📚 Momentum: Key Points Review for IB & CIE Physics | 动量 考点精讲
Momentum is a cornerstone of mechanics that appears frequently in IB and CIE Physics examinations. Understanding how to apply momentum conservation to collisions, explosions and impulse problems is essential for top marks. This article compiles the most critical definitions, laws, problem-solving methods and common pitfalls you need to master.
动量是力学中的基石，在 IB 和 CIE 物理考试中频繁出现。理解如何将动量守恒应用于碰撞、爆炸和冲量问题是获得高分的关键。本文汇集了你必须掌握的最关键定义、定律、解题方法以及常见陷阱。
1. Definition of Momentum | 动量的定义
Momentum, often symbolised by p, is the product of an object’s mass and velocity: p = m v. Because velocity is a vector, momentum is also a vector quantity – its direction is identical to the direction of the velocity. In the International System, the unit is kg m s⁻¹.
动量常用符号 p 表示，是物体的质量与速度的乘积：p = m v。由于速度是矢量，动量也是矢量——其方向与速度方向相同。在国际单位制中，动量的单位是 kg m s⁻¹。
For a system of particles, the total momentum is the vector sum of the individual momenta. A 2 kg cart moving at 3 m s⁻¹ to the right has a momentum of 6 kg m s⁻¹ to the right; if a second cart of 1 kg moves at 4 m s⁻¹ to the left, the total system momentum is 2 kg m s⁻¹ to the right.
对于多个质点组成的系统，总动量是各个质点动量的矢量和。一辆 2 kg 的小车以 3 m s⁻¹ 向右运动，动量为 6 kg m s⁻¹ 向右；若另一辆 1 kg 的小车以 4 m s⁻¹ 向左运动，则系统总动量为 2 kg m s⁻¹ 向右。
2. Impulse–Momentum Theorem | 冲量–动量定理
Impulse J measures the effect of a net force acting over a time interval Δt. It is defined as J = Fnet Δt. The theorem states that the impulse delivered to an object equals its change in momentum: J = Δp = m v − m u, where u is initial velocity and v is final velocity.
冲量 J 衡量力在一段时间间隔 Δt 内的作用效果，定义为 J = Fnet Δt。冲量–动量定理指出，作用在物体上的冲量等于其动量的变化量：J = Δp = m v − m u，其中 u 是初速度，v 是末速度。
J = F Δt = Δp
If the force varies, the impulse equals the area under a force–time graph. This relationship is extremely useful for finding the average force during a collision when the contact time is known. For example, a 500 N force acting for 0.02 s gives an impulse of 10 N s, changing a 2 kg mass’s velocity by 5 m s⁻¹.
如果力随时间变化，冲量等于力–时间图下的面积。当已知碰撞接触时间时，这一关系对于求平均力极为有用。例如，一个 500 N 的力作用 0.02 s 产生 10 N s 的冲量，使 2 kg 物体的速度改变 5 m s⁻¹。
3. Law of Conservation of Momentum | 动量守恒定律
For a closed, isolated system (no net external force), the total linear momentum remains constant. Mathematically: Σ pinitial = Σ pfinal. The law is a direct consequence of Newton’s third law: internal forces between objects occur in equal and opposite pairs, cancelling out when summing the system’s momentum change.
对于一个封闭、孤立的系统（无合外力），总线性动量保持恒定。数学表达式为：Σ pinitial = Σ pfinal。该定律是牛顿第三定律的直接推论：物体间的内力成对出现，大小相等、方向相反，在考虑系统总动量变化时相互抵消。
This principle applies to any event – collisions, explosions or separations – as long as external forces are negligible or balanced. Even if external forces act, momentum may be considered approximately conserved when the collision time is very short, such as in a snooker ball impact.
该原理适用于任何事件——碰撞、爆炸或分离——只要外力可忽略或互相平衡。即使有外力作用，若碰撞时间极短（如斯诺克球撞击），也可近似认为动量守恒。
4. Elastic Collisions | 弹性碰撞
An elastic collision is one in which both momentum and total kinetic energy are conserved. The two conditions for two bodies of masses m₁ and m₂ are:
弹性碰撞是指动量和总动能均守恒的碰撞。对于质量分别为 m₁ 和 m₂ 的两个物体，条件为：
m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂
½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂²
Idealised collisions between gas molecules, perfectly hard billiard balls or atomic particles can be modelled as elastic. A special one‑dimensional case: when two identical masses collide elastically with one initially at rest, they exchange velocities – the incident mass stops, and the target moves away with the incident speed.
理想化的气体分子碰撞、完全刚性的台球或原子粒子碰撞可模拟为弹性碰撞。一个特殊的一维情形：当两个相同质量的物体弹性碰撞且其中一个初始静止时，它们交换速度——入射物体停止，目标物体以入射速度离开。
5. Inelastic Collisions | 非弹性碰撞
In an inelastic collision, momentum is conserved but kinetic energy is not. The ‘missing’ kinetic energy is transformed into other forms such as internal energy (heat, sound, permanent deformation). Most everyday collisions – car crashes, a ball hitting a wall – are inelastic to some degree.
在非弹性碰撞中，动量守恒但动能不守恒。“消失”的动能转化为其他形式，如内能（热能、声能、永久形变）。日常生活中的大多数碰撞——汽车撞车、球撞击墙壁——在某种程度上都是非弹性的。
The coefficient of restitution e, defined as the ratio of relative speed after collision to relative speed before collision, quantifies the elasticity. For a perfectly elastic collision e = 1, for a perfectly inelastic collision e = 0, and real collisions have 0 < e < 1. Although e is often examined conceptually, you should know how it relates to energy loss.
恢复系数 e 定义为碰撞后相对速度与碰撞前相对速度的比值，用以量化弹性程度。完全弹性碰撞 e = 1，完全非弹性碰撞 e = 0，实际碰撞的 e 在 0 到 1 之间。虽然 IB 和 CIE 通常只要求定性地理解 e，但了解其与能量损失的关系很有帮助。
6. Perfectly Inelastic Collisions | 完全非弹性碰撞
A perfectly inelastic collision is a limiting case where the colliding bodies stick together after impact and move with a common velocity. Only momentum conservation is used; kinetic energy is not conserved. If a mass m₁ moving at u collides with a stationary mass m₂ and they coalesce, the final velocity v is given by:
完全非弹性碰撞是一种极限情况，碰撞后两物体粘在一起以共同速度运动。只应用动量守恒，动能不守恒。若质量为 m₁ 的物体以速度 u 撞击静止的质量 m₂ 并合为一体，末速度 v 为：
m₁ u = (m₁ + m₂) v
This type of collision loses the maximum possible kinetic energy, making it the opposite extreme to an elastic collision. An example is a bullet embedding itself in a wooden block in a ballistic pendulum – a classic problem to determine the bullet’s speed.
这类碰撞损失的动能最大，是与弹性碰撞相对的另一个极端。一个典型例子是子弹嵌入木块形成弹道摆——这是测定子弹速度的经典问题。
7. Explosions | 爆炸
An explosion can be viewed as the reverse of a perfectly inelastic collision. A single object, often initially at rest, breaks into two or more fragments. The total momentum before the explosion is zero, so the vector sum of the fragments’ momenta after the explosion must also be zero.
爆炸可以看作完全非弹性碰撞的逆过程。一个物体（通常最初静止）分裂成两个或多个碎片。爆炸前的总动量为零，因此爆炸后碎片动量的矢量和也应为零。
If a stationary firework splits into two parts of masses m₁ and m₂ moving with velocities v₁ and v₂, then m₁ v₁ + m₂ v₂ = 0 (vector equation). The parts move in opposite directions with speeds inversely proportional to their masses. This principle also underpins the recoil of a gun when a bullet is fired.
如果一枚静止的烟花分裂成质量为 m₁ 和 m₂ 的两部分，速度分别为 v₁ 和 v₂，则矢量关系为 m₁ v₁ + m₂ v₂ = 0。两部分反向运动，速率与质量成反比。这一原理同样解释了子弹射出时枪身的反冲。
8. Newton’s Second Law in Momentum Form | 动量形式的牛顿第二定律
Newton originally formulated his second law in terms of momentum: the net force on an object equals the rate of change of its momentum.
牛顿最初用动量表述他的第二定律：物体受到的合外力等于其动量的变化率。
Fnet = dp/dt
For a system with constant mass, this reduces to Fnet = m a. The momentum form is more general because it remains valid when mass changes – vital for understanding rocket propulsion. It also explains why ‘following through’ in sports (increasing the contact time) can produce a larger change in momentum for the same average force, or why a sudden stop generates a large force.
当质量恒定时，该式简化为 Fnet = m a。动量形式更为普遍，因为它在质量变化时仍然成立——这对理解火箭推进至关重要。它也解释了为什么运动中“随挥”（增加接触时间）能在相同平均力下产生更大的动量变化，或者为什么突然停止会产生很大的力。
9. Momentum in Two Dimensions | 二维动量问题
When collisions occur in a plane, momentum conservation must be applied to two perpendicular directions, typically x and y. The total momentum components before the collision equal the components after.
当碰撞发生在平面内时，动量守恒必须应用于两个相互垂直的方向，通常取 x 和 y 方向。碰撞前的总动量分量等于碰撞后的分量。
Σpx before = Σpx after and Σpy before = Σpy after
To solve such problems: sketch the situation, resolve initial velocities into components, apply conservation separately, and use trigonometry to find unknown speeds or angles. A common example is a moving puck striking a stationary one at a glancing angle; the two move off at different angles, and you must check that the vector momentum diagram closes.
解题步骤为：画出示意图，将初速度分解为分量，分别应用动量守恒，再利用三角关系求未知速度或角度。一个常见例子是运动中的冰球侧击静止冰球，两者以不同角度散开，你需要检查动量矢量图是否闭合。
10. Rocket Propulsion | 火箭推进
Rockets accelerate by ejecting exhaust gases at high velocity. The system (rocket + ejected gas) is isolated in deep space, so momentum is conserved. As the gas is expelled backwards, the rocket gains forward momentum.
火箭通过高速喷出燃气而加速。在深空，系统（火箭 + 喷出气体）是孤立的，因此动量守恒。当气体向后喷出时，火箭获得向前的动量。
The thrust force is the rate at which momentum is transferred to the exhaust: Thrust = vex × (Δm/Δt), where vex is the exhaust velocity relative to the rocket and Δm/Δt is the mass flow rate. Even though the rocket’s mass decreases as fuel burns, the thrust can remain constant if vex and Δm/Δt are steady. The velocity of the rocket can be derived using calculus, but at IB/CIE level you are typically asked to explain the principle or apply conservation of momentum at an instant.
推力是排气获得动量的速率：推力 = vex × (Δm/Δt)，其中 vex 是相对于火箭的排气速度，Δm/Δt 是质量流率。尽管火箭质量随燃料燃烧而减小，如果 vex 和 Δm/Δt 保持恒定，推力就恒定。火箭速度可以通过微积分推导，但在 IB 和 CIE 层面，你通常只需解释原理或在某一时刻应用动量守恒。
11. Experimental Verification of Momentum Conservation | 动量守恒的实验验证
A classic laboratory investigation uses a linear air track to minimise friction. Two gliders of known masses are fitted with light gates or motion sensors. In an elastic collision, one glider is pushed and collides with a second glider; velocities before and after are recorded. The product m v is calculated for each glider and summed. For an inelastic collision, the gliders have Velcro or magnets so they stick; again, total momentum is compared. For explosions, a loaded spring between two stationary gliders is released, and their recoil velocities are measured.
一个经典的实验室研究使用线性气垫导轨以最小化摩擦。两个已知质量的滑行器配有光门或运动传感器。在弹性碰撞中，推动一个滑行器与另一滑行器碰撞；记录碰撞前后的速度。计算每个滑行器的 m v 并求和。对于非弹性碰撞，滑行器上装有魔术贴或磁铁，使其粘在一起；同样比较总动量。对于爆炸，释放两静止滑行器之间压缩的弹簧，测量它们的反冲速度。
In all cases, within experimental uncertainty, the total momentum before equals the total momentum after. Sources of error include residual friction, non‑horizontal track, timing inaccuracies and external vibrations. You should be prepared to describe the procedure, identify independent/dependent variables and explain why the track must be level.
在实验不确定度范围内，所有情况都显示碰撞前总动量等于碰撞后总动量。误差来源包括残余摩擦、轨道不水平、计时不准确以及外部振动。你应该准备好描述实验步骤，识别自变量和因变量，并解释为什么轨道必须水平。
12. Common Mistakes and Exam Tips | 常见错误与考试要点
Mistake 1: Treating momentum as a scalar. Always define a positive direction and assign signs to velocities accordingly. The sign of momentum indicates direction.
错误 1：把动量当作标量。务必规定正方向，并相应为速度赋予正负号。动量的符号表示方向。
Mistake 2: Assuming a collision is elastic without checking kinetic energy. Unless the problem states it is elastic, verify by calculating KE before and after.
错误 2：未经验证就假设碰撞是弹性的。除非题目明确说明是弹性碰撞，否则应通过计算碰撞前后动能来验证。
Mistake 3: Confusing speed with velocity. In momentum problems, direction matters. An object reversing direction has a velocity of opposite sign, which changes its momentum significantly.
错误 3：混淆速率与速度。在动量问题中，方向至关重要。若物体反向运动，速度符号相反，这会显著改变动量。
Mistake 4: Applying momentum conservation when external forces are not negligible. Look for phrases like ‘smooth surface’ or ‘immediately after’ to justify neglecting friction or gravity during the short interaction.
错误 4：在外力不可忽略时应用动量守恒。寻找“光滑表面”或“紧接着”等表述，以证明在短暂相互作用过程中可忽略摩擦或重力。
Mistake 5: Forgetting to include the mass of all components in a perfectly inelastic collision. The final mass is the sum of the sticking bodies.
错误 5：在完全非弹性碰撞中遗漏了某些组份的质量。末质量是粘合体的总质量。
Mistake 6: Drawing a vector diagram where the arrows do not form a closed triangle when total momentum after must equal zero. Always sketch and check.
错误 6：在末总动量必须为零时，矢量图画出的箭头不能构成闭合三角形。始终画草图并检查。
Exam tip: Always state the law of conservation of momentum in words if you use it. Show your working stepwise: define the system, write the conservation equation, substitute values with correct signs and solve. When dealing with two‑dimensional collisions, resolve velocities explicitly before applying conservation.
考试技巧：如果使用动量守恒定律，务必用文字表述。步骤清晰地展示解题过程：定义系统，写出守恒方程，代入带正确符号的数值，求解。处理二维碰撞时，先明确分解速度，再应用守恒。
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