📚 Vectors in Kinematics | 运动学中的向量
Vectors turn kinematics from a one-dimensional sign problem into a powerful, direction-aware tool. In A-Level Mechanics we treat displacement, velocity and acceleration as vector quantities written with i and j components, and we use calculus or the constant-acceleration equations on each component. This article covers everything Edexcel and AQA expect you to know: from constant acceleration vector equations to projectile motion, relative velocity, closest approach, and how to differentiate and integrate vectors.
向量将运动学从一维的正负号问题转变为强大的、具有方向性的工具。在 A-Level 力学中，我们使用 i 和 j 分量把位移、速度和加速度作为向量来处理，并对每个分量分别使用微积分或恒加速度公式。本文涵盖 Edexcel 与 AQA 要求你掌握的全部内容：从恒加速度向量方程到抛体运动、相对速度、最近距离，以及向量的微分与积分。
1. From Scalars to Vectors | 从标量到向量
In one-dimensional kinematics, direction is simply + or −. In two dimensions we need both magnitude and direction, which is why we work with vectors. The standard unit vectors are i (horizontal, usually east) and j (vertical, usually north). A velocity v = 3i + 4j means 3 units east and 4 units north every second.
在一维运动学中，方向只是 + 或 −。在二维中我们需要大小和方向，因此我们使用向量。标准单位向量是 i（水平，通常向东）和 j（垂直，通常向北）。速度 v = 3i + 4j 表示每秒向东 3 个单位、向北 4 个单位。
The magnitude of that velocity comes from Pythagoras: |v| = √3² + 4² e, speed, mass and time have size only; vectors such as displacement, velocity, acceleration, momentum and force carry direction as well.
该速度的大小由勾股定理给出：|v| = √3² + 4² = 5 个单位每秒。距离、速率、质量、时间等标量只有大小；位移、速度、加速度、动量、力等向量还带有方向。
Never leave a vector answer as a bare number. A velocity of 5 m s⁻¹ is incomplete; a velocity of 5 m s⁻¹ on a bearing of 053° is a full answer, and exam mark schemes usually award a mark for the direction.
不要把向量答案写成一个孤立的数字。只写速度 5 m s⁻¹ 是不完整的；写成速度为 5 m s⁻¹、方位角 053° 才是完整答案，评分标准通常会给方向单独一分。
2. Position and Displacement Vectors | 位置向量与位移
The position vector r of a particle relative to an origin O is r = xi + yj, where x and y are the coordinates of the particle, so it sits at the point (x, y). Its length is |r| = √x² + y², the straight-line distance from the origin.
粒子相对于原点 O 的位置向量为 r = xi + yj，其中 x 与 y 就是粒子的坐标，即粒子位于点 (x, y)。它的长度为 |r| = √x² + y²，也就是到原点的直线距离。
r = x i + y j
Δr = r₂ − r₁ = (x₂ − x₁) i + (y₂ − y₁) j
Displacement is the change in position, Δr = r₂ − r₁. Its magnitude gives the straight-line distance between the two points, but the actual path length (the distance travelled) equals that magnitude only when the motion is in a straight line and never reverses.
位移是位置的变化：Δr = r₂ − r₁。位移的大小给出两点间的直线距离，但实际路径长度（路程）仅在运动始终沿同一直线且不折返时才等于该大小。
For example, if a particle moves from r₁ = 2i + 3j to r₂ = 7i + 9j, then Δr = 5i + 6j and the distance between the two points is √5² + 6² = √61 81 m. Subtract coordinates in the same order: final minus initial, component by component.
例如，粒子从 r₁ = 2i + 3j 运动到 r₂ = 7i + 9j，则 Δr = 5i + 6j，两点间距离为 √5² + 6² = √61 ≈ 7.81 m。相减时次序要一致：末位置减初位置，逐分量计算。
3. Velocity, Speed and Acceleration | 速度、速率与加速度
Velocity is the rate of change of position: v = dr/dt. In components, v = vxi + vyj, where vx = dx/dt and vy = dy/dt. Speed is the magnitude of velocity: |v| = √vx² + vy². Speed is a scalar: it tells you how fast the particle moves but says nothing about where it is heading.
速度是位置的变化率：v = dr/dt。用分量表示为 v = vxi + vyj，其中 vx = dx/dt，vy = dy/dt。速率是速度的大小：|v| = √vx² + vy²。速率是标量：它只说明运动有多快，不说明朝哪个方向。
Acceleration is the rate of change of velocity: a = dv/dt = d²r/dt², again a vector, with ax = dvx/dt and ay = dvy/dt. Average velocity over a time interval is Δr/Δt, while instantaneous velocity is the limit of that quotient as Δt → 0.
加速度是速度的变化率：a = dv/dt = d²r/dt²，同样是向量，其中 ax = dvx/dt，ay = dvy/dt。一段时间内的平均速度是 Δr/Δt，而瞬时速度是当 Δt → 0 时该比值的极限。
v = dr/dt a = dv/dt = d²r
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