📚 A.2 Forces and Momentum | 力和动量
Forces and momentum lie at the heart of classical mechanics. In IB Physics Topic A.2, you will explore how forces cause changes in motion, the meaning of momentum, Newton's three laws, the impulse-momentum relationship, and the principle of conservation of momentum. These ideas allow us to analyse collisions, explosions, recoil and everyday situations involving moving objects, and they also underpin the later study of energy, circular motion and fields.
力和动量是经典力学的核心。在 IB 物理 A.2 主题中，你将探索力如何引起运动的变化、动量的含义、牛顿三大定律、冲量与动量的关系以及动量守恒原理。这些概念使我们能够分析碰撞、爆炸、反冲以及涉及运动物体的日常场景，它们也是后续学习能量、圆周运动和场的重要基础。
1. Introduction to Forces and Momentum | 力与动量导论
A force is a push or a pull that can change an object's speed, shape or direction. It is a vector quantity, having both magnitude and direction, and is measured in newtons (N). One newton is the force that gives a 1 kg mass an acceleration of 1 m s⁻².
力是可以改变物体速度、形状或方向的推或拉。它是一个矢量，既有大小又有方向，以牛顿 (N) 为单位。1 N 是使 1 kg 质量产生 1 m s⁻² 加速度的力。
Momentum is defined as the product of an object's mass and its velocity: p = m v. Like velocity, momentum is a vector and has units of kg m s⁻¹. Its direction is always the same as the direction of the velocity.
动量定义为物体质量与速度的乘积：p = m v。与速度一样，动量是矢量，单位为 kg m s⁻¹；它的方向始终与速度方向相同。
p = m v
动量 = 质量 × 速度
An object with larger mass or higher speed has greater momentum, making it harder to stop. The net force on an object determines the rate of change of its momentum, which links the study of forces directly to the study of momentum.
质量越大或速度越高的物体动量越大，因此更难停止。物体所受的净力决定了其动量变化的速率，这就把力的研究与动量的研究直接联系起来。
- Force: vector, unit N, changes motion or shape. | 力：矢量，单位 N，改变运动状态或形状。
- Momentum: vector, unit kg m s⁻¹, p = m v. | 动量：矢量，单位 kg m s⁻¹，p = m v。
- Net force controls the rate of change of momentum. | 净力决定动量变化的速率。
2. Newton's First Law and Inertia | 牛顿第一定律与惯性
Newton's first law states that an object remains at rest, or moves with constant velocity in a straight line, unless acted upon by a net external force. This property is called inertia, and it explains why seatbelts are needed in cars: a passenger keeps moving forward at constant velocity when a car brakes suddenly.
牛顿第一定律指出，物体将保持静止或匀速直线运动状态，除非受到净外力的作用。这一性质称为惯性，也解释了为什么汽车需要安全带：当汽车突然刹车时，乘客会以恒定速度继续向前运动。
If the resultant force on an object is zero, the object is said to be in translational equilibrium. The velocity remains constant, which means the momentum is also constant. Note that "constant velocity" includes the case of zero velocity, and that a body in equilibrium may still be moving.
如果作用在物体上的合力为零，则称该物体处于平动平衡状态。速度保持不变，这意味着动量也保持不变。要注意，"速度恒定"也包括速度为零的情形，而且处于平衡状态的物体仍可能在运动。
Inertia depends on mass: the greater the mass of an object, the greater its inertia and the more difficult it is to change its state of motion. Inertia is not a force, and mass is a scalar quantity measured in kilograms.
惯性取决于质量：物体的质量越大，惯性越大，其运动状态就越难被改变。惯性不是一种力，而质量是标量，以千克为单位。
- Zero resultant force ⇌ constant velocity ⇌ constant momentum. | 合力为零 ⇌ 速度恒定 ⇌ 动量不变。
- Equilibrium does not mean the object is stationary. | 平衡并不意味着物体静止。
- Larger mass means larger inertia. | 质量越大，惯性越大。
3. Newton's Second Law and Rate of Change of Momentum | 牛顿第二定律与动量变化率
Newton's second law can be written in two equivalent forms. In terms of acceleration, the resultant force equals mass times acceleration: F = m a. In terms of momentum, the resultant force equals the rate of change of momentum: F = Δp ÷ Δt.
牛顿第二定律可以写成两种等价形式。用加速度表示时，合力等于质量乘以加速度：F = m a。用动量表示时，合力等于动量变化的速率：F = Δp ÷ Δt。
F = m a and F = Δp ÷ Δt = (m v − m u) ÷ Δt
forms are consistent, because if mass is constant then Δp = m Δv and F = m Δv ÷ Δt = m a.
其中 u 是初速度，v 是末速度，Δt 是发生变化所用的时间间隔。两种形式是一致的：当质量不变时，Δp = m Δv，于是 F = m Δv ÷ Δt = m a。
city change is easier to measure than the acceleration. The unit check is straightforward: N = kg m s⁻², and Δp ÷ Δt also gives kg m s⁻¹ ÷ s = kg m s⁻².
当质量发生变化（例如火箭燃烧燃料）时，或者当力作用时间极短、测量速度变化比测量加速度更容易时，动量形式尤其有用。单位检验很简单：N = kg m s⁻²，而 Δp ÷ Δt 也得到 kg m s⁻¹ ÷ s = kg m s⁻²。
- F = m a requires a constant mass. | F = m a 要求质量不变。
- F = Δp ÷ Δt works even when mass changes. | F = Δp ÷ Δt 在质量变化时仍然成立。
- Force and acceleration are always in the same direction. | 力与加速度的方向始终相同。
4. Newton's Third Law | 牛顿第三定律
Newton's third law states that if body A exerts a force on body B, then body B exerts a force on body A of the same type, equal in magnitude and opposite in direction. These two forces are often called an action-reaction pair.
牛顿第三定律指出，如果物体 A 对物体 B 施加一个力，那么物体 B 也会对物体 A 施加一个同类型、大小相等、方向相反的力。这两个力通常称为作用力与反作用力。
Three conditions must be satisfied by a genuine third-law pair: the forces are of the same type (for example both gravitational, or both contact forces), they act on two different bodies, and they act along the same line. Because they act on different bodies, they never cancel each other out and must never be drawn on the same free-body diagram.
真正的第三定律力对必须满足三个条件：两力类型相同（例如都是引力或都是接触力）；作用在两个不同的物体上；沿同一直线作用。由于它们作用在不同物体上，所以永远不会相互抵消，也绝不能画在同一个受力分析图上。
Typical examples include the Earth pulling a book downwards while the book pulls the Earth upwards with an equal gravitational force; a swimmer pushing water backwards while the water pushes the swimmer forwards; and a rocket expelling gas downwards while the gas pushes the rocket upwards.
典型例子包括：地球向下吸引书本，同时书本以同样大小的引力向上吸引地球；游泳者向后推水，水则向前推游泳者；火箭向下喷出气体，气体则向上推火箭。
5. Free-Body Diagrams and Translational Equilibrium | 受力分析图与平动平衡
ould label its type and, where possible, its magnitude. Common forces include weight (W = m g acting downwards), normal contact force (perpendicular to a surface), friction (along a surface, opposing relative motion), tension and drag.
受力分析图展示作用在某一选定的单个物体上的所有力，用从某点或质心出发的箭头表示。每个力都应标注类型，并在可能时标注大小。常见的力包括重力（W = m g，方向向下）、法向接触力（垂直于表面）、摩擦力（沿表面，阻碍相对运动）、张力和阻力。
librium, the vector sum of the forces is zero. In component form this gives ΣFₓ = 0 and ΣFy omponent down the slope, m g sin θ, and a component perpendicular to the slope, m g cos θ.
对于处于平动平衡的物体，各力的矢量和为零。写成分量形式就是 ΣFₓ = 0 和 ΣFy = 0。在倾角为 θ 的斜面上，通常方便的做法是把重力分解为沿斜面向下的分量 m g sin θ 和垂直于斜面的分量 m g cos θ。
- Draw only the forces acting on the chosen object. | 只画作用在所选定物体上的力。
- Do not include the object's own force on other things. | 不要画该物体对其他物体施加的力。
- Choose axes along the direction of likely motion to simplify algebra. | 沿可能的运动方向选取坐标轴以简化运算。
6. Impulse and Force-Time Graphs | 冲量与力-时间图像
Impulse is defined as the product of a force and the time for which it acts: J = F Δt. From Newton's second law in momentum form, impulse equals the change in momentum: J = Δp = m v − m u. Impulse is a vector and has units of N s, which is identical to kg m s⁻¹.
冲量定义为力与该力作用时间的乘积：J = F Δt。由动量形式的牛顿第二定律可知，冲量等于动量变化：J = Δp = m v − m u。冲量是矢量，单位为 N s，与 kg m s⁻¹ 完全相同。
J = F Δt = Δp = m v − m u
On a force-time graph, the impulse is equal to the area under the graph. If the force varies, you must find the area (or estimate it by counting squares) rather than simply multiplying the peak force by the time; the average force is then the impulse divided by the total time.
在力-时间图像中，冲量等于图像下方的面积。如果力是变化的，就必须求面积（或通过数格子估算），而不能简单地用峰值力乘以时间；平均力则等于冲量除以总时间。
e force is reduced, since F = Δp ÷ Δt. Conversely, a short contact time such as a bat hitting a ball produces a large force.
这解释了许多安全设计。安全带、安全气囊或防撞栏延长了碰撞时间，因此在动量变化一定时，由于 F = Δp ÷ Δt，平均力被减小。相反，像球棒击球那样极短的接触时间则会产生很大的力。
7. Conservation of Momentum | 动量守恒
the system. For two bodies interacting along a line, this is written as: total momentum before = total momentum after.
动量守恒原理指出，只要系统不受合外力作用，系统的总动量就保持不变。对于沿同一直线相互作用的两物体，可写作：碰撞前的总动量 = 碰撞后的总动量。
m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂
Here u₁ and u₂ are the velocities before the interaction, and v₁ and v₂ are the velocities after it. Momentum is a vector, so velocities to the right may be taken as positive and velocities to the left as negative. Keeping signs consistent is the single most important step in these calculations.
其中 u₁ 和 u₂ 是相互作用前的速度，v₁ 和 v₂ 是相互作用后的速度。动量是矢量，因此可规定向右的速度为正、向左的速度为负。保持符号一致是这类计算中最重要的一步。
The condition "no resultant external force" is often met approximately: during a short collision the internal forces are far larger than external ones such as friction or air resistance, so momentum is conserved to a good approximation even if the system is not perfectly isolated.
"不受合外力"这一条件通常是近似满足的：在短暂碰撞过程中，内力远大于摩擦力、空气阻力等外力，因此即使系统并非完全孤立，动量也在很好的近似下守恒。
- Momentum is conserved in all collisions and explosions. | 动量在一切碰撞与爆炸中守恒。
- Kinetic energy is not always conserved. | 动能不一定守恒。
- Always define a positive direction before substituting values. | 代入数据前必须先规定正方向。
8. Elastic and Inelastic Collisions | 弹性碰撞与非弹性碰撞
the relative speed of approach equals the relative speed of separation: u₁ − u₂ = v₂ − v₁.
在弹性碰撞中，动量和动能都守恒。在两物体的一维弹性碰撞中，接近时的相对速度等于分离时的相对速度：u₁ − u₂ = v₂ − v₁。
In an inelastic collision, momentum is still conserved but kinetic energy is not: some of it is transferred to thermal energy, sound and deformation. A perfectly inelastic collision is one in which the bodies stick together and move with a common final velocity, producing the maximum possible loss of kinetic energy.
在非弹性碰撞中，动量仍然守恒，但动能不守恒：一部分动能转化为热能、声能和形变能。完全非弹性碰撞是指两物体粘在一起、以共同末速度运动的情况，此时动能损失最大。
c: m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ and ½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂²
| Type | 类型 | Momentum | 动量 | Kinetic energy | 动能 | Example | 例子 |
| Elastic 弹性 | Conserved 守恒 | Conserved 守恒 | Gas molecule collisions 气体分子碰撞 |
| Inelastic 非弹性 | Conserved 守恒 | Reduced 减少 | Car crash 汽车碰撞 |
| Perfectly inelastic 完全非弹性 | Conserved 守恒 | Maximum loss 损失最大 | Clay balls sticking 黏土球粘连 |
9. Explosions, Recoil and Rockets | 爆炸、反冲与火箭
In an explosion or a recoil event, the total momentum before the event is zero, so the total momentum afterwards must also be zero. Two bodies fly apart in opposite directions with momenta of equal magnitude:
在爆炸或反冲事件中，事件前的总动量为零，因此事件后的总动量也必须为零。两个物体会沿相反方向分开，动量大小相等：
0 = m₁ v₁ + m₂ v₂ so m₁ v₁ = −m₂ v₂
A rifle recoiling when fired and a rocket accelerating by expelling exhaust gas are both applications of the same idea. Notice that the lighter body always receives the greater speed, and that kinetic energy is not conserved in these events: chemical or elastic potential energy is converted into kinetic energy.
步枪射击时的后坐和火箭通过喷出燃气而加速，都是同一思想的应用。要注意，较轻的物体总是获得较大的速度；而且这类事件中动能不守恒，化学能或弹性势能被转化为动能。
- Total momentum stays zero at every instant. | 总动量在任何时刻都保持为零。
- Speeds are inversely proportional to masses. | 速度与质量成反比。
- Kinetic energy increases, supplied by stored energy. | 动能增加，由储存的能量提供。
10. Momentum in Two Dimensions | 二维动量问题
two perpendicular directions. In two dimensions we write Σpₓ (before) = Σpₓ (after) and Σpy (before) = Σpy (after), using pₓ = m v cos θ and py = m v sin θ.
由于动量是矢量，守恒必须分别在两个互相垂直的方向上应用。在二维情形中，我们写出 Σpₓ（前）= Σpₓ（后）与 Σpy（前）= Σpy（后），其中 pₓ = m v cos θ，py = m v sin θ。
A useful technique for two-dimensional collisions is to draw a vector triangle or a closed vector polygon: the sum of the momentum vectors before the collision must equal the sum of the momentum vectors afterwards.
处理二维碰撞的一个有用技巧是画出矢量三角形或封闭的矢量多边形：碰撞前的动量矢量之和必须等于碰撞后的动量矢量之和。
For example, a ball striking a wall at an angle and rebounding with the same speed but a reversed perpendicular component experiences a change in momentum of 2 m v cos θ perpendicular to the wall, directed away from the wall. This change, divided by the contact time, gives the average force exerted by the wall.
例如，一个球以一定角度撞击墙壁，并以相同速率反弹、垂直分量反向，则它经历垂直于墙面、背离墙面的动量变化 2 m v cos θ。该变化除以接触时间，即得墙施加的平均力。
11. Practical Work and Common Exam Mistakes | 实验技能与常见考试错误
Typical practical work uses a linear air track or a low-friction runway with two trolleys and light gates to measure velocities before and after a collision. The mass of each trolley is measured with a balance, and the momentum values are compared. Video analysis or a motion sensor can be used instead of light gates.
典型的实验使用直线气垫导轨或低摩擦滑轨，配合两辆小车和光电门来测量碰撞前后的速度。用天平测量每辆小车的质量，然后比较动量数值。也可以用视频分析或运动传感器代替光电门。
- Forgetting that momentum is a vector and losing negative signs. | 忘记动量是矢量而丢失负号。
- Confusing momentum p = m v with kinetic energy Eₖ = ½ m v². | 把动量 p = m v 与动能 Eₖ = ½ m v² 混淆。
- Assuming kinetic energy is conserved in every collision. | 误以为每一次碰撞动能都守恒。
- Using F = m a when the mass changes during the event. | 在事件中质量变化时仍使用 F = m a。
- Quoting impulse in joules instead of N s. | 把冲量的单位写成焦耳而不是 N s。
When writing answers, always state the principle you are using, define the positive direction, show the substitution with units, and give the final answer with a sensible number of significant figures and a direction where relevant.
作答时，务必说明所依据的原理、规定正方向、带单位写出代入过程，并给出有效数字合理、必要时标注方向的最终答案。
12. Summary of Key Equations | 核心公式总结
The following table collects the equations you should be able to recall and use confidently in questions on forces and momentum.
下表汇总了在力与动量相关题目中你应当能够熟练记起并运用的公式。
| Quantity | 物理量 | Equation | 公式 | Unit | 单位 |
| Force 力 | F = m a | N |
| Momentum 动量 | p = m v | kg m s⁻¹ |
| Rate of change of momentum 动量变化率 | F = Δp ÷ Δt | N |
| Impulse 冲量 | J = F Δt = Δp | N s |
| Conservation 动量守恒 | Σ p (before) = Σ p (after) | kg m s⁻¹ |
| Kinetic energy 动能 | Eₖ = ½ m v² | J |
| Weight 重力 | W = m g | N |
A final revision tip: practise problems that combine several ideas, such as a ball bouncing off a wall (impulse, then average force), a trolley collision (conservation, then energy check), and a rocket or gun (recoil). If you can explain each step in words as well as in symbols, you have mastered Topic A.2.
最后一条复习建议：多练习综合多个知识点的题目，例如球撞墙反弹（先求冲量、再求平均力）、小车碰撞（先用守恒、再检验动能）以及火箭或枪的后坐。如果你既能用符号计算、又能用语言解释每一步，就说明你已经掌握了 A.2 主题。
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