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Vectors in Mechanics | 力学中的向量

2026-08-02 15:14 · 本文部分内容由AI辅助创作

📚 Vectors in Mechanics | 力学中的向量

In mechanics, quantities like displacement, velocity, acceleration, and force are not just about magnitude—they also have direction. This makes them vectors, and understanding how to manipulate vectors is the foundation of solving almost every mechanics problem in A-Level Physics. From resolving forces on a slope to calculating the resultant velocity of a projectile, vector methods give you the tools to break down complex scenarios into manageable parts. This article covers everything an Edexcel A-Level student needs: vector representation, addition, resolution, equilibrium, free-body diagrams, motion vectors, relative velocity, and projectile motion, all explained with clear, bilingual step‑by‑step guidance.

在力学中，位移、速度、加速度和力这些量不仅仅有大小，还有方向。这使它们成为向量，而掌握向量的运算是解决 A-Level 物理几乎所有力学问题的基础。从斜面上力的分解到抛体合速度的计算，向量方法能帮你把复杂情境拆解成容易处理的部分。本文涵盖 Edexcel A-Level 学生需要掌握的全部内容：向量的表示、加法、分解、平衡条件、受力图、运动向量、相对速度以及抛体运动，全部采用清晰的双语逐步讲解。

1. Scalars and Vectors | 标量与向量

A scalar is a physical quantity that has magnitude only. Examples include mass (kg), time (s), distance (m), speed (m s⁻¹), and energy (J). Scalars obey ordinary arithmetic; two masses of 3 kg and 4 kg simply add to make 7 kg.

标量是只有大小的物理量。例如质量（kg）、时间（s）、距离（m）、速率（m s⁻¹）和能量（J）。标量遵从普通的算术运算；3 kg 与 4 kg 的两个质量加起来就是 7 kg。

A vector is a quantity that has both magnitude and direction. Displacement, velocity, acceleration, force, and momentum are all vectors. The direction is an essential part of the quantity; a velocity of 5 m s⁻¹ east is completely different from 5 m s⁻¹ west. When adding vectors, direction must be taken into account.

向量是既有大小又有方向的量。位移、速度、加速度、力和动量都是向量。方向是这个量不可分割的一部分；5 m s⁻¹ 向东的速度与 5 m s⁻¹ 向西的速度完全不同。在对向量做加法时，必须考虑方向。

Specification focus: Edexcel expects you to distinguish between vector and scalar quantities and to combine vectors either graphically or by calculation. Make sure you can quote common examples from mechanics.

考纲重点：Edexcel 要求你能区分向量与标量，并能通过作图或计算来合成向量。务必能举出力学的常见例子。

2. Representing Vectors | 向量的表示

A vector can be drawn as an arrow. The length of the arrow represents the magnitude (drawn to a chosen scale), and the arrowhead shows the direction. The starting point is the tail, and the tip is the head. In print, vectors are often written in bold, e.g. F, or with an arrow above, e.g. F⃗ (though here we mainly use bold). In handwriting, you would underline the letter or add an arrow.

向量可以用带箭头的线段表示。线段的长度代表大小（按选定的比例尺画出），箭头表示方向。起点为尾部，终点为头部。在印刷体中，向量常用粗体表示，例如 F，或者加箭头上标，如 F⃗（本文中我们主要用粗体）。手写时，可在字母下方加波浪线或画箭头。

We also use unit vectors to describe direction in a coordinate system. A unit vector has a magnitude of exactly 1 and points along a particular axis. In two dimensions, the unit vector along the x‑axis is written as i, and along the y‑axis as j. So a vector v with components 3 m s⁻¹ in the x‑direction and 4 m s⁻¹ in the y‑direction is written as v = 3i + 4j m s⁻¹.

我们还会用单位向量来描述坐标系中的方向。单位向量的大小恰好为 1，并指向某特定坐标轴。在二维情况下，沿 x 轴的单位向量记作 i，沿 y 轴的记作 j。因此，一个在 x 方向分量为 3 m s⁻¹、在 y 方向分量为 4 m s⁻¹ 的向量 v 可写为 v = 3i + 4j m s⁻¹。

3. Vector Addition: Triangle and Parallelogram Laws | 向量加法：三角形法则与平行四边形法则

When two vectors are added, the result is called the resultant vector. Graphically, the triangle law places the tail of the second vector at the head of the first. The resultant is drawn from the tail of the first vector to the head of the second. For vectors A and B, R = A + B.

两个向量相加的结果称为合向量。作图时，三角形法则把第二个向量的尾部放在第一个向量的头部。从第一个向量的尾部画向第二个向量的头部的向量即为合向量。对于向量 A 与 B，R = A + B。

The parallelogram law places both vectors tail‑to‑tail, completes a parallelogram, and the resultant is the diagonal starting from the common tail. Both methods give the same answer. Vector addition is commutative: A + B = B + A.

平行四边形法则将两个向量的尾部对齐，作平行四边形，从公共尾部引出的对角线即为合向量。两种方法给出相同的结果。向量加法满足交换律：A + B = B + A。

To subtract vectors, add the negative: A – B = A + (–B). The negative of a vector has the same magnitude but opposite direction.

向量的减法相当于加上负向量：A – B = A + (–B)。负向量大小相同，方向相反。

In mechanics, vector addition is used whenever two forces act at a point, or when a boat moves relative to water while the water itself moves. The resultant is the overall effect.

在力学中，只要有两个力作用于同一点，或一艘船在水中运动而水本身也在流动，都会用到向量加法。合向量就是总效果。

4. Resolving Vectors into Components | 向量的分解

Any vector can be split into two perpendicular components, typically along the x‑ and y‑axes. This process is called resolution. If a vector of magnitude V makes an angle θ with the x‑axis, then:

任何向量都可以分解为两个互相垂直的分量，通常沿 x 轴和 y 轴。这个过程叫做分解。若向量大小为 V，与 x 轴夹角为 θ，则：

Vₓ = V cos θ, Vᵧ = V sin θ

orizontal component = V cos θ; vertical component = V sin θ. Check the triangle carefully.

邻角分量总是余弦，对角分量为正弦。这一点至关重要：若 θ 从水平线量起，水平分量 = V cos θ；竖直分量 = V sin θ。做题时要仔细检查三角形。

Resolving vectors is one of the most frequently used skills in mechanics. You will use it to find net force along a slope, to add non‑right‑angled vectors by components, and to apply Newton’s second law in two directions independently.

向量分解是力学中最常使用的技能之一。你会用它来求沿斜面的合力、用分量法相加不在同一直线上的向量，以及独立地在两个方向上分别应用牛顿第二定律。

Once resolved, the original vector is completely represented by its two perpendicular components. To reverse the process and find magnitude and direction from components:

一旦完成分解，原来的向量完全由它的两个垂直分量表示。要从分量反求大小和方向：

V = √Vₓ² + Vᵧ², θ = tan⁻¹(Vᵧ / Vₓ)

5. Unit Vectors and Component Form | 单位向量与分量形式

Writing a vector in component form using i and j makes vector addition analytical and precise. For example, a force F₁ = (5i + 3j) N and a force F₂ = (2i − 4j) N can be added by simply adding their components: F₁ + F₂ = (5+2)i + (3−4)j = 7i − j N.

用 i 和 j 表示向量的分量形式可以解析地、精确地进行向量加法。例如，力 F₁ = (5i + 3j) N 与力 F₂ = (2i − 4j) N 的加法只需将分量对应相加：F₁ + F₂ = (5+2)i + (3−4)j = 7i − j N。

The magnitude of the resultant is √(7² + (–1)²) = √50 ≈ 7.07 N, and its direction relative to the positive x‑axis is θ = tan⁻¹(–1/7) ≈ –8.13˚, meaning it points slightly below the x‑axis.

合向量的大小为 √(7² + (–1)²) = √50 ≈ 7.07 N，方向相对于正 x 轴的夹角 θ = tan⁻¹(–1/7) ≈ –8.13˚，表示它指向 x 轴略下方。

Using unit vectors is especially helpful in three dimensions (adding k for the z‑axis), but at A-Level most problems are confined to two dimensions. Edexcel often includes forces expressed in i‑j notation, so practice converting between the graphical form and the component form.

使用单位向量在三维空间尤为方便（增加代表 z 轴的 k），但在 A-Level 阶段，绝大多数问题限制在二维。Edexcel 常考以 i‑j 形式表示的力，因此要练习在图形形式和分量形式之间互相转换。

6. Adding Vectors by Resolution | 用分解法做向量加法

When more than two vectors need to be added, or when the vectors are not at right angles to each other, resolving each vector into components along perpendicular axes is the most reliable method. Follow these steps:

当需要相加的向量超过两个，或彼此不垂直时，将每个向量沿相互垂直的坐标轴分解是最可靠的方法。遵循以下步骤：

Step 1: Choose a convenient set of axes, often horizontal (x) and vertical (y), or parallel and perpendicular to a slope.

第一步：选择一组方便的坐标轴，通常为水平方向（x）和竖直方向（y），或平行于斜面和垂直于斜面的方向。

Step 2: Resolve each vector into x‑ and y‑components. For a vector of magnitude F at angle θ to the x‑ os θ, Fᵧ = F sin θ. Pay attention to signs: components pointing left or down are negative.

第二步：将每个向量分解为 x‑分量与 y‑分量。对于大小为 F 且与 x 轴夹角为 θ 的向量：Fₓ = F cos θ, Fᵧ = F sin θ。注意符号：指向左或下的分量为负。

Step 3: Sum all the x‑components to get the total x‑component Rₓ, and all the y‑components to get Rᵧ.

第三步：将所有 x‑分量相加得到总 x‑分量 Rₓ，将所有 y‑分量相加得到 Rᵧ。

Step 4: The resultant vector is R = Rₓ i + Rᵧ j. If required, find the magnitude |R| = √Rₓ² + Rᵧ² and direction θ = tan⁻¹(Rᵧ / Rₓ).

第四步：合向量为 R = Rₓ i + Rᵧ j。如有需要，求大小 |R| = √Rₓ² + Rᵧ² 和方向 θ = tan⁻¹(Rᵧ / Rₓ)。

This method avoids scale‑drawing errors and works for any number of vectors. In mechanics, it is essential when finding the net force on a body or the resultant velocity of an object moving in a flowing medium.

这种方法避免作图带来的误差，可用于任意数目的向量。在力学中，求物体所受合力或求在流体中运动的物体的合速度时，它必不可少。

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7. Forces in Equilibrium | 力的平衡

An object is in equilibrium when the net force acting on it is zero. This means the vector sum of all forces is zero. In component terms:

当作用在物体上的合力为零时，物体处于平衡状态。这意味着所有力的向量和为零。用分量语言表述：

ΣFₓ = 0 and ΣFᵧ = 0

For a body at rest or moving with constant velocity (translational equilibrium), these two conditions must both be satisfied. You can therefore solve for unknown forces by setting up two independent equations.

对于静止或匀速直线运动的物体（平动平衡），这两个条件必须同时满足。因此，你可以通过建立两个独立的方程来求解未知力。

A common A‑Level problem involves three forces in equilibrium. They can be represented as a closed triangle when added head‑to‑tail. This “triangle of forces” provides a graphical way to check equilibrium or find an unknown force. If the three force vectors form a closed triangle, the resultant is zero.

A‑Level 的常见题型涉及三力平衡。当按首尾相接相加时，它们可以构成一个闭合三角形。这个“力三角形”提供了一种检验平衡或求解未知力的作图方法。若三个力向量形成闭合三角形，则合力为零。

More complex problems use Lami’s theorem (for three concurrent forces in equilibrium) or simply resolve forces in two perpendicular directions. Lami’s theorem states: if three forces are in equilibrium, each force is proportional to the sine of the angle between the other two. However, Edexcel tends to favour resolution into components, as it works in all situations.

更复杂的问题会用到拉密定理（适用于三力汇交平衡）或直接将力沿两个互相垂直的方向分解。拉密定理指出：若三个力平衡，则每个力的大小与另外两个力夹角的正弦成正比。不过，Edexcel 更偏爱用分解法，因为它在所有情况下都适用。

8. Free‑Body Diagrams | 受力分析图

A free‑ dy is usually drawn as a dot or a simple shape, and each force vector is labelled. Common forces include weight (W = m g), normal reaction (N), tension (T), friction (f), and applied forces.

受力分析图（FBD）是将一个物体从其周围环境中隔离出来，用向量标出所有作用在该物体上的力的示意图。物体通常画成一个点或简单形状，每个力向量都加上标注。常见的力包括重力（W = m g）、法向反作用力（N）、拉力（T）、摩擦力（f）和外加力。

Drawing a correct FBD is the crucial first step for any mechanics problem. It helps you visualise directions and angles, and ensures you account for all forces before starting calculations. Remember: only include forces acting on the body, not forces the body exerts on other objects. That is the key to Newton’s third law clarity.

画出一张正确的受力图是解决任何力学问题的关键第一步。它能帮你直观看到方向和角度，并确保你在开始计算前已经考虑了所有力。记住：只包括作用在该物体上的力，而不包括该物体施加给其他物体的力。这正是理解牛顿第三定律的秘诀。

Once the FBD is drawn, choose a coordinate system—often align one axis with the direction of motion or with a slope— ΣF = m a in each coordinate direction. For equilibrium problems, set ΣFₓ = 0 and ΣFᵧ = 0.

画出受力图后，选择一个坐标系——通常将某一轴对准运动方向或斜面方向——然后对力进行分解。接着在每个坐标方向上应用 ΣF = m a。对于平衡问题，令 ΣFₓ = 0 且 ΣFᵧ = 0。

9. Motion Vectors: Displacement, Velocity, Acceleration | 运动向量：位移、速度、加速度

In kinematics, displacement s, velocity v, and acceleration a are vectors. Displacement is the change in position, pointing from start to finish. Velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. All have both magnitude and direction.

在运动学中，位移 s、速度 v 和加速度 a 都是向量。位移是位置的改变量，方向由起点指向终点。速度是位移的变化率，加速度是速度的变化率。它们都既有大小又有方向。

If a particle moves from position r₁ = 2i + 3j m to r₂ = 5i + 1j m, its displacement is Δr = (5–2)i + (1–3)j = 3i − 2j m. The average velocity over a time interval Δt is vₐᵥ = Δr / Δt. Note that the direction of average velocity is the same as the direction of the displacement.

假如一个质点从位置 r₁ = 2i + 3j m 运动到 r₂ = 5i + 1j m，其位移为 Δr = (5–2)i + (1–3)j = 3i − 2j m。在时间间隔 Δt 内的平均速度 vₐᵥ = Δr / Δt。注意平均速度的方向与位移的方向相同。

Instantaneous velocity is the derivative of the position vector with respect to time. If the position vector is given as a function of time, say r(t) = 2ti + (3t²–t)j m, then velocity v(t) = dr/dt = 2i + (6t–1)j m s⁻¹, and acceleration a(t) = dv/dt = 6j m s⁻². While calculus is not heavily tested in Edexcel AS mechanics, it appears in A‑Level when vectors are written in i, j form.

瞬时速度是位置向量对时间的导数。若位置向量以时间函数给出，比如 r(t) = 2ti + (3t²–t)j m，则速度 v(t) = dr/dt = 2i + (6t–1)j m s⁻¹，加速度 a(t) = dv/dt = 6j m s⁻²。尽管 Edexcel AS 力学不重点考查微积分，但在 A‑Level 中当向量以 i、j 形式书写时会出现。

10. Relative Velocity | 相对速度

Relative velocity is the velocity of one object as observed from another moving object. If object A has velocity vₐ and object B has velocity vᴮ, the velocity of A relative to B is:

相对速度是从一个运动物体上观察另一个物体时的速度。若物体 A 的速度为 vₐ，物体 B 的速度为 vᴮ，则 A 相对于 B 的速度为：

vA/B = vₐ − vᴮ

This vector equation is immensely useful in navigation and collision problems. For instance, a boat trying to cross a river with a current. The boat’s velocity relative to the water, plus the water’s velocity relative to the ground, gives the boat’s velocity relative to the ground.

这个向量方程在导航和碰撞问题中非常有用。例如，一艘船要横渡河水有流速的河流。船相对于水的速度加上水相对于地面的速度，就得到船相对于地面的速度。

Worked example: A boat heads due north at 4 m s⁻¹ relative to the water. The river flows east at 3 m s⁻¹. What is the boat’s resultant velocity relative to the bank?
Vector addition: vᵦₐₙₖ = vₐₒₐₜ/ₐₐₜₑᵣ + vₐₐₜₑᵣ/ₐₐₙₖ = 4j + 3i m s⁻¹.
Magnitude = √4²+3² = 5 m s⁻¹, direction θ = tan⁻¹(3/4) ≈ 36.9˚ east of north.

例题：一船相对于水以 4 m s⁻¹ 向正北行驶。河水以 3 m s⁻¹ 向东流。问船相对于岸的合速度。
向量加法：vₐᵦₐₙₖ = vₐₒₐₜ/ₐₐₜₑᵣ + vₐₐₜₑᵣ/ₐₐₙₖ = 4j + 3i m s⁻¹。
大小 = √4²+3² = 5 m s⁻¹，方向 θ = tan⁻¹(3/4) ≈ 36.9˚，北偏东。

Always draw a vector triangle and label velocities carefully to avoid sign errors. Remember that relative velocity problems are simply vector addition or subtraction.

务必画出向量三角形并仔细标注速度，以避免符号错误。记住相对速度问题本质上就是向量的加法或减法。

11. Projectile Motion – Vectors in Action | 抛体运动 – 向量的实际应用

Projectile motion is a beautiful illustration of vector resolution. A projectile launched with initial velocity u n due to gravity (g = 9.81 m s⁻² downwards). The two components are treated independently.

抛体运动是对向量分解的极好展示。以初速 u、与水平方向夹角 θ 抛出的物体，其水平速度恒定，竖直加速度恒定（重力加速度 g = 9.81 m s⁻² 向下）。这两个方向的分量独立处理。

ty components are: uₓ = u cos θ, uᵧ = u sin θ. At any time t, the velocity components are:

初速度分量为：uₓ = u cos θ，uᵧ = u sin θ。在任意时刻 t，速度分量为：

vₓ = u cos θ (constant), vᵧ = u sin θ − g t

The displacement components are:

位移分量为：

x = (u cos θ) t, y = (u sin θ) t − ½ g t²

The path is a parabola because y is a quadratic function of x. All standard projectile results (time of flight, maximum height, range) follow from these vector equations. When solving problems, always resolve initial velocity first, then apply equations of motion separately in each direction.

轨迹为抛物线，因为 y 是 x 的二次函数。所有标准抛体结果（飞行时间、最大高度、射程）都可以从这些向量方程推导出来。解题时，一定要先分解初速度，然后在每个方向上单独应用运动学公式。

At the highest point, the vertical component of velocity is zero, but the horizontal component remains unchanged. The speed at any point is the magnitude of the resultant velocity: √vₓ² + vᵧ². This is a classic example of how vectors give a complete picture of motion in two dimensions.

在最高点，竖直分速度为零，但水平分速度保持不变。任意一点的速度大小是合速度的量值：√vₓ² + vᵧ²。这完美地展示了向量如何完整描绘二维运动。

12. Summary and Exam Tips | 总结与应试技巧

Vectors are the language of mechanics. Mastering them means you can handle any force or motion problem with confidence. Remember these key points:

向量是力学的语言。掌握它们意味着你可以自信地处理任何力或运动问题。记住以下要点：