Combining Velocities | 速度的合成

📚 Combining Velocities | 速度的合成

Velocity is a fundamental concept in kinematics that tells us not only how fast an object moves but also the direction of motion. In many real-world situations, an object experiences two or more velocities at the same time — a boat crossing a flowing river, an aircraft flying through a side wind, or a person walking on a moving walkway. To find the overall motion, we must combine these velocities correctly. This article explains the classical rules for combining velocities in one and two dimensions, including relative velocity, vector diagrams, resolution into components, and graphical methods. We also touch on the fascinating limit of relativistic velocity addition.

速度是运动学中的一个基本概念，它不仅告诉我们物体运动的快慢，也包含了运动的方向。在许多实际情况中，物体会同时受到两个或多个速度的影响——小船在流动的河水中过河、飞机在侧风中飞行、人在移动步道上行走。要得到物体的整体运动，我们必须正确地将这些速度合成。本文将介绍在一维和二维空间中组合速度的经典规则，涵盖相对速度、矢量图、正交分解以及图解法，最后还会简要提及相对论速度合成的奇妙之处。

1. Velocity as a Vector | 速度——一种矢量

Velocity is defined as the rate of change of displacement. Unlike speed, which is a scalar, velocity is a vector quantity — it possesses both magnitude (the speed) and direction. When we talk about combining velocities, we must use vector addition, not ordinary arithmetic. Adding two speeds of 5 m s⁻¹ and 5 m s⁻¹ might give 10 m s⁻¹ if they are in the same direction, but if they act at 180° to each other, the resultant speed is 0 m s⁻¹. Direction always matters.

速度被定义为位移的变化率。和标量速率不同，速度是矢量——既有大小（即速率）也有方向。当我们说合成速度时，必须使用矢量加法，而不是普通的算术相加。如果两个 5 m s⁻¹ 的速率在同一直线上同向，合成速率是 10 m s⁻¹；但如果它们方向相反（夹角 180°），合成速率就会是 0 m s⁻¹。方向永远是关键。

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2. One-Dimensional Addition | 一维速度的合成

The simplest case occurs when all velocities lie along the same straight line. We choose a positive direction, for example to the right or east. Velocities pointing in that direction are assigned positive values; those pointing opposite are negative. The resultant velocity is the algebraic sum of all the signed velocities.

最简单的情况是所有速度都在同一条直线上。我们会选取一个正方向，比如向右或向东。指向正方向的速度取正值，反向的速度取负值。合速度就是这些带符号速度的代数和。

v_res = v₁ + v₂ + v₃ + …

Example: moving walkway. A person walks on a travelator. Let the velocity of the belt relative to the ground be +2 m s⁻¹. The person walks forward relative to the belt at +1 m s⁻¹. The velocity of the person relative to the ground is v_{person,ground} = v_person,belt + v_belt,ground = +1 + (+2) = +3 m s⁻¹. If the person walks backward at 1 m s⁻¹ relative to the belt, the ground speed becomes −1 + 2 = +1 m s⁻¹ (still moving forward, but slowly).

例题：移动步道。 一个人在传送带上行走。传送带相对地面的速度为 +2 m s⁻¹，人相对传送带以 +1 m s⁻¹ 向前走。人相对地面的速度就是 v_人,地 = v_人,带 + v_带,地 = +1 + (+2) = +3 m s⁻¹。如果人相对传送带向后走，速度为 −1 m s⁻¹，那么地面速度变为 −1 + 2 = +1 m s⁻¹（仍向前运动，但速度慢了）。

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3. Relative Velocity in One Dimension | 一维相对速度

The velocity of object A as seen from object B is called the relative velocity of A with respect to B, written as v_A/B. In one dimension it is simply the difference of their velocities relative to the ground:

从物体 B 上观察物体 A 的速度称为 A 相对于 B 的速度，记作 v_A/B。在一维情况下，它等于两者相对地面速度的差：

v_A/B = v_A − v_B

For example, car A travels at 30 m s⁻¹ east and car B travels at 20 m s⁻¹ east. The velocity of A relative to B is 30 − 20 = +10 m s⁻¹ east, meaning A is pulling away from B at 10 m s⁻¹. If car B is travelling at 20 m s⁻¹ west (taken as −20 m s⁻¹), then v_A/B = 30 − (−20) = 50 m s⁻¹ east; the relative speed is 50 m s⁻¹, which is the speed at which they approach each other.

例如，一辆汽车 A 以 30 m s⁻¹ 向东行驶，汽车 B 以 20 m s⁻¹ 向东行驶。A 相对于 B 的速度为 30 − 20 = +10 m s⁻¹ 向东，表明 A 正以 10 m s⁻¹ 的速度远离 B。如果 B 以 20 m s⁻¹ 向西行驶（取 −20 m s⁻¹），则 v_A/B = 30 − (−20) = 50 m s⁻¹ 向东；它们的相对速率为 50 m s⁻¹，也是它们互相靠近的速率。

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4. Vector Addition in Two Dimensions | 二维矢量加法

When velocities are not aligned along a single line, simple algebraic addition fails. We must draw velocity vectors as arrows and combine them using the rules of vector geometry. There are two equivalent graphical methods: the parallelogram law and the triangle law.

当速度不沿同一直线时，简单的代数加减就不适用了。我们必须将速度矢量画成箭头，再用矢量几何规则合成。常用的方法有两种：平行四边形法则和三角形法则。

In the parallelogram law, draw vectors v₁ and v₂ from a common origin, making an angle θ between them. Complete the parallelogram. The diagonal starting from the common origin represents the resultant velocity v_R. Its magnitude is given by the cosine rule:

用平行四边形法则时，从同一点出发画出矢量 v₁ 和 v₂，两者夹角为 θ。再完成平行四边形，从同一点引出的对角线就是合速度 v_R。其大小由余弦定理给出：

v_R = √(v₁² + v₂² + 2 v₁ v₂ cosθ)

The direction of the resultant relative to v₁ is found from the sine rule or by calculating the angle φ where:

合速度相对于 v₁ 的方向可由正弦定理或计算角度 φ 得到：

tan φ = (v₂ sinθ) / (v₁ + v₂ cosθ)

These formulas are valid for any angle θ between 0° and 180°.

这些公式对 0° 至 180° 之间的任意夹角 θ 都成立。

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5. Triangle Law of Vector Addition | 三角形法则

The triangle law is often simpler when adding two velocities: draw the first velocity vector, then place the tail of the second vector at the head of the first. The resultant velocity is the vector drawn from the tail of the first to the head of the second. If you have more than two velocities, continue placing vectors head-to-tail; the resultant is the vector from the starting point to the final point.

三角形法则在合成两个速度时往往更简单：先画出第一个速度矢量，再把第二个速度的起点放在第一个速度的箭头末端。合速度就是从第一个矢量的起点指向第二个矢量末端的箭头。如果速度超过两个，就继续首尾相连；合速度就是从初始起点指向最终末端的矢量。

This method is mathematically equivalent to the parallelogram law and is very helpful for sketching vector addition quickly in an exam.

这一方法与平行四边形法则在数学上等价，在考试中能帮助快速画出矢量合成的草图。

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6. Resolving Velocities into Components | 速度的分解

An even more powerful approach, especially when dealing with several velocities at various angles, is to resolve each velocity into perpendicular components. Usually we choose horizontal (x) and vertical (y) axes. For a velocity v making an angle θ with the x-axis:

一个更强大的方法——尤其当需要处理多个角度各不相同的速度时——是将每个速度分解为相互垂直的分量。通常选择水平 (x) 和竖直 (y) 轴。对于一个与 x 轴成 θ 角的速度 v：

v_x = v cosθ, v_y = v sinθ

After resolving all velocities, we sum all x-components to obtain the resultant x-component R_x, and all y-components to obtain R_y. The resultant magnitude and direction are then:

将所有速度分解后，我们分别求出所有 x 分量的和 R_x，以及所有 y 分量的和 R_y。合成速度的大小和方向就是：

R = √(R_x² + R_y²), θ = tan⁻¹(R_y / R_x)

This component method avoids the need to remember the cosine and sine rules for every case and works elegantly with any number of velocities.

这种分解法避免了对每个情况都回忆余弦和正弦定理的麻烦，而且能优雅地处理任意数量的速度。

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7. Perpendicular Case: Boat Crossing a River | 垂直情况：小船过河

One of the most common examination problems involves a boat trying to cross a river that flows with a constant current. The boat’s velocity relative to the water, v_bw, is directed straight across the river (perpendicular to the banks). The water velocity relative to the ground, v_w, is parallel to the banks. These two velocities are perpendicular, so the resultant velocity relative to the ground is:

考试中最常见的题目之一是小船过河，河水以恒定水流流动。船相对于水的速度 v_bw 垂直于河岸，水流相对于地面的速度 v_w 平行于河岸。这两个速度相互垂直，因此船对地的合速度为：

v_bg = √(v_bw² + v_w²)

The time taken to cross the river of width d depends only on the perpendicular component:

穿越宽度为 d 的河流所需的时间只和垂直于河岸的速度分量有关：

t = d / v_bw

The boat drifts downstream by a distance x = v_w × t. Many students mistakenly think that the water speed affects the crossing time — it does not, because the downstream drift is independent of the motion across the river.

船向下游漂移的距离为 x = v_w × t。很多学生误以为水流速度会影响过河时间——实际上并不会，因为向下游的漂移独立于横穿河流的运动。

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8. Non-Perpendicular Example: Aircraft in Wind | 非垂直示例：风中飞机

An aircraft’s engines give it a velocity relative to the air, known as airspeed v_a. However, the air itself moves relative to the ground as wind with velocity v_w. The ground velocity v_g is the vector sum v_a + v_w. If the wind direction makes an angle θ with the aircraft’s heading, we can use the cosine rule:

飞机的发动机给予它一个相对于空气的速度，称为空速 v_a。但空气本身也以风速 v_w 相对地面运动。地速 v_g 就是 v_a + v_w

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