Distance & Displacement, Scalars & Vectors | 距离与位移、标量与矢量

📚 Distance & Displacement, Scalars & Vectors | 距离与位移、标量与矢量

In A-Level Physics, one of the first and most important distinctions you must master is the difference between distance and displacement, and the broader classification of physical quantities into scalars and vectors. This foundation underpins everything from kinematics to forces and fields.

在 A-Level 物理中，你首先必须掌握且最重要的区分之一，就是距离与位移之间的差别，以及将物理量广义划分为标量与矢量的方法。这一基础贯穿从运动学到力与场的全部内容。

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1. Distance and Displacement | 距离与位移

Distance is the total length of the path travelled by an object, regardless of direction. It is a scalar quantity, meaning it has magnitude only. Its SI unit is the metre (m).

距离是物体运动路径的总长度，与方向无关。它是一个标量，意味着只有大小。其国际单位是米（m）。

Displacement is the straight-line distance from the starting point to the final position, measured in a specific direction. It is a vector quantity, meaning it has both magnitude and direction.

位移是从起点到终点位置的直线距离，并且沿特定方向测量。它是一个矢量，意味着既有大小又有方向。

displacement = final position − initial position

位移 = 末位置 − 初位置

Consider a student who walks 3 km east, then 4 km north. The distance travelled is 3 km + 4 km = 7 km. However, the displacement is the straight-line separation between the start and end points, which is √(3² + 4²) = 5 km in a direction 53.1° north of east.

设想一名学生先向东走 3 km，再向北走 4 km。走过的距离为 3 km + 4 km = 7 km。然而，位移是起点与终点之间的直线距离，即 √(3² + 4²) = 5 km，方向为东偏北 53.1°。

It is possible for displacement to be zero while distance is non-zero. If you run once around a 400 m track and return to the start, your distance is 400 m but your displacement is zero, because your final position is identical to your initial position.

可能出现位移为零而距离不为零的情况。如果你绕 400 m 跑道跑一圈并回到起点，你的距离是 400 m，但位移为零，因为你的末位置与初位置完全相同。

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2. Scalars and Vectors | 标量与矢量

A scalar is a physical quantity that has magnitude only. A vector is a physical quantity that has both magnitude and direction.

标量是只有大小的物理量。矢量是既有大小又有方向的物理量。

Scalars | 标量	Vectors | 矢量
Distance | 距离	Displacement | 位移
Speed | 速率	Velocity | 速度
Mass | 质量	Acceleration | 加速度
Time | 时间	Force | 力
Temperature | 温度	Momentum | 动量
Energy | 能量	Weight | 重力
Work | 功	Electric field strength | 电场强度
Power | 功率	Magnetic flux density | 磁通量密度

From this table, notice that speed is the scalar counterpart of velocity. Speed is the rate of change of distance, while velocity is the rate of change of displacement. Their formulae are:

由上表可见，速率是速度的标量对应量。速率是距离的变化率，而速度是位移的变化率。其公式为：

speed = distance ÷ time

velocity = displacement ÷ time

速率 = 距离 ÷ 时间

速度 = 位移 ÷ 时间

A useful exam point: if an object returns to its starting point, its average velocity is zero, but its average speed is not zero. Many students lose marks by confusing these two ideas in multiple-choice questions.

一个实用的考点：若物体回到起点，其平均速度为零，但平均速率不为零。许多学生在选择题中因混淆这两个概念而失分。

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3. Representing Vectors | 矢量的表示

A vector is represented graphically by an arrow. The length of the arrow is proportional to the magnitude of the vector, and the direction of the arrow shows the direction of the vector.

矢量在图形上用箭头表示。箭头的长度与矢量的大小成比例，箭头所指的方向表示矢量的方向。

In printed text, vectors are often written in bold, such as F or v. In handwritten work, a vector is indicated with an arrow above the symbol, such as F⃗ or v⃗. The magnitude of a vector is written without the bold or arrow, for example |F| or simply F.

在印刷文本中，矢量通常用粗体书写，如 F 或 v。在手写中，矢量用符号上方的箭头表示，如 F⃗ 或 v⃗。矢量的大小不用粗体或箭头书写，例如 |F| 或直接写 F。

A negative vector has the same magnitude as the original vector but points in the exact opposite direction. If vector A points east with magnitude 5 N, then −A points west with magnitude 5 N.

负矢量与原矢量大小相同，但方向完全相反。若矢量 A 向东且大小为 5 N，则 −A 向西且大小为 5 N。

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4. Vector Addition | 矢量加法

Scalars can be added with ordinary arithmetic: 3 kg + 4 kg = 7 kg. Vectors cannot be added this way unless they point in the same direction, because direction matters.

标量可以用普通算术相加：3 kg + 4 kg = 7 kg。矢量则不能这样相加，除非它们方向相同，因为方向至关重要。

Triangle rule (nose-to-tail method): Draw the first vector. Then draw the second vector starting from the tip (nose) of the first. The resultant vector is drawn from the tail of the first to the nose of the second.

三角形法则（首尾相接法）：先画出第一个矢量，再从第一个矢量的箭头（首）端画出第二个矢量。合矢量从第一个矢量的尾端指向第二个矢量的首端。

Parallelogram rule: Draw both vectors from the same starting point. Complete the parallelogram. The resultant is the diagonal drawn from the common starting point.

平行四边形法则：从同一点画出两个矢量。补全平行四边形。合矢量是从公共起点出发的对角线。

Both methods give the same resultant. Consider two perpendicular forces: F₁ = 3 N east and F₂ = 4 N north. The resultant magnitude is √(3² + 4²) = 5 N. The angle θ above the east direction is given by tan θ = 4/3, so θ = 53.1°.

两种方法给出的合矢量相同。考虑两个互相垂直的力：F₁ = 3 N 向东，F₂ = 4 N 向北。合矢量大小为 √(3² + 4²) = 5 N。与东方向的夹角 θ 由 tan θ = 4/3 给出，故 θ = 53.1°。

resultant = √(F₁² + F₂²) when F₁ ⊥ F₂

当 F₁ ⊥ F₂ 时，合力 = √(F₁² + F₂²)

If two vectors point in the same direction, the resultant is the arithmetic sum. If they point in opposite directions, the resultant is the arithmetic difference, directed along the larger vector.

若两个矢量方向相同，合力为算术和。若方向相反，合力为算术差，方向沿较大的矢量。

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5. Vector Subtraction | 矢量减法

To subtract vector B from vector A, add the negative of B to A:

要用 A 减去矢量 B，只需将 A 与 B 的负矢量相加：

A − B = A + (−B)

Graphically, this means reversing the direction of B, then applying the triangle or parallelogram rule. Vector subtraction is essential when calculating relative velocity or relative displacement.

在图形上，这意味着先将 B 的方向反转，再应用三角形或平行四边形法则。矢量减法在计算相对速度或相对位移时至关重要。

For example, if a car A moves east at 20 m s⁻¹ and car B moves east at 15 m s⁻¹, the velocity of A relative to B is 20 − 15 = 5 m s⁻¹ east. In vector notation, v(AB) = v(A) − v(B).

例如，若汽车 A 以 20 m s⁻¹ 向东行驶，汽车 B 以 15 m s⁻¹ 向东行驶，则 A 相对 B 的速度为 20 − 15 = 5 m s⁻¹ 向东。用矢量记号表示，v(AB) = v(A) − v(B)。

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6. Resolving Vectors into Components | 矢量的正交分解

Just as two vectors can be added to give one resultant, a single vector can be resolved into two perpendicular components. This is called resolving a vector. The standard convention uses horizontal (x) and vertical (y) components.

正如两个矢量可以合成为一个合矢量，一个矢量也可以分解为两个互相垂直的分量。这称为矢量的分解。标准惯例使用水平（x）分量和竖直（y）分量。

For a vector F making an angle θ with the horizontal:

对于与水平方向成夹角 θ 的矢量 F：

Fₓ = F cos θ

Fᵧ = F sin θ

水平分量 Fₓ = F cos θ

竖直分量 Fᵧ = F sin θ

It is critical to choose the correct trigonometric function. The component adjacent to the angle always uses cosine; the component opposite the angle uses sine. A common trick: if the component “points toward” the angle, it is F cos θ; if it “points away” from the angle, it is F sin θ.

选择正确的三角函数至关重要。与角相邻的分量用余弦；与角相对的分量用正弦。一个常用技巧：若分量“指向”该角，则为 F cos θ；若分量“远离”该角，则为 F sin θ。

Resolution is particularly useful when dealing with forces on inclined planes, projectiles, and objects on slopes. Always draw a clear diagram before resolving.

分解在处理斜面上的力、抛体运动和斜坡上的物体时特别有用。分解前务必画出清晰的受力图。

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7. Adding Vectors by Components | 用分量法合成矢量

When adding two or more vectors, resolving each into components is often the most reliable method, especially when the vectors are not perpendicular to each other.

当相加两个或多个矢量时，将每个矢量分解为分量往往是最可靠的方法，尤其是当矢量之间不垂直时。

The method has four steps:

该方法分为四步：

• Resolve every vector into its x and y components. | 将每个矢量分解为 x 和 y 分量。
• Sum all x components: Rₓ = ΣFₓ. | 将所有 x 分量求和：Rₓ = ΣFₓ。
• Sum all y components: Rᵧ = ΣFᵧ. | 将所有 y 分量求和：Rᵧ = ΣFᵧ。
• Combine the resultants: R = √(Rₓ² + Rᵧ²) and θ = tan⁻¹(Rᵧ / Rₓ). | 合成结果：R = √(Rₓ² + Rᵧ²)，方向 θ = tan⁻¹(Rᵧ / Rₓ)。

R = √(Rₓ² + Rᵧ²)

θ = tan⁻¹(Rᵧ / Rₓ)

Remember to pay attention to signs. Components pointing left or down are negative. Careless sign errors are among the most common reasons for losing marks in CIE vector questions.

务必注意正负号。指向左或向下的分量为负。粗心的符号错误是 CIE 矢量题失分最常见的原因之一。

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8. Worked Example 1: Walking Journey | 例题一：行走路径

Problem: A student walks 6.0 km due north, then 8.0 km due east. Calculate (a) the total distance travelled, (b) the displacement from the starting point, including direction.

题目：一名学生先向正北走 6.0 km，再向正东走 8.0 km。求（a）总路程；（b）相对于起点的位移，包括方向。

Solution:

解答：

(a) Distance = 6.0 + 8.0 = 14.0 km (scalar addition).

（a）距离 = 6.0 + 8.0 = 14.0 km（标量相加）。

(b) The two displacement vectors are perpendicular, so the resultant magnitude is:

（b）两位移矢量互相垂直，因此合位移大小为：

s = √(6.0² + 8.0²) = √100 = 10 km

The direction is found from tan θ = 8.0 / 6.0, giving θ = 53.1° east of north. So the displacement is 10 km at 53.1° east of north.

方向由 tan θ = 8.0 / 6.0 求得，即 θ = 53.1°（北偏东）。因此位移为 10 km，方向北偏东 53.1°。

Notice that distance (14 km) is greater than displacement (10 km). This is always true unless the motion is in a straight line without changing direction.

注意距离（14 km）大于位移（10 km）。除非物体沿直线且不改变方向运动，否则总是如此。

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9. Worked Example 2: Resolving a Force | 例题二：力的分解

Problem: A box is pulled by a force of 50 N at an angle of 30° above the horizontal. Calculate the horizontal and vertical components of the force.

题目：一箱子受到 50 N 的拉力，方向与水平方向成 30° 角。求该力的水平分量与竖直分量。

Solution:

解答：

Fₓ = 50 cos 30° = 50 × 0.866 = 43.3 N

Fᵧ = 50 sin 30° = 50 × 0.500 = 25.0 N

The horizontal component is 43.3 N and the vertical component is 25.0 N. The vertical component reduces the effective normal contact force between the box and the ground, which is a common application in mechanics questions.

水平分量为 43.3 N，竖直分量为 25.0 N。竖直分量会减小箱子与地面之间的有效支持力，这是力学题中常见的应用。

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10. Common Mistakes and Exam Tips | 常见错误与应试技巧

Students frequently make the following errors in CIE examinations:

学生在 CIE 考试中经常出现以下错误：

• Treating distance and displacement as identical. Always ask: does this quantity require a direction? | 将距离与位移视为等同。始终问自己：这个量是否需要方向？
• Using scalar addition for vectors that are not parallel. Use the triangle, parallelogram, or component method instead. | 对不平行矢量使用标量加法。应使用三角形法、平行四边形法或分量法。
• Forgetting that average velocity uses displacement, not distance. | 忘记平均速度用的是位移而非距离。
• Writing a vector answer without a direction. A vector answer without direction is incomplete and loses marks. | 写出矢量答案却不带方向。没有方向的矢量答案是不完整的，会被扣分。
• Misapplying sine and cosine when resolving. Check whether the component is adjacent or opposite to the given angle. | 分解时误用正弦与余弦。检查该分量是与给定角相邻还是相对。
• Ignoring negative signs when summing components. | 分量求和时忽略负号。

Finally, always quote units and give directions using both an angle and a reference direction, such as “30° above the horizontal” or “south of east”. This makes your answer unambiguous and ensures full marks.

最后，始终写上单位，并用角度加参考方向给出方向，例如“与水平成 30° 角”或“南偏东”。这使你的答案清晰无歧义，确保获得满分。

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11. Summary | 小结

Distance is the total path length and is a scalar; displacement is the straight-line change in position and is a vector. Scalars have magnitude only, while vectors have both magnitude and direction. Vectors are added using the triangle or parallelogram rule, or by resolving into perpendicular components and summing those components separately.

距离是路径总长度，属于标量；位移是位置的直线变化量，属于矢量。标量只有大小，矢量既有大小又有方向。矢量用三角形法则或平行四边形法则相加，或分解为互相垂直的分量后分别求和。

Mastering these ideas will support your understanding of velocity, acceleration, forces, momentum, and even electric and magnetic fields. Practise drawing vector diagrams and always check whether your final answer includes a direction when a vector is required.

掌握这些概念将帮助你理解速度、加速度、力、动量乃至电场和磁场。多练习画矢量图，并在需要矢量时始终检查最终答案是否包含了方向。

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