📚 Distance and Displacement, Scalar and Vector | 距离与位移、标量与矢量
In CIE A-Level Physics, the terms 'distance' and 'displacement' are often used in everyday language, but they have precise and different meanings. Understanding the distinction between scalar and vector quantities is the key skill for describing motion accurately. This revision note explains the definitions, notation, vector addition, and exam-style calculations.
在 CIE A-Level 物理中，'distance'（距离）和 'displacement'（位移）在日常用语中常被混用，但在物理学中它们有严格且不同的含义。区分标量与矢量是准确描述运动的关键能力。本复习资料将讲解定义、符号、矢量加减法以及考试常见计算。
1. Defining Distance and Displacement | 距离与位移的定义
Distance is the total length of the path travelled by an object, regardless of its direction. It is a scalar quantity, so it only has magnitude and no direction. Displacement is the straight-line change in position from the starting point to the finishing point, in a specified direction. It is a vector quantity, so it has both magnitude and direction.
距离是物体运动路径的总长度，不涉及方向。它是一种标量，只有大小，没有方向。位移是从起点到终点的直线位置变化，并具有指定方向。它是一种矢量，既有大小又有方向。
For example, if a runner completes one full lap of a 400 m circular track, the distance travelled is 400 m, but the displacement is 0 m because the start and finish positions are identical.
例如，如果一名跑步者绕 400 m 圆形跑道跑完一整圈，所经过的距离是 400 m，但位移是 0 m，因为起点和终点位置相同。
2. Scalar and Vector Quantities | 标量与矢量
A scalar quantity is fully described by a magnitude and a unit. Examples in mechanics include distance, speed, mass, energy and time. A vector quantity requires both magnitude and direction for a complete description. Examples include displacement, velocity, acceleration, force and momentum.
标量只需用大小和单位就能完整描述。力学中的例子包括距离、速率、质量、能量和时间。矢量则需要同时用大小和方向才能完整描述。例子包括位移、速度、加速度、力和动量。
The table below summarises the contrast between distance and displacement.
下表总结了距离和位移的对比。
| Quantity | Type | Direction? | SI unit |
|---|---|---|---|
| Distance | Scalar | No | metre (m) |
| Displacement | Vector | Yes | metre (m) |
3. Notation and Units | 符号与单位
Distance is usually denoted by d or s, and its SI unit is the metre (m). Displacement is usually written as s with an arrow above it in printed textbooks, or as Δr for a change in position vector. In calculations, you can simply use 's' or 'Δx' with a sign and direction stated.
距离通常用 d 或 s 表示，SI 单位是米 (m)。位移在教材中通常写作上方带箭头的 s，或写作 Δr 表示位置矢量的变化。在计算中，你可以直接使用 's' 或 'Δx'，并标明正负号和方向。
Displacement can be calculated from coordinates by:
Δx = x₂ − x₁
位移可以由坐标计算：Δx = x₂ − x₁，其中 x₁ 是初位置，x₂ 是末位置。
4. Distance as Path Length | 距离作为路径长度
Distance is always positive or zero; it can never be negative. It accumulates as the object moves, even if the object changes direction. In a journey with several stages, total distance is the sum of the magnitudes of each stage.
距离总是正值或零，永远不会为负。即使物体改变方向，距离也会不断累积。在包含多个阶段的路程中，总距离等于各阶段路程大小的总和。
e, if a car drives 8 km north, then 6 km south, the total distance is 8 km + 6 km = 14 km.
例如，如果一辆汽车向北行驶 8 km，然后向南行驶 6 km，总距离为 8 km + 6 km = 14 km。
5. Displacement as Change in Position | 位移作为位置变化
Displacement depends only on the initial and final positions, not on the path taken. It is the shortest directed line from the initial point to the final point.
位移只取决于初、末位置，与运动路径无关。它是从初位置指向末位置的最短有向直线。
In the same car journey of 8 km north then 6 km south, the final position is 2 km north of the starting point, so the resultant displacement has magnitude 2 km and direction north.
在同一段汽车行程中（先向北 8 km，再向南 6 km），末位置在起点以北 2 km 处，因此合位移的大小为 2 km，方向为北。
If the car then drives another 4 km south, the final position is 2 km south of the start. The resultant displacement is 2 km south, while the total distance becomes 18 km.
如果汽车继续向南行驶 4 km，末位置在起点以南 2 km 处。合位移为 2 km 向南，而总距离变为 18 km。
6. Positive and Negative Displacement | 正位移与负位移
When motion is constrained to a straight line, we usually choose one direction as positive. Displacement in that direction is positive, and displacement in the opposite direction is negative. The sign therefore carries directional information.
当运动被限制在一条直线上时，我们通常选择一个方向为正。沿该方向的位移为正，沿相反方向的位移为负。因此，正负号带有方向信息。
For vertical motion, the upward direction is often taken as positive, so a ball thrown upward has a positive initial displacement, while the gravitational acceleration is negative. In horizontal motion, 'to the right' is a common positive convention.
对于竖直运动，通常取向上为正，因此向上抛出的球初位移为正，而重力加速度为负。在水平运动中，通常取“向右”为正。
In one dimension, the resultant displacement of sequential movements is the algebraic sum:
Δxtotal = Δx₁ + Δx₂ + Δx₃
在一维运动中，连续运动的合位移是代数和：Δxtotal = Δx₁ + Δx₂ + Δx₃。
7. Adding Vectors: Tip-to-Tail and Triangle Method | 矢量加法：首尾相接法与三角形法
Two displacement vectors can be added by drawing the first vector, then drawing the second vector starting from the head of the first. The resultant displacement is the vector from the tail of the first to the head of the second. This is called the tip-to-tail method.
两个位移矢量相加时，先画出第一个矢量，再从第一个矢量的末端（箭头处）开始画第二个矢量。合位移就是从第一个矢量的起点指向第二个矢量末端的矢量。这种方法称为首尾相接法。
If the two displacements are perpendicular, the magnitude of the resultant can be found using Pythagoras' theorem:
r = √a² + b²
如果两个位移互相垂直，合位移的大小可用勾股定理求得：r = √a² + b²。
The direction angle is given by:
θ = tan⁻¹(b / a)
方向角为：θ = tan⁻¹(b / a)。
For non-perpendicular vectors, a scale diagram can be used, or the vectors can be resolved into components and then added algebraically.
对于不垂直的矢量，可以使用比例图，或者将矢量分解为分量后再进行代数相加。
8. Resolving Vectors into Components | 矢量分解
Any vector can be split into perpendicular components, usually horizontal and vertical. If a displacement vector has magnitude A and makes an angle θ with the horizontal, then:
Ax = A cos θ
Ay = A sin θ
任何矢量都可以分解为相互垂直的分量，通常是水平分量和竖直分量。若位移矢量的大小为 A，与水平方向的夹角为 θ，则：Ax = A cos θ，Ay = A sin θ，其中 Ax 和 Ay 分别是水平分量和竖直分量。
The original vector can be recovered by using Pythagoras' theorem and trigonometry:
A = √Ax² + Ay²
θ = tan⁻¹(Ay / Ax)
原始矢量可以通过勾股定理和三角函数还原：A = √Ax² + Ay²，θ = tan⁻¹(Ay / Ax)。
Component addition is especially useful in CIE exam questions where several vectors act in different directions.
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