📚 Momentum and Impulse in AS Mathematics | AS数学：动量与冲量考点精讲
Momentum and impulse lie at the heart of AS Mathematics Mechanics, linking Newton’s laws to the analysis of collisions, explosions and everyday interactions. A firm grasp of these concepts allows you to solve problems with confidence, whether objects are sticking together, recoiling apart or experiencing varying forces. This article breaks down every key idea, formula and common pitfall you need for exam success.
动量与冲量是AS数学力学部分的核心内容，把牛顿运动定律与碰撞、爆炸及日常相互作用的分析紧密联系起来。扎实掌握这些概念能让你从容解决各种问题，无论物体是粘在一起、反冲分离还是受到变力作用。本文为你拆解考试必需的所有关键概念、公式和常见误区。
1. What is Momentum? | 什么是动量？
The momentum of an object is defined as the product of its mass and its velocity. It is a measure of ‘how much motion’ the object possesses and always acts in the same direction as the velocity.
物体的动量定义为其质量与速度的乘积。它衡量物体“运动量”的大小，方向始终与速度方向相同。
p = m × v
Momentum is a vector quantity with SI units of kilogram metres per second, written as kg m s&supmin;¹ or, equivalently, newton seconds (N s). In the AS specification, you will most often see it expressed in N s because of its direct link to impulse.
动量是矢量，国际单位制为千克·米/秒（kg m s&supmin;¹），也等价于牛顿秒（N s）。在AS大纲中，由于动量与冲量的直接联系，它更常以N s为单位出现。
Because momentum depends on velocity, a heavier object moving slowly can have the same momentum as a lighter object moving quickly. Always be ready to substitute mass in kilograms and velocity in metres per second.
由于动量取决于速度，一个缓慢运动的沉重物体可以与快速运动的轻物体具有相同的动量。务必始终以千克代入质量、以米/秒代入速度。
2. Momentum as a Vector Quantity | 动量作为矢量
Treating momentum as a vector is essential when objects move in opposite directions. Choosing a positive direction at the start of a problem and assigning positive and negative signs to velocities ensures that momentum adds correctly.
当物体沿相反方向运动时，将动量视为矢量至关重要。解题伊始选定正方向，并给速度赋予正负号，才能确保动量正确相加。
For example, a 2 kg mass moving to the right at 3 m s&supmin;¹ has momentum +6 N s; if it were moving to the left, its momentum would be -6 N s. Failing to account for sign is one of the most common mistakes in momentum problems.
例如，一个2 kg的物体以3 m s&supmin;¹的速度向右运动，动量为+6 N s；如果它向左运动，动量则为-6 N s。未能正确使用符号是动量问题中最常见的错误之一。
In calculations involving impulse and change in momentum, the vector nature appears in the formula I = m(v – u) where u and v are velocities that may have opposite signs, producing a large change in momentum.
在涉及冲量和动量变化的计算中，矢量的特性体现在公式 I = m(v – u) 中，其中 u 和 v 可能具有相反的符号，从而产生很大的动量改变。
3. Impulse and the Impulse-Momentum Theorem | 冲量和冲量-动量定理
Impulse is defined as the product of a constant force and the time for which it acts. It measures the overall effect of a force applied over an interval and is a vector with the same direction as the force.
冲量定义为一个恒力与其作用时间的乘积。它衡量一段时间内力施加的总效果，是矢量，方向与力的方向相同。
I = F × t
The impulse-momentum theorem states that the impulse acting on a body equals the change in its momentum. This follows directly from Newton’s second law: F = ma = m(v – u)/t, giving Ft = mv – mu.
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。该结论直接源于牛顿第二定律：F = ma = m(v – u)/t，因此 Ft = mv – mu。
I = mv – mu
This relationship is often written as I = Δp. It allows you to find the change in velocity caused by a force, or to calculate the force if the contact time and change in velocity are known. Impulse has units of N s, which are dimensionally identical to those of momentum.
该关系常写作 I = Δp。通过它，你可以求出一个力产生的速度变化，或者在已知接触时间和速度变化时计算力的大小。冲量的单位N s与动量的单位量纲相同。
For a variable force, the impulse cannot be found simply by F×t; instead we turn to a force-time graph. This is covered in the next section.
对于变力，不能简单地用 F×t 求冲量，此时需要借助力-时间图像，下一节将详述。
4. Impulse from Force-Time Graphs | 从力-时间图求冲量
When the force acting on an object varies with time, the impulse is equal to the area under the force-time graph. This is a direct consequence of impulse being the integral of force over time, but at AS level you will work with simple shapes such as rectangles, triangles and trapeziums.
当作用在物体上的力随时间变化时，冲量等于力-时间图像下的面积。这是因为冲量是力对时间的积分，但在AS阶段你只需处理矩形、三角形和梯形等简单图形。
For example, a force that rises linearly from 0 to 10 N over 2 seconds produces a triangular area of ½ × 2 × 10 = 10 N s. If the force stays constant at 10 N for a further 1 second, the additional rectangular area is 10 × 1 = 10 N s, giving a total impulse of 20 N s.
例如，一个力在2秒内从0线性增加到10 N，产生的三角形面积为½ × 2 × 10 = 10 N s。如果该力再保持10 N恒定1秒，额外的矩形面积为10 × 1 = 10 N s，总冲量为20 N s。
Questions may also ask you to calculate the average force from an impulse found by area, or to interpret a force-time graph to determine the duration of an impact. Always check the axes: time is on the horizontal axis and force on the vertical.
考题还可能要求根据面积求得的冲量计算平均力，或根据力-时间图像推断碰撞的持续时间。务必查看坐标轴：横轴为时间，纵轴为力。
5. Conservation of Linear Momentum | 线动量守恒
The principle of conservation of momentum states that when no external resultant force acts on a system, the total momentum of the system remains constant. This is one of the most powerful tools in mechanics for analysing collisions and explosions.
动量守恒定律指出，当系统不受外力（合外力为零）时，系统的总动量保持不变。它是力学中分析碰撞与爆炸最有力的工具之一。
For two interacting objects, with masses m&sub1; and m&sub2;, initial velocities u&sub1; and u&sub2;, and final velocities v&sub1; and v&sub2;, the conservation law is expressed as:
对于两个相互作用的物体，质量分别为 m&sub1; 和 m&sub2;，初速度分别为 u&sub1; 和 u&sub2;，末速度分别为 v&sub1; 和 v&sub2;，守恒定律可表示为：
m&sub1;u&sub1; + m&sub2;u&sub2; = m&sub1;v&sub1; + m&sub2;v&sub2;
This vector equation must be applied with consistent sign conventions. It is vital to remember that momentum is conserved in any single direction provided no external forces act in that direction. For AS problems, you will usually consider motion along a straight line.
这个矢量方程必须配合一致的符号规则来使用。关键要记住，只要在某一方向上没有外力作用，该方向上的动量就守恒。在AS问题中，通常只需考虑沿一条直线的运动。
Conservation of momentum can be used together with the impulse-momentum theorem: while the impulse changes the momentum of an individual object, the total momentum change of an isolated system is zero.
动量守恒可与冲量-动量定理结合使用：冲量改变单个物体的动量，但孤立系统的总动量变化为零。
6. Direct Collisions and Conservation of Momentum | 正碰与动量守恒
In a direct collision, two objects move along the same straight line and make contact. The conservation of momentum equation allows you to link their velocities before and after the impact.
在正碰中，两个物体沿同一直线运动并接触。利用动量守恒方程可以将碰撞前后的速度联系起来。
To solve a collision problem:
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Draw a clear diagram showing the direction of each velocity and label masses.
画出清晰的示意图，标出速度的方向并注明质量。
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Choose a positive direction and assign positive or negative signs to velocities accordingly.
选定正方向，并据此给速度赋予正号或负号。
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Write the conservation of momentum equation and substitute known values.
写出动量守恒方程，代入已知值。
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Solve for the unknowns.
求解未知量。
If only one unknown velocity needs to be found, the equation will suffice. If two final velocities are unknown, you will usually be given an additional condition, such as the objects sticking together or the fact that one object comes to rest.
如果只需求一个未知速度，该方程便已足够。若有两个未知末速度，题目通常会给出附加条件，比如物体粘在一起，或某物体静止。
Always interpret your final answer in context: a negative velocity means the object moves in the opposite direction to your chosen positive axis.
始终结合题意解释最终答案：负速度表示物体的实际运动方向与你选定的正方向相反。
7. Perfectly Inelastic Collisions | 完全非弹性碰撞
In a perfectly inelastic collision, the colliding objects stick together after impact and move with a common velocity v. This is the simplest type of collision to analyse because the number of unknowns reduces to one final speed.
在完全非弹性碰撞中，碰撞后物体粘连在一起并以共同速度 v 运动。这是最容易分析的一类碰撞，因为未知量减少为一个末速度。
The conservation equation simplifies to:
此时守恒方程简化为：
m&sub1;u&sub1; + m&sub2;u&sub2; = (m&sub1; + m&sub2;)v
Kinetic energy is not conserved in such collisions — some of it transforms into heat, sound or deformation. However, at AS level you are not required to calculate kinetic energy changes unless the question specifically asks.
在这种碰撞中动能并不守恒——一部分能量转化为热、声或形变能。不过，除非题目明确要求，AS阶段通常不需要计算动能的变化。
Perfectly inelastic collisions appear frequently in exam questions, for example in shunting problems involving railway trucks, or in situations where a bullet embeds itself in a block of wood. Always verify that the objects genuinely stick together before applying this simplification.
完全非弹性碰撞在考试题中很常见，比如铁路车厢的溜放问题，或子弹嵌入木块的情形。在应用上述简化公式之前，务必确认物体确实粘在一起。
8. Explosions and Recoil Problems | 爆炸与反冲问题
Explosions can be thought of as the reverse of collisions: a single object breaks into two or more fragments. Internal forces cause separation, but the total momentum of the system remains zero (or equal to the initial momentum before explosion) because no external horizontal forces act.
爆炸可以视为碰撞的逆过程：一个物体分裂成两个或更多碎片。内力引起分离，但因为没有水平方向外力，系统的总动量保持为零（或等于爆炸前的初始动量）。
For an initially stationary object that explodes into two fragments of masses m&sub1; and m&sub2;:
对于原本静止的物体爆炸成质量为 m&sub1; 和 m&sub2; 的两块：
0 = m&sub1;v&sub1; + m&sub2;v&sub2;
This implies that the fragments move in opposite directions, with speeds inversely proportional to their masses: |v&sub1;/v&sub2| = m&sub2;/m&sub1;.
这表明碎片沿相反方向运动，速度大小与质量成反比：|v&sub1;/v&sub2| = m&sub2;/m&sub1;。
Recoil problems work similarly: when a gun fires a bullet, the gun recoils in the opposite direction. If the bullet and gun are initially at rest, the forward momentum of the bullet equals the backward momentum of the gun. The impulse on the bullet and the gun are equal in magnitude and opposite in direction.
反冲问题类似：当枪发射子弹时，枪身向后反冲。若子弹和枪初始静止，则子弹向前的动量等于枪向后的动量。子弹和枪受到的冲量大小相等、方向相反。
If the object has an initial velocity before exploding, simply include it in the conservation equation: mtotal u = m&sub1;v&sub1; + m&sub2;v&sub2;.
如果物体在爆炸前已有初速度，只需将其纳入守恒方程：m总 u = m&sub1;v&sub1; + m&sub2;v&sub2;。
9. Impulse in Vector Form (i, j notation) | 冲量的矢量形式 (i, j 表示法)
In two-dimensional situations, velocity is expressed in terms of the unit vectors i and j. Impulse is then calculated component by component using I = m(v – u), treating i and j directions independently.
在二维情况下，速度用单位向量 i 和 j 表示。此时冲量可利用 I = m(v – u) 逐个分量进行计算，i 方向和 j 方向独立处理。
For a particle of mass 0.5 kg whose velocity changes from (3i + 6j) m s&supmin;¹ to (i – 2j) m s&supmin;¹, the change in velocity is (i – 2j) – (3i + 6j) = -2i – 8j. Multiplying by 0.5 gives the impulse: -1i – 4j N s.
一个质量为0.5 kg的质点，速度从 (3i + 6j) m s&supmin;¹ 变为 (i – 2j) m s&supmin;¹，速度变化量为 (i – 2j) – (3i + 6j) = -2i – 8j。乘以0.5得到冲量：-1i – 4j N s。
The impulse vector has both magnitude and direction. Its magnitude is found using Pythagoras, and its direction can be given as a bearing or an angle relative to i or j. The impulse vector points in the same direction as the change in momentum.
冲量矢量既有大小又有方向。其大小可用勾股定理求得，方向则可用方位角或相对于 i、j 的角度表示。冲量矢量的方向与动量变化的方向一致。
When writing answers, it is often sufficient to leave the impulse in component form, such as -1i – 4j N s, though some questions may ask for the magnitude. Always check the requirement.
作答时，通常保留冲量的分量形式（如 -1i – 4j N s）即可，但有些题目会要求计算冲量的大小。务必看清题目要求。
10. Common Pitfalls and Exam Advice | 常见错误与应试建议
Even straightforward momentum questions can trip up students if they lose focus on direction, units or the definition of the system. Here are the most frequent errors and how to avoid them.
即使简单的动量问题，如果对方向、单位或系统的界定掉以轻心，也可能让学生栽跟头。以下是最高频的错误及避免方法。
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Neglecting sign conventions: Always choose a positive direction and apply it consistently. A forgotten negative sign can completely reverse an answer.
忽略符号规则：始终选定正方向并一致运用。一个漏掉的负号会让答案彻底反转。
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Confusing mass and weight: Momentum uses mass in kilograms, never weight in newtons. Make sure to convert any given weights into mass using m = W/g if necessary.
混淆质量与重力：动量中使用以千克为单位的质量，而非以牛顿为单位的重力。如有需要，务必用 m = W/g 将给出的重力转换为质量。
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Incorrect unit conversions: Velocities must be in m s&supmin;¹ and time in seconds. If a question gives km h&supmin;¹ or minutes, convert immediately.
单位转换
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