Momentum Key Points for IB Edexcel Physics | IB Edexcel 物理：动量 考点精讲

📚 Momentum Key Points for IB Edexcel Physics | IB Edexcel 物理：动量 考点精讲

Momentum is a fundamental concept in physics that combines mass and velocity to describe the ‘quantity of motion’ of an object. It is a vector quantity central to understanding collisions, explosions, and the action of forces over time. This article covers the essential principles, equations, and applications of momentum for IB and Edexcel Physics, including impulse, conservation of momentum, collision types, and two‑dimensional problems.

动量是物理学的基本概念，结合质量和速度来描述物体的“运动量”。它是矢量，对理解碰撞、爆炸及力随时间的作用至关重要。本文涵盖IB和Edexcel物理中动量的核心原理、公式及应用，包括冲量、动量守恒、碰撞类型和二维问题。

1. What is Momentum? | 什么是动量？

Momentum (p) is defined as the product of an object’s mass m and its velocity v. It is a vector quantity with the same direction as velocity.

动量 (p) 定义为物体质量 m 与速度 v 的乘积，是矢量，方向与速度相同。

p = m v

The SI unit of momentum is kilogram metre per second (kg m s⁻¹), which is equivalent to a newton second (N s). For a system of several particles, the total momentum is the vector sum of the individual momenta: P = Σpᵢ.

动量的国际单位是千克米每秒 (kg·m·s⁻¹)，等效于牛秒 (N·s)。对于多个质点组成的系统，总动量是各质点动量的矢量和：P = Σpᵢ。

Momentum provides a measure of how difficult it is to stop a moving object. A heavier object or a faster object has greater momentum.

动量衡量使运动物体停止的难易程度。质量越大或速度越大，动量越大。

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2. Impulse and the Impulse–Momentum Theorem | 冲量与动量定理

Impulse (J) is the change in momentum of an object when a resultant force F acts on it for a time interval Δt.

冲量 (J) 是合力 F 在一段时间间隔 Δt 内作用引起的动量变化。

J = F Δt = Δp = m v − m u

Impulse is a vector quantity with unit N s. The impulse–momentum theorem states that the impulse delivered to an object equals the change in its momentum. This follows directly from Newton’s second law: F = Δp/Δt.

冲量是矢量，单位 N·s。动量定理指出，物体所受冲量等于其动量的变化。这直接由牛顿第二定律 F = Δp/Δt 推导而来。

When a force varies with time, the impulse is the area under a force–time graph. A large force acting for a short time can produce the same impulse as a small force acting for a long time.

当力随时间变化时，冲量等于力-时间图下的面积。大力作用短时间与小力作用长时间可以产生相同的冲量。

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3. Law of Conservation of Momentum | 动量守恒定律

In a closed system (no external forces act), the total momentum before an interaction equals the total momentum after the interaction. This principle is a direct consequence of Newton’s third law.

在封闭系统（无外力作用）中，相互作用前的总动量等于相互作用后的总动量。该原理是牛顿第三定律的直接推论。

Σ p_initial = Σ p_final

Internal forces between objects in the system occur in equal and opposite pairs. Their impulses cancel, leaving the total momentum unchanged. The law applies to all types of collisions and explosions, provided no net external force is present.

系统内物体间的内力以等大反向的成对形式出现，它们的冲量相互抵消，使总动量保持不变。只要没有净外力，该定律适用于所有碰撞和爆炸。

For a two‑body system, this is often written as m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where u is initial velocity and v is final velocity.

对于两个物体组成的系统，常写作 m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂，其中 u 为初速度，v 为末速度。

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4. Elastic Collisions | 弹性碰撞

An elastic collision is one in which both momentum and kinetic energy are conserved. There is no loss of kinetic energy to heat, sound or permanent deformation.

弹性碰撞是指动量和动能都守恒的碰撞。动能不会转化为热、声或永久形变。

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂²

A useful relation derived from these equations is that the relative speed of approach equals the relative speed of separation: u₁ − u₂ = v₂ − v₁. In a head‑on elastic collision between equal masses, the two objects simply exchange velocities.

由以上方程组可导出一个有用关系：相对靠近速度等于相对分离速度：u₁ − u₂ = v₂ − v₁。在两等质量物体的一维弹性碰撞中，它们会交换速度。

In IB and Edexcel exams, you may need to solve simultaneous equations or use the relative speed shortcut for one‑dimensional elastic collisions.

在IB和Edexcel考试中，你可能需要求解方程组，或利用相对速度关系快速解决一维弹性碰撞问题。

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5. Inelastic Collisions | 非弹性碰撞

In an inelastic collision, momentum is conserved but kinetic energy is not conserved. Some of the initial kinetic energy is transformed into other forms such as internal energy, sound or plastic deformation.

在非弹性碰撞中，动量守恒但动能不守恒。部分初始动能转化为其他形式的能量，如内能、声音或塑性形变。

The coefficient of restitution e quantifies the elasticity of a collision:

恢复系数 e 用于量化碰撞的弹性程度：

e = (relative speed of separation) / (relative speed of approach)

For a perfectly elastic collision, e = 1. For an inelastic collision, 0 < e < 1. In Edexcel specifications, the coefficient of restitution is often used to analyse collisions; IB students should understand the concept of energy loss without necessarily referring to e.

完全弹性碰撞 e = 1，非弹性碰撞 0 < e < 1。Edexcel 考试中常使用恢复系数分析碰撞；IB 学生则应理解能量损失的概念，不一定引入 e。

Calculating the kinetic energy before and after the collision reveals the loss: ΔKE = KE_initial − KE_final.

计算碰撞前后的动能差可得到损失量：ΔKE = KE_初 − KE_末。

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6. Perfectly Inelastic Collisions | 完全非弹性碰撞

A perfectly inelastic collision is a special case where the colliding bodies stick together and move with a common velocity after impact. The coefficient of restitution is zero (e = 0), and the loss of kinetic energy is maximum.

完全非弹性碰撞是一种特殊情形，碰撞后物体粘在一起并以共同速度运动。恢复系数为零 (e = 0)，动能损失最大。

m₁u₁ + m₂u₂ = (m₁ + m₂) v

The common final velocity v is found from the momentum conservation equation above. The kinetic energy after the collision is always less than before, and the ‘lost’ energy can be calculated as ΔKE = ½ m₁u₁² + ½ m₂u₂² − ½ (m₁+m₂)v².

共同末速度 v 由上方的动量守恒方程求得。碰撞后动能总小于碰撞前，“损失” 的动能可用 ΔKE = ½ m₁u₁² + ½ m₂u₂² − ½ (m₁+m₂)v² 计算。

Classic examples include a bullet embedding itself in a wooden block (ballistic pendulum) or two lumps of clay colliding. Such problems often combine momentum conservation with energy considerations.

典型例子包括子弹嵌入木块（冲击摆）或两块黏土碰撞。这类问题常将动量守恒与能量分析相结合。

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7. Explosions and Recoil | 爆炸与反冲

An explosion can be viewed as the reverse of a perfectly inelastic collision. Before the explosion, the system is often stationary with zero total momentum. After the explosion, the fragments move apart, but the vector sum of their momenta remains zero.

爆炸可视为完全非弹性碰撞的逆过程。爆炸前系统通常静止，总动量为零。爆炸后碎片飞散，但它们的动量矢量和仍为零。

0 = m₁v₁ + m₂v₂ + … (all vectors)

In a gun recoil problem, the bullet and the gun gain equal and opposite momenta. For a bullet of mass m and velocity v, the recoil speed of the gun (mass M) is V = − (m/M) v. Kinetic energy is not conserved; it comes from the chemical energy of the propellant.

在枪械反冲问题中，子弹和枪获得大小相等、方向相反的动量。对于质量为 m 的子弹以速度 v 射出，枪（质量 M）的反冲速度为 V = − (m/M) v。动能并不守恒，它来自推进剂的化学能。

Explosion problems are solved by applying conservation of momentum exactly as with collisions, but with the initial total momentum being zero.

爆炸问题的解法与碰撞完全相同，只是初始总动量为零。

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8. Momentum in Two Dimensions | 二维动量

When collisions or explosions occur in a plane, momentum must be conserved independently in perpendicular directions. Choose a convenient coordinate system (e.g. x‑axis along the direction of one object’s initial velocity) and resolve all momenta into components.

当碰撞或爆炸发生在一个平面内时，动量必须在相互垂直的方向上分别守恒。选择合适的坐标系（如取某一物体初速度方向为 x 轴），将所有动量分解为分量。

Σ p_x (before) = Σ p_x (after)

Σ p_y (before) = Σ p_y (after)

Use the component equations to find unknown speeds or directions. Vector diagrams and trigonometry (sine/cosine rules) are often helpful, especially when dealing with right‑angled collisions or equal‑mass objects in an oblique elastic collision, where the two objects move off at 90° to each other.

利用分量方程求解未知速度或方向。矢量图和三角法（正弦/余弦定理）常有用，尤其是处理直角碰撞或等质量物体的斜向弹性碰撞，此时两物体分开的角度互成 90°。

Typical IB and Edexcel exam problems involve billiard ball collisions, explosions of stationary objects into two or three fragments, or nuclear decay scenarios where a nucleus splits into particles.

典型的IB和Edexcel考题包括台球碰撞、静止物体爆炸成两三个碎片，或原子核衰变分裂为粒子的情景。

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9. Force–Time Graphs and Impulse | 力-时间图像与冲量

The impulse experienced by an object can be determined from the area under a force–time (F–t) graph. For a constant force, this is simply the area of a rectangle: Impulse = F × Δt.

物体受到的冲量可由力-时间 (F–t) 图下的面积得出。对于恒力，就是矩形的面积：冲量 = F × Δt。

For a variable force, the area must be found by counting squares or by integration if the function is known. The impulse–momentum theorem links this area directly to Δp = mv − mu, allowing determination of final velocity or change in motion.

对于变力，需通过数格或用已知函数积分求面积。动量定理将该面积与 Δp = mv − mu 直接关联，可据此求末速度或运动变化。

Impulse = Area under F–t graph = Δp

Safety devices such as airbags and crumple zones are designed to increase the collision time Δt. For a given change in momentum, a longer time produces a smaller average force, reducing injury.

安全气囊和溃缩区等装置能延长碰撞时间 Δt。对于给定的动量变化，时间越长平均力越小，从而减轻伤害。

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10. Experimental Verification of Momentum Conservation | 实验验证动量守恒

A classic laboratory experiment uses a linear air track with two gliders and light gates to verify momentum conservation. Velocities before and after a collision or explosion are recorded, and the total momenta are compared.

经典实验室实验使用气垫导轨、两个滑行器和光门来验证动量守恒。记录碰撞或爆炸前后的速度，并比较总动量。

One glider can be given an initial push and allowed to collide with a stationary glider with or without a Velcro strip (for perfectly inelastic collisions). Light gates measure the interruption time for a card of known length, giving v = d/t. Alternatively, ticker timers or video motion analysis can be used.

一个滑行器被推动后，与静止滑行器发生碰撞（可贴尼龙搭扣实现完全非弹性碰撞）。光门测量已知长度的遮光板通过时间，得出 v = d/t。也可使用打点计时器或视频运动分析。

The initial total momentum, m₁u₁ + m₂u₂, is compared with the final total momentum, m₁v₁ + m₂v₂. Within experimental uncertainties (friction, timing errors), they should be equal. This directly supports the law of conservation of momentum.

将初始总动量 m₁u₁ + m₂u₂ 与末总动量 m₁v₁ + m₂v₂ 比较。在实验误差范围内（摩擦、计时误差），它们应相等，直接支持动量守恒定律。

In an explosion experiment, the two gliders are initially at rest and held together by a compressed spring or small explosive charge. After release, their speeds are measured, and the momenta are expected to be equal in magnitude but opposite in direction, summing to zero.

在爆炸实验中，两滑行器起初静止并用压缩弹簧或小炸药包相连。释放后，测量其速度，预期动量大小相等方向相反，总合为零。

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