📚 The Velocity of a Particle as a Vector | 质点速度的矢量表示
In AS Mechanics, velocity is treated as a vector quantity, meaning it has both magnitude and direction. Many motion problems become much clearer when we write velocity in component form using unit vectors i and j, rather than just quoting a speed.
在 AS 力学中，速度被视为矢量，既有大小又有方向。当我们用单位矢量 i 和 j 以分量形式表示速度时，许多运动问题会变得更加清晰，而不是仅仅给出速率。
1. Scalar vs Vector: Speed and Velocity | 标量与矢量：速率和速度
Speed is a scalar quantity; it only tells us how fast a particle is moving. Velocity is a vector; it tells us how fast and in which direction. In everyday language the two words are often mixed, but in mechanics the distinction is essential.
速率是标量，它只告诉我们质点运动有多快。速度是矢量，它告诉我们运动有多快以及朝哪个方向。日常语言中这两个词常被混用，但在力学中这一区别至关重要。
- Speed = distance ÷ time (scalar)
- Velocity = displacement ÷ time (vector)
速率 = 路程 ÷ 时间（标量）；速度 = 位移 ÷ 时间（矢量）。
2. Position Vector and Displacement | 位置矢量与位移
A particle’s position is described by a position vector r = x i + y j. If the particle moves from initial position r₀ = x₀ i + y₀ j to final position r₁ = x₁ i + y₁ j, its displacement is Δr = r₁ − r₀.
质点的位置由位置矢量 r = x i + y j 描述。若质点从初位置 r₀ = x₀ i + y₀ j 运动到末位置 r₁ = x₁ i + y₁ j，其位移为 Δr = r₁ − r₀。
Δr = (x₁ − x₀) i + (y₁ − y₀) j
3. Defining Velocity as a Vector | 将速度定义为矢量
Average velocity is displacement divided by time, so it is a vector parallel to the displacement. Instantaneous velocity is the limit of average velocity as the time interval tends to zero, written as the derivative of position with respect to time.
平均速度是位移除以时间，因此它是与位移平行的矢量。瞬时速度是当时间间隔趋于零时平均速度的极限，写作位置对时间的导数。
v = dr/dt
4. Components of Velocity in Two Dimensions | 二维速度的分量
In the i, j basis, velocity can be written as v = v₁ i + v₂ j. The horizontal component v₁ and vertical component v₂ are independent in projectile-type problems, so equations of motion can be applied separately to each direction.
在 i、j 基底下，速度可写作 v = v₁ i + v₂ j。水平分量 v₁ 和竖直分量 v₂ 在抛体类问题中相互独立，因此运动方程可以分别应用于每个方向。
v = v₁ i + v₂ j
5. Magnitude and Direction of Velocity | 速度的大小与方向
The magnitude of velocity is the speed, found by Pythagoras’ theorem: |v| = √(v₁² + v₂²). Direction is usually given as an angle θ from the positive i direction, where tan θ = v₂ ÷ v₁.
速度的大小就是速率，可由勾股定理求得：|v| = √(v₁² + v₂²)。方向通常用与正 i 方向的夹角 θ 表示，其中 tan θ = v₂ ÷ v₁。
|v| = √(v₁² + v₂²)
tan θ = v₂ ÷ v₁
6. Unit Vectors and Velocity | 单位矢量与速度
A unit vector has length 1 and is used to indicate direction. The velocity vector can be written as its magnitude multiplied by a unit vector in the direction of motion: v = |v| e. In two dimensions e = cos θ i + sin θ j.
单位矢量长度为 1，用于表示方向。速度矢量可以写成其大小乘以运动方向上的单位矢量：v = |v| e。在二维情形下 e = cos θ i + sin θ j。
v = |v| (cos θ i + sin θ j)
7. Velocity from Position: Differentiation | 由位置求速度：微分法
If a position vector is given as a function of time, differentiating each component with respect to t gives the velocity. For example, if r = t² i + 3t j, then v = 2t i + 3 j.
若位置矢量作为时间的函数给出，对每个分量关于 t 求导即可得速度。例如，若 r = t² i + 3t j，则 v = 2t i + 3 j。
r = t² i + 3t j ⇒ v = 2t i + 3 j
8. Position from Velocity: Integration | 由速度求位置：积分法
Conversely, if velocity is known as a function of time, integrating each component gives displacement. Initial position must be included to find the full position vector. If v = 2t i + 3 j and r(0) = i + j, then r = (t² + 1) i + (3t + 1) j.
反之，若速度作为时间的函数已知，对每个分量积分即可得到位移。必须包含初始位置才能求出完整位置矢量。若 v = 2t i + 3 j 且 r(0) = i + j，则 r = (t² + 1) i + (3t + 1) j。
r = r₀ + ∫ v dt
9. Constant Velocity Motion | 匀速直线运动
When velocity is a constant vector, the particle moves in a straight line with constant speed. The position vector at time t is r = r₀ + v t. This is the vector form of ‘distance = speed × time’, but with direction included.
当速度为常矢量时，质点以恒定速率沿直线运动。t 时刻的位置矢量为 r = r₀ + v t。这是“距离 = 速率 × 时间”的矢量形式，但包含了方向。
r = r₀ + v t
10. Relative Velocity | 相对速度
The velocity of particle A relative to particle B is v(A/B) = v(A) − v(B). This vector tells us how A appears to move from B’s point of view. River crossing and overtaking problems often use relative velocity.
质点 A 相对于质点 B 的速度为 v(A/B) = v(A) − v(B)。该矢量告诉我们在 B 看来 A 如何运动。过河和超车问题常使用相对速度。
v(A/B) = v(A) − v(B)
11. Acceleration as the Rate of Change of Velocity Vector | 作为速度矢量变化率的加速度
Acceleration is also a vector: a = dv/dt. It can arise from a change in speed, a change in direction, or both. A particle moving in a circle at constant speed still has acceleration because its velocity direction is changing.
加速度也是矢量：a = dv/dt。它可以由速度大小变化、方向变化或两者同时变化引起。以恒定速率做圆周运动的质点仍有加速度，因为其速度方向在不断变化。
a = dv/dt = a₁ i + a₂ j
12. Exam-style Worked Example | 考试风格例题
A particle moves with velocity v = 3 i + 4 j m/s. Find (a) its speed, (b) the angle its direction makes with the positive i-axis.
质点以速度 v = 3 i + 4 j m/s 运动。求 (a) 其速率；(b) 其方向与正 i 轴的夹角。
Solution: speed = √(3² + 4²) = 5 m/s. The direction is θ = arctan(4/3) ≈ 53.1°.
解：速率 = √(3² + 4²) = 5 m/s。方向为 θ = arctan(4/3) ≈ 53.1°。
| Quantity | Type | Example |
| Speed | Scalar | 5 m/s |
| Velocity | Vector | 3 i + 4 j m/s |
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导