Vectors in Kinematics | 运动学中的向量

📚 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 using i and j components. This article covers everything Edexcel expects you to know: from constant acceleration vector equations to projectile motion, relative velocity, closest approach, and how to integrate or differentiate 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 分量将位移、速度和加速度作为向量来处理。本文涵盖 Edexcel 要求的所有内容：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积分与微分。

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 vector v = 3i + 4j means 3 units east and 4 units north every second if it is a velocity.

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）。向量 v = 3i + 4j 如果是速度，则表示每秒向东 3 个单位、向北 4 个单位。

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 = x i + y j. Displacement is the change in position: Δr = r₂ − r₁. Magnitude of displacement gives the straight‑line distance between two points, but the actual path length (distance travelled) is only equal to this magnitude when motion is in a straight line without reversing.

粒子相对于原点 O 的位置向量为 r = x i + y j。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）仅在直线运动且不折返时才等于该大小。

r = x i + y j

Δr = (x₂ − x₁) i + (y₂ − y₁) j

3. Velocity and Speed | 速度与速率

Velocity v = dr/dt. In components, v = v_x i + v_y j where v_x = dx/dt and v_y = dy/dt. Speed is the magnitude of velocity: |v| = √(v_x² + v_y²). Always remember that speed is a scalar; it tells you ‘how fast’ but not the direction.

速度 v = dr/dt。用分量表示为 v = v_x i