📚 Further Mechanics 1: Momentum, Collisions, Work, Energy and Centres of Mass | 进阶力学1：动量、碰撞、功、能量与质心
Further Mechanics 1 builds on basic mechanics by introducing impulse, restitution, work-energy principles, elastic potential energy and centres of mass. This article covers the core ideas, standard formulas and common exam techniques needed for A-level Further Mathematics.
进阶力学1在基础力学之上引入冲量、恢复系数、功与能量原理、弹性势能以及质心等内容。本文覆盖A-level进阶数学所需的核心概念、标准公式和常见解题技巧。
1. Impulse and Momentum in One Dimension | 一维冲量与动量
he change in momentum caused by a force acting over a time interval, so J = FΔt = mv − mu.
动量定义为p = mv，其中m为质量，v为速度。冲量是力在一段时间内产生的动量变化，因此J = FΔt = mv − mu。
In exam questions, you will often be given a force-time graph. The impulse is the area under the graph, taking signed areas into account when the force changes direction.
在考试题中，常会给出力-时间图像。冲量等于图像下方的面积，当力的方向改变时需要考虑正负面积。
J = FΔt = Δp = m(v − u)
Remember that momentum and impulse are vector quantities. In one-dimensional problems, choose a positive direction and apply signs consistently to u and v.
请记住动量和冲量都是矢量。在一维问题中，选择一个正方向并对u和v始终一致地使用正负号。
2. Conservation of Momentum and Collisions | 动量守恒与碰撞
₂u₂ = m₁v₁ + m₂v₂, where u are velocities before impact and v are velocities after impact.
当系统不受外力作用时，总动量守恒。对于两个物体直接碰撞，m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂，其中u为碰撞前速度，v为碰撞后速度。
This principle applies even if the collision is elastic or inelastic. It is the first key equation to write down in any collision problem, and it must be used with a clear positive direction.
无论碰撞是弹性还是非弹性，该原理均成立。它是任何碰撞问题中首先要写出的关键方程，并且必须配合明确的正方向使用。
with a common velocity v. The conservation equation then becomes m₁u₁ + m₂u₂ = (m₁ + m₂)v.
对于完全非弹性碰撞，两个物体粘在一起并以共同速度v运动。动量守恒方程变为m₁u₁ + m₂u₂ = (m₁ + m₂)v。
3. Newton's Law of Restitution | 牛顿恢复系数定律
Newton's experimental law of restitution states that for a direct collision between two bodies, the relative speed after collision is e times the relative speed before collision, but in the opposite direction.
牛顿的实验性恢复定律指出，两个物体直接碰撞时，碰撞后的相对速度大小等于碰撞前相对速度大小的e倍，但方向相反。
e = (v₂ − v₁) / (u₁ − u₂)
y elastic and kinetic energy is conserved. If e = 0, the collision is perfectly inelastic.
恢复系数e满足0 ≤ e ≤ 1。若e = 1，碰撞为完全弹性，动能守恒；若e = 0，碰撞为完全非弹性。
ration / speed of approach. The wall is treated as having zero velocity and infinite mass.
当球撞击固定墙面时，公式简化为e = 分离速度 / 接近速度。墙面被视为速度为零且质量无穷大。
4. Oblique Collisions and Vector Impulse | 斜碰与矢量冲量
In an oblique collision with a smooth wall, the velocity component parallel to the wall is unchanged, while the perpendicular component is reversed and multiplied by e.
在与光滑墙面发生斜碰时，平行于墙面的速度分量保持不变，而垂直于墙面的分量方向反转并乘以e。
Write the velocity as a vector using unit vectors i and j. If the wall is vertical and the ball approaches with velocity u i + v j, after impact the perpendicular component becomes −e u i, while v j remains the same.
使用单位向量i和j将速度写成矢量形式。如果墙面竖直且球以速度u i + v j接近，碰撞后垂直分量变为−e u i，而v j保持不变。
the change in momentum in the perpendicular direction: J = m(−e u) − m(u) = −m u(1 + e) i.
墙对球施加的冲量等于垂直方向上动量的变化：J = m(−e u) − m(u) = −m u(1 + e) i。
For oblique collisions between two spheres, resolve along the line of centres and perpendicular to it. Apply restitution along the line of centres and conserve momentum in both directions.
对于两个球体的斜碰，沿连心线方向和垂直方向分解。沿连心线应用恢复系数，并在两个方向上分别应用动量守恒。
5. Work, Kinetic Energy and Power | 功、动能与功率
application. For a constant force F acting at an angle θ to displacement d, W = F d cos θ.
当力使其作用点发生位移时，力做了功。对于与位移d夹角为θ的恒力F，W = F d cos θ。
Kinetic energy is the energy possessed by a moving body: Ek inciple states that the total work done on a particle equals its change in kinetic energy.
动能是运动物体具有的能量：Ek = ½mv²。功-能量原理指出，对质点所做的总功等于其动能的变化量。
Wtotal = ½mv² − ½mu²
on a body moving with velocity v, P = Fv. Power is measured in watts, where 1 W = 1 J s⁻¹.
功率是做功的速率。对于作用在速度为v的物体上的恒力，P = Fv。功率的单位是瓦特，1 W = 1 J s⁻¹。
force equals the resistance force, so P = Fv can be used to find speed or tractive force.
当车辆匀速行驶时，驱动力等于阻力，因此可使用P = Fv求速度或牵引力。
6. Gravitational Potential Energy and Energy Conservation | 重力势能与能量守恒
Gravitational potential energy is the energy a body has due to its height above a chosen reference level: Ep = mgh.
重力势能是物体由于相对于选定参考平面的高度而具有的能量：Ep = mgh。
If the only forces doing work are gravity and other conservative forces, total mechanical energy is conserved: Ek + Ep = constant.
如果做功的力只有重力和其他保守力，总机械能守恒：Ek + Ep = 常数。
ss in gravitational potential energy equals the gain in kinetic energy: mgh = ½mv² − ½mu².
对于沿光滑斜面下滑的质点，减少的重力势能等于增加的动能：mgh = ½mv² − ½mu²。
must be included: total energy before = total energy after + work done against resistance.
当摩擦力或空气阻力等非保守力做功时，必须计入克服这些阻力所做的功：初始总能量 = 末态总能量 + 克服阻力所做的功。
7. Elastic Strings and Hooke's Law | 弹性绳与胡克定律
An elastic string or spring obeys Hooke's law until its elastic limit. The tension T is proportional to the extension x measured from its natural length L.
弹性绳或弹簧在弹性限度内遵循胡克定律。张力T与从原长L开始测量的伸长量x成正比。
T = λx / L
For a spring, the same law is often written as T = kx, where k is the stiffness constant.
其中λ为弹性模量，单位为牛顿。对于弹簧，同一定律通常写成T = kx，其中k为劲度系数。
An elastic string only exerts tension when stretched, so it goes slack when its length is less than or equal to its natural length. A spring can exert both tension and thrust.
弹性绳只有在被拉伸时才产生张力，因此当长度小于或等于原长时绳子会松弛。弹簧则可以产生拉力和推力。
other applied forces. Start by writing the extension x = current length − natural length.
在平衡问题中，弹性绳或弹簧的张力与重力或其他外力平衡。解题时首先写出伸长量x = 当前长度 − 原长。
8. Elastic Potential Energy | 弹性势能
When an elastic string or spring is stretched by an amount x, it stores elastic potential energy. The formula is:
当弹性绳或弹簧被拉伸x时，会储存弹性势能。公式为：
Ee = λx² / (2L) = ½kx²
This energy is scalar and is measured in joules. It represents the work done in stretching the material from its natural length to the extended length.
该能量为标量，单位为焦耳。它表示材料从原长拉伸到伸长后长度所做的功。
In conservation of energy problems, include elastic potential energy alongside kinetic and gravitational potential energy. For example, a particle attached to a stretched string may convert elastic potential energy into kinetic energy and gravitational potential energy.
在能量守恒问题中，应将弹性势能与动能和重力势能一起考虑。例如，连接在拉伸绳子上的质点可将弹性势能转化为动能和重力势能。
Be careful: EPE is always positive and depends on extension squared. If a spring is compressed, the same formula applies with x representing compression.
注意：弹性势能始终为正，并与伸长量的平方有关。如果弹簧被压缩，同一公式也适用，此时x表示压缩量。
9. Centres of Mass of Composite Bodies | 组合体质心
The centre of mass of a system of particles is the average position weighted by mass. For particles in a plane:
质点系的质心是各质点位置按质量加权的平均位置。对于平面内的质点系：
x̄ = Σ mᵢ xᵢ / Σ mᵢ, ȳ = Σ mᵢ yᵢ / Σ mᵢ
For uniform laminae, the mass is proportional to area, so you can use area instead of mass. The centre of mass of a uniform rod lies at its midpoint; for a uniform rectangular lamina, it lies at the intersection of the diagonals.
对于均匀薄片，质量与面积成正比，因此可以用面积代替质量。均匀杆的质心在其中点；均匀矩形薄片的质心在对角线交点处。
For a composite body, split it into simple shapes, find the position and mass or area of each part, then apply the weighted average formula. Use a table to organise calculations.
对于组合体，将其拆分成简单形状，找出每一部分的位置和质量或面积，然后应用加权平均公式。可使用表格来组织计算。
For a uniform triangular lamina, the centre of mass lies at the intersection of the medians, two-thirds of the way from each vertex to the midpoint of the opposite side.
对于均匀三角形薄片，质心位于中线的交点处，即从每个顶点到对边中点距离的三分之二处。
10. Equilibrium, Toppling and Sliding | 平衡、倾覆与滑动
ent of weight down the plane exceeds limiting friction: mg sin θ > μR, where R = mg cos θ.
对于静止在粗糙斜面上的物体，当下滑分力超过极限摩擦力时开始滑动：mg sin θ > μR，其中R = mg cos θ。
Toppling occurs when the line of action of the weight falls outside the base of the body. For a uniform block on an inclined plane, toppling occurs when tan θ > b/h, where b is the base width and h is the height of the centre of mass above the base.
当重力的作用线超出物体底面时会发生倾覆。对于斜面上的均匀长方体，当tan θ > b/h时发生倾覆，其中b为底面宽度，h为质心距离底面的高度。
In exam questions, compare the angle at which sliding would occur with the angle at which toppling would occur. The smaller angle determines which happens first.
在考试题中，比较发生滑动和发生倾覆的角度。较小的角度决定哪种情况先发生。
Always draw a clear diagram showing the weight, normal reaction, friction and any applied forces, with the centre of mass marked accurately.
始终画出清晰的受力图，标出重力、法向反作用力、摩擦力和任何外力，并准确标出质心位置。
11. Connected Particles and Energy Methods | 连接体与能量法
When two particles are connected by a light inextensible string passing over a smooth pulley, each particle has the same acceleration and the same speed at any instant.
当两个质点通过跨过光滑滑轮的轻质不可伸长绳连接时，每个质点在任意时刻具有相同的加速度和相同的速率。
You can solve such systems using Newton's second law for each particle separately, or by using the energy method. In the energy method, the loss in potential energy of the heavier particle equals the increase in kinetic energy of both particles plus any work done against resistance.
可以分别对每个质点使用牛顿第二定律求解此类系统，也可以使用能量法。在能量法中，较重质点减少的势能等于两个质点增加的动能加上克服阻力所做的功。
between two points and combine it with the radial equation of motion T − mg cos θ = mv²/r.
对于系在绳上在竖直平面内做圆周运动的质点，在两点之间使用能量守恒，并结合径向运动方程T − mg cos θ = mv²/r。
When an elastic string is involved, include its elastic potential energy. The total energy of the system is then kinetic energy + gravitational potential energy + elastic potential energy.
当涉及弹性绳时，应计入其弹性势能。系统的总能量为动能 + 重力势能 + 弹性势能。
12. Exam Strategy and Common Errors | 备考策略与常见错误
Start every collision problem by drawing a diagram and defining a positive direction. Write down conservation of momentum and restitution equations before substituting numbers.
解决每个碰撞问题时，先画图并规定正方向。在代入数值之前，先写出动量守恒方程和恢复系数方程。
t collisions, energy is lost as heat and sound, so apply conservation of momentum instead.
常见错误1：在碰撞中使用能量守恒。动能仅在e = 1时守恒；在大多数碰撞中，能量会以热和声的形式损失，因此应使用动量守恒。
Common error 2: forgetting that impulse is a vector. In oblique collisions, only the perpendicular component of velocity changes for a smooth wall.
常见错误2：忘记冲量是矢量。在斜碰中，对于光滑墙面，只有垂直于墙面的速度分量发生变化。
Common error 3: using the wrong sign for extension or velocity. Always check the physical direction and keep signs consistent throughout the calculation.
常见错误3：伸长量或速度的正负号使用错误。始终检查物理方向，并在整个计算过程中保持符号一致。
Finally, practise past-paper questions under timed conditions. Focus on setting up equations clearly, using a table for centres of mass, and checking that final answers have correct units and signs.
最后，在限时条件下练习历年真题。重点在于清晰地建立方程、使用表格处理质心问题，并检查最终答案的单位和符号是否正确。
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