Combining Velocities | 速度的合成

📚 Combining Velocities | 速度的合成

In CIE A-Level Physics, combining velocities is a fundamental skill that appears in kinematics, vector analysis and relative motion problems. You must be able to add, subtract and resolve velocity vectors, and apply the results to real situations such as boats crossing rivers and aircraft flying in crosswinds. This article explains the key concepts, methods and worked examples you need for the exam.

在CIE A-Level物理中，速度的合成是一项基本技能，出现在运动学、矢量分析和相对运动问题中。你必须能够对速度矢量进行相加、相减和分解，并将结果应用于真实情境，例如小船渡河和飞机在侧风中飞行。本文讲解考试所需的核心概念、方法和例题。

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1. Scalars and Vectors | 标量与矢量

Physical quantities are divided into scalars and vectors. A scalar is fully described by its magnitude only, for example distance, speed, mass, time and energy. A vector requires both magnitude and direction, for example displacement, velocity, acceleration, force and momentum. This distinction is the starting point for combining velocities correctly.

物理量分为标量和矢量。标量只需大小即可完全描述，例如距离、速率、质量、时间和能量。矢量则需要大小和方向，例如位移、速度、加速度、力和动量。这一区别是正确合成速度的起点。

When we say a runner has a speed of 8 m/s, we give only a scalar. When we say the runner moves at 8 m/s due north, we give a velocity vector. Adding speeds without direction is meaningless when directions differ, so you must always treat velocity as a vector.

当我们说一名跑步者的速率为8 m/s时，我们只给出了一个标量。当我们说这名跑步者以8 m/s向正北运动时，我们给出的是一个速度矢量。如果方向不同，仅将速率相加是没有意义的，因此你必须始终将速度视为矢量。

Scalar | 标量	Vector | 矢量
speed | 速率	velocity | 速度
distance | 距离	displacement | 位移
mass | 质量	force | 力

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2. What is Velocity? | 什么是速度

Velocity is defined as the rate of change of displacement. It is a vector quantity, so its direction is essential. Average velocity is calculated by dividing total displacement by total time. Instantaneous velocity is the velocity at a specific moment, and its magnitude equals the instantaneous speed.

速度定义为位移的变化率。它是一个矢量，因此其方向至关重要。平均速度等于总位移除以总时间。瞬时速度是某一瞬间的速度，其大小等于瞬时速率。

If an object moves from A to B and then returns to A, its total displacement is zero, so its average velocity for the whole journey is zero. However, its average speed is not zero because the total distance travelled is positive. This example shows why speed and velocity cannot be treated as the same thing.

如果一个物体从A点运动到B点，然后又返回A点，它的总位移为零，因此整个过程的平均速度为零。但是，它的平均速率不为零，因为它所经过的总距离是正的。这个例子说明了为什么速率和速度不能混为一谈。

v = Δs / Δt

In this equation, Δs is the displacement vector, Δt is the time interval, and v is the average velocity. The SI unit of velocity is metre per second (m/s).

在这个公式中，Δs是位移矢量，Δt是时间间隔，v是平均速度。速度的国际单位是米每秒（m/s）。

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3. Representing Velocity as a Vector | 用矢量表示速度

A velocity vector is represented graphically by an arrow. The length of the arrow is drawn to scale and represents the magnitude of the velocity, while the arrowhead indicates the direction. For example, a scale of 1 cm = 10 m/s allows a 30 m/s velocity to be drawn as a 3 cm arrow.

速度矢量用箭头表示。箭头的长度按比例绘制，表示速度的大小，而箭头指向表示方向。例如，比例尺为1 cm = 10 m/s时，30 m/s的速度可以画成3 cm长的箭头。

When drawing vector diagrams, always label the scale and use a ruler and protractor. Directions are usually measured from the horizontal axis or from north, using degrees or bearings. In calculations, a vector may be written in component form, such as v = 30 m/s at 40° above the horizontal.

绘制矢量图时，务必标注比例尺，并使用直尺和量角器。方向通常从水平轴或正北方向测量，使用角度或方位角。在计算中，矢量可以用分量形式表示，例如 v = 30 m/s，与水平方向成40°角。

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4. Adding Velocities in One Dimension | 一维速度相加

In one dimension, velocities are added algebraically. First choose a positive direction, such as right or north. Velocities in the positive direction are positive, and velocities in the opposite direction are negative. The resultant velocity is the algebraic sum of all individual velocities.

在一维情况下，速度用代数方法相加。首先选择一个正方向，例如向右或向北。正方向的速度为正，相反方向的速度为负。合速度是所有单个速度的代数和。

v_resultant = v₁ + v₂ (same direction)

v_resultant = v₁ − v₂ (opposite directions)

For example, a boat moves at 3 m/s relative to the water. The river flows at 2 m/s. When the boat travels downstream, the resultant velocity relative to the bank is 3 m/s + 2 m/s = 5 m/s. When it travels upstream, the resultant velocity is 3 m/s − 2 m/s = 1 m/s relative to the bank.

例如，一艘船相对于水以3 m/s的速度行驶。河水以2 m/s流动。当船顺流而下时，相对于河岸的合速度为 3 m/s + 2 m/s = 5 m/s。当船逆流而上时，合速度为 3 m/s − 2 m/s = 1 m/s 相对于河岸。

This also works for vehicles on a straight road. If a car travels at 20 m/s east and the wind blows at 5 m/s east, the wind velocity relative to the car is 5 m/s − 20 m/s = −15 m/s, meaning 15 m/s west. Choosing and stating the positive direction avoids sign errors.

这也适用于直线道路上的车辆。如果一辆汽车以20 m/s向东行驶，风以5 m/s向东吹，那么相对于汽车的风速为 5 m/s − 20 m/s = −15 m/s，即15 m/s向西。选择并声明正方向可以避免符号错误。

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5. Adding Velocities in Two Dimensions | 二维速度合成

When velocities are not along the same line, they must be added as vectors. Two common graphical methods are the triangle method and the parallelogram method. In the triangle method, place the tail of the second vector at the head of the first vector. The resultant is drawn from the tail of the first vector to the head of the second vector.

当速度不在同一直线上时，必须作为矢量相加。两种常见的图解法是三角形法和平行四边形法。在三角形法中，将第二个矢量的尾部放在第一个矢量的头部。合矢量从第一个矢量的尾部画到第二个矢量的头部。

If the two velocities are perpendicular, the magnitude of the resultant velocity is found using Pythagoras’ theorem. The direction is calculated with the inverse tangent function. If the angle between the velocities is not 90°, the cosine rule or component method must be used.

如果两个速度互相垂直，合速度的大小用勾股定理求得。方向用反正切函数计算。如果两个速度之间的夹角不是90°，则必须使用余弦定理或分量法。

v_resultant = √(v₁² + v₂²)

θ = tan⁻¹(v₂ / v₁)

Here θ is the angle between the resultant and the direction of v₁. For example, a swimmer moves at 2 m/s east relative to the water, and the water moves at 1.5 m/s north. The swimmer’s velocity relative to the ground has magnitude √(2² + 1.5²) = 2.5 m/s and direction θ = tan⁻¹(1.5/2) ≈ 36.9° north of east.

这里θ是合速度与v₁方向之间的夹角。例如，一名游泳者相对于水以2 m/s向东游，水以1.5 m/s向北流动。游泳者相对于地面的速度大小为 √(2² + 1.5²) = 2.5 m/s，方向为东偏北约36.9°。

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6. Resolving Velocity Components | 速度分量分解

Resolving a velocity is the reverse of vector addition. A velocity v at an angle θ to a chosen axis can be split into two perpendicular components. The component along the axis is v cos θ, and the component perpendicular to the axis is v sin θ.

速度分解是矢量相加的逆过程。与选定轴成θ角的速度v可以分解为两个互相垂直的分量。沿轴方向的分量为 v cos θ，垂直于轴的分量为 v sin θ。

vₓ = v cos θ

vᵧ = v sin θ

This technique is extremely useful in projectile motion, where the horizontal velocity remains constant and the vertical velocity changes due to gravity. It is also used in river crossing and aircraft problems when the velocity must be split into perpendicular components relative to the bank or flight path.

这种技术在抛体运动中非常有用，其中水平速度保持不变，垂直速度因重力而变化。它也用于渡河和飞机问题中，当速度必须

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