📚 Impulse and Momentum Conservation Formula Derivation | 冲量与动量守恒公式推导
In IB Physics, understanding impulse and momentum is essential for analysing interactions between objects, from billiard balls to rocket propulsion. This article provides a step-by-step derivation of the impulse-momentum theorem and the principle of conservation of momentum. You will learn how these concepts arise from Newton’s laws, how to apply them in collisions, and how to avoid common mistakes. The explanations are paired in English and Chinese to support international learners.
在IB物理中，理解冲量与动量对于分析物体间的相互作用（从台球碰撞到火箭推进）至关重要。本文逐步推导冲量-动量定理以及动量守恒原理。你将了解这些概念如何从牛顿定律产生，如何在碰撞中应用，以及如何避免常见错误。中英双语解释为国际学习者提供支持。
1. Defining Momentum | 定义动量
Momentum is a fundamental quantity in mechanics that describes an object’s quantity of motion. It depends on two factors: how much mass is moving and how fast it is moving. Momentum is a vector, meaning it has both magnitude and direction. The symbol for momentum is p, and it is defined as the product of mass m and velocity v.
动量是力学中描述物体运动量的基本物理量。它取决于两个因素：运动物体的质量以及它运动的速度。动量是矢量，既有大小也有方向。动量的符号为 p，其定义为质量 m 与速度 v 的乘积。
p = m v
Mass is measured in kilograms (kg), velocity in metres per second (m s⁻¹), and therefore the SI unit of momentum is kg m s⁻¹. Because velocity has direction, momentum also points in the same direction as the velocity of the object. When solving problems, always assign a positive direction and treat velocities with the correct signs.
质量以千克 (kg) 为单位，速度以米每秒 (m s⁻¹) 为单位，因此动量的国际单位是 kg m s⁻¹。由于速度有方向，动量也指向物体速度的方向。解题时，始终要指定一个正方向，并用正确的符号处理速度。
2. Understanding Impulse | 理解冲量
When a force acts on an object over a period of time, it can change the object’s momentum. The effect of this force acting over time is captured by the quantity called impulse. Impulse J is defined as the product of the average net force F and the time interval Δt during which the force acts.
当力在一段时间内作用在物体上时，它可以改变物体的动量。这种力随时间作用的效应由称为冲量的物理量描述。冲量 J 定义为平均净力 F 与力作用的时间间隔 Δt 的乘积。
J = F Δt
Like momentum, impulse is a vector; it has the same direction as the net force. The SI unit for impulse is the newton-second (N s), which is dimensionally equivalent to kg m s⁻¹, matching the unit of momentum. If the force is not constant, the impulse can be found by determining the area under a force–time graph. For a constant force, the area is simply a rectangle of height F and width Δt.
像动量一样，冲量是矢量，其方向与净力相同。冲量的国际单位是牛顿·秒 (N s)，量纲等同于 kg m s⁻¹，与动量的单位一致。如果力不是恒定的，可以通过求力–时间图下的面积来计算冲量。对于恒力，该面积只是一个高为 F、宽为 Δt 的矩形。
3. Deriving the Impulse-Momentum Theorem | 推导冲量-动量定理
The impulse-momentum theorem directly connects impulse to the change in momentum. To derive it, we start from Newton’s second law in its most common form, but expressed using acceleration.
冲量-动量定理将冲量与动量的变化直接联系起来。为了推导，我们从牛顿第二定律最常见的形式开始，但用加速度表示。
F = m a
Acceleration a is defined as the rate of change of velocity: a = Δv / Δt. Substituting this into Newton’s law gives F = m (Δv / Δt). Multiplying both sides by the time interval Δt yields F Δt = m Δv. Since the mass is constant in classical mechanics, the change in momentum Δp is equal to m Δv. Therefore, we obtain the impulse-momentum theorem.
加速度 a 定义为速度的变化率：a = Δv / Δt。代入牛顿定律得到 F = m (Δv / Δt)。两边同时乘以时间间隔 Δt 得出 F Δt = m Δv。因为在经典力学中质量不变，动量的变化 Δp 等于 m Δv。由此我们得到冲量-动量定理。
J = Δp
This theorem tells us that the impulse delivered to an object equals the change in its momentum. It is particularly useful when a large force acts for a very short time, such as during a collision or a bat striking a ball. Since the theorem is a vector equation, it must be applied with careful attention to sign conventions.
该定理告诉我们，作用在物体上的冲量等于其动量的变化。当一个大大的力作用在很短的时间时（例如碰撞或球棒击球过程中），它特别有用。由于这一定理是矢量方程，使用时必须仔细注意符号规定。
4. The Principle of Conservation of Momentum | 动量守恒原理
The principle of conservation of momentum states that for a system of objects, if no external resultant force acts, the total momentum remains constant. In other words, momentum cannot be created or destroyed within the system; it can only be transferred between objects.
动量守恒原理指出，对于物体系统，如果没有外力的合作用，总动量保持不变。换句话说，动量不能在系统内产生或消灭，只能在物体之间传递。
Σpinitial = Σpfinal
For a collision between two objects A and B, this is written as mAuA + mBuB = mAvA + mBvB, where u represents velocities before the interaction and v represents velocities after. This is a powerful tool for analysing collisions, explosions, and other interactions, provided the condition of zero net external force is met.
对于两个物体 A 和 B 之间的碰撞，该式写作 mAuA + mBuB = mAvA + mBvB，其中 u 表示相互作用前的速度，v 表示相互作用后的速度。只要满足合外力为零的条件，这就是分析碰撞、爆炸和其他相互作用的强有力工具。
5. Derivation of Momentum Conservation from Newton’s Third Law | 从牛顿第三定律推导动量守恒
Momentum conservation is not an independent law but arises from Newton’s laws, particularly the third law. Consider two objects A and B that interact (e.g., collide). During the interaction, object A exerts a force FAB on B, and object B exerts a force FBA on A.
动量守恒并非独立定律，而是源自牛顿定律，特别是第三定律。考虑两个相互作用（例如碰撞）的物体 A 和 B。在相互作用过程中，物体 A 对 B 施加力 FAB，物体 B 对 A 施加力 FBA。
According to Newton’s third law, these forces are equal in magnitude and opposite in direction: FAB = -FBA. They also act for exactly the same time interval Δt. We apply the impulse-momentum theorem to each object separately. For object A: FBA Δt = mAvA – mAuA. For object B: FAB Δt = mBvB – mBuB.
根据牛顿第三定律，这两个力大小相等、方向相反：FAB = -FBA。它们作用的时间间隔 Δt 也完全相同。我们对每个物体分别应用冲量-动量定理。对于物体 A：FBA Δt = mAvA – mAuA。对于物体 B：FAB Δt = mBvB – mBuB。
Adding these two equations gives (FBA + FAB)Δt = (mAvA + mBvB) – (mAuA + mBuB). Since FAB + FBA = 0, the left-hand side is zero. Therefore, the total final momentum equals the total initial momentum: mAuA + mBuB = mAvA + mBvB. This derivation shows that momentum conservation holds as long as the only forces present are internal to the system.
将这两个方程相加得到 (FBA + FAB)Δt = (mAvA + mBvB) – (mAuA + mBuB)。由于 FAB + FBA = 0，左边为零。因此，末总动量等于初总动量：mAuA + mBuB = mAvA + mBvB。这一推导表明，只要系统中仅存在内力，动量守恒就成立。
6. Types of Collisions | 碰撞的类型
In the IB syllabus, collisions are classified according to whether kinetic energy is conserved. It is essential to remember that momentum is conserved in all collisions, provided no external forces act.
在IB 教学大纲中，碰撞根据动能是否守恒进行分类。必须记住，只要没有外力作用，所有碰撞中动量都是守恒的。
Elastic collisions: Both momentum and kinetic energy are conserved. The objects bounce apart without permanent deformation or generation of heat. The two governing equations are:
弹性碰撞：动量和动能都守恒。物体弹开，没有永久形变或发热。两个控制方程是：
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂²
Inelastic collisions: Momentum is conserved, but kinetic energy is not. Some kinetic energy is transformed into other forms such as thermal energy or sound. In a perfectly inelastic collision, the objects stick together after impact and move with a common velocity v. The momentum equation simplifies to:
非弹性碰撞：动量守恒，但动能不守恒。部分动能转化为其他形式的能量，如热能或声能。在完全非弹性碰撞中，物体碰撞后粘在一起并以共同速度 v 运动。动量方程简化为：
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