📚 Motion in a Two-Dimensional Plane | 二维平面内的运动分析
In A-level Mathematics, ‘Motion in a Two-Dimensional Plane’ is the study of how particles move using vectors to describe position, velocity and acceleration. This topic links pure vector geometry with kinematics, and it forms the foundation for solving projectile motion, relative motion and interception problems.
在 A-level 数学中，“二维平面内的运动分析”是利用向量描述质点位置、速度与加速度的运动学研究。该主题将纯向量几何与运动学相结合，是解决抛体运动、相对运动与追及问题的基础。
1. Position Vectors and Displacement | 位置向量与位移
Take a fixed origin O. The position of a particle P is described by the position vector r = x i + y j, where x and y are its coordinates in a Cartesian plane.
取固定原点 O，质点 P 的位置由位置向量 r = x i + y j 描述，其中 x、y 是其笛卡尔平面内的坐标。
The displacement from A to B is the vector AB = r_B − r_A. It represents the net change of position, and its magnitude is the straight-line distance between A and B.
从 A 到 B 的位移是向量 AB = r_B − r_A，它表示位置的总变化，其大小等于 A、B 两点间的直线距离。
In handwritten work, underline vector symbols to distinguish them from scalars. For example, write r with a solid underline.
在手写答卷中，请给向量符号添加下划线以与标量区分，例如在 r 下方画一条实线。
2. Velocity and Acceleration Vectors | 速度与加速度向量
The average velocity over a time interval Δt is the displacement divided by Δt: v_avg = (r(t + Δt) − r(t)) / Δt. This is a vector quantity.
在时间间隔 Δt 内，平均速度等于位移除以 Δt：v_avg = (r(t + Δt) − r(t)) / Δt，它是一个向量量。
The instantaneous velocity is the limit of the average velocity as Δt tends to zero: v = dr/dt. Its components are v_x = dx/dt and v_y = dy/dt.
瞬时速度是 Δt 趋于零时平均速度的极限：v = dr/dt。其分量为 v_x = dx/dt、v_y = dy/dt。
Acceleration is the rate of change of velocity: a = dv/dt = d²r/dt². In component form, a_x = d²x/dt² and a_y = d²y/dt².
加速度是速度的变化率：a = dv/dt = d²r/dt²。分量形式为 a_x = d²x/dt²、a_y = d²y/dt²。
3. Motion with Constant Acceleration | 恒定加速度运动
When the acceleration a is constant, the vector equations of motion are:
当加速度 a 恒定时，运动的向量方程为：
v = u + a t, r = r₀ + u t + ½ a t², v·v = u·u + 2 a·(r − r₀)
Here u is the initial velocity, v is the final velocity, and r₀ is the initial position. These equations are only valid when a is constant.
其中 u 是初速度，v 是末速度，r₀ 是初始位置。这些方程仅在 a 恒定时成立。
In practice, resolve the equation into horizontal and vertical components. For example, v_x = u_x + a_x t and v_y = u_y + a_y t.
实际计算中，需要将方程分解为水平与竖直分量。例如 v_x = u_x + a_x t、v_y = u_y + a_y t。
4. Projectile Motion: Initial Velocity Components | 抛体运动：初速度分量
For a projectile launched with speed u at angle θ above the horizontal, ignoring air resistance and taking g as the acceleration due to gravity, the initial velocity is u = (u cos θ) i + (u sin θ) j.
对于以速率 u、与水平方向成角 θ 抛出的抛体，忽略空气阻力并取重力加速度为 g，初速度为 u = (u cos θ) i + (u sin θ) j。
Because there is no horizontal acceleration, the horizontal displacement after time t is x = u cos θ · t.
由于水平方向无加速度，t 时刻的水平位移为 x = u cos θ · t。
Vertically, the acceleration is −g, so the vertical displacement is y = u sin θ · t − ½ g t².
竖直方向加速度为 −g，因此竖直位移为 y = u sin θ · t − ½ g t²。
5. Equation of the Trajectory | 轨迹方程
Eliminate t from the horizontal equation: t = x / (u cos θ). Substitute this into the vertical displacement equation to obtain the trajectory.
由水平方程得 t = x / (u cos θ)。将其代入竖直位移方程，即可得到轨迹方程。
y = x tan θ − (g x²) / (2 u² cos²θ)
This is a quadratic in x, so the path of a projectile is a parabola. The term x tan θ would be the straight-line path without gravity.
这是关于 x 的二次式，因此抛体运动的路径为抛物线。其中 x tan θ 表示没有重力时质点应沿直线经过的位置。
6. Maximum Height and Time of Flight | 最大高度与飞行时间
At the maximum height, the vertical velocity is zero. Using v_y² = u_y² − 2g y with v_y = 0, we get H = (u² sin²θ) / (2g).
在最高点，竖直速度为零。利用 v_y² = u_y² − 2g y 并令 v_y = 0，得最大高度 H = (u² sin²θ) / (2g)。
The time taken to reach the top is t_p = u sin θ / g. If the projectile lands at the same height as it was launched, the total flight time is T = 2 u sin θ / g.
到达最高点所需时间为 t_p = u sin θ / g。若落点与发射点同高，则总飞行时间为 T = 2 u sin θ / g。
7. Horizontal Range and Maximum Range Angle | 水平射程与最大射程角
The horizontal range R is the horizontal distance covered during the flight time T. Substituting T into x = u cos θ · t gives R = (u² sin 2θ) / g.
水平射程 R 是飞行时间 T 内经过的水平距离。将 T 代入 x = u cos θ · t，得到 R = (u² sin 2θ) / g。
Since sin 2θ ≤ 1, the maximum range occurs when sin 2θ = 1, i.e. θ = 45°. At this angle, R_max = u² / g.
因为 sin 2θ ≤ 1，当 sin 2θ = 1，即 θ = 45° 时射程最大，此时 R_max = u² / g。
8. Relative Motion in Two Dimensions | 二维相对运动
For two particles A and B with velocities v_A and v_B, the velocity of A relative to B is v_rel = v_A − v_B. This vector shows how A appears to move from B’s point of view.
对于速度分别为 v_A、v_B 的两质点 A 和 B，A 相对于 B 的速度为 v_rel = v_A − v_B，它描述了从 B 观察时 A 的运动。
If r_A and r_B are their position vectors, then r_rel = r_A − r_B is the position of A relative to B. Differentiating this expression gives the relative velocity.
若 r_A、r_B 分别是它们的位置向量，则 r_rel = r_A − r_B 表示 A 相对于 B 的位置，对其求导即得相对速度。
Relative motion is particularly useful for interception problems: if A is to reach B, the relative position vector must point directly toward B.
相对运动特别适用于追及问题：若 A 要追上 B，则相对位置向量必须始终指向 B。
9. Problem-Solving Tips and Common Mistakes | 解题技巧与常见错误
Always draw a clear diagram, resolve vectors into components, and treat horizontal motion separately from vertical motion. Choose a consistent sign convention, e.g. upwards as positive for vertical motion.
务必画示意图，将向量分解为分量，并将水平运动与竖直运动分开处理。选择一致的符号约定，例如竖直方向取向上为正。
Common mistakes include forgetting the minus sign on gravity, mixing up x and y components, and applying constant-acceleration equations when acceleration is not constant. Always check units and state whether the answer is a scalar or a vector.
常见错误包括忘记重力加速度取负号、混淆 x 与 y 分量，以及在加速度不恒定时误用恒定加速度公式。请始终检查单位，并明确最终答案应表示标量还是向量。
For projectile problems, if the launch speed u and angle θ are given, write u_x = u cos θ and u_y = u sin θ before applying any formula.
在抛体问题中，若已知初速率 u 和角度 θ，应先写出 u_x = u cos θ、u_y = u sin θ，再代入相关公式。
10. Worked Example | 典型例题
A particle is projected
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