时间的导数。
Relative velocity problems are solved by subtracting the velocity vectors: vB relative to A = vB – vA.
相对速度问题通过速度向量相减来解决：vB相对于A=vB–vA。
Using vectors can simplify problems where motion occurs on inclined planes or in two dimensions, as the SUVAT equations can be applied vectorially.
使用向量可简化斜面或二维运动问题，因为SUVAT方程可以直接用向量形式应用。
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