📚 Motion in Two Dimensions | 二维运动
In real-world physics, very few objects move strictly in a straight line. From a football arching through the air to a planet orbiting the Sun, motion often unfolds across a plane involving both horizontal and vertical changes. This topic explores how to analyse such motion by treating perpendicular directions independently, focusing primarily on projectile motion under constant gravitational acceleration.
在现实物理世界中，很少有物体严格地沿直线运动。从空中飞行的足球到绕太阳运行的行星，运动往往在一个平面内展开，同时涉及水平和垂直方向的变化。本专题探讨如何通过独立处理相互垂直的方向来分析这类运动，重点讲解在恒定重力加速度作用下的抛体运动。
1. Introduction to Motion in Two Dimensions | 二维运动简介
Motion in two dimensions means an object's position changes along both an x‑axis and a y‑axis simultaneously. To describe such movement, we use vectors for displacement, velocity and acceleration. Resolving these vectors into perpendicular components is the key technique – it lets us apply one‑dimensional equations separately in each direction.
二维运动意味着物体的位置同时沿 x 轴和 y 轴发生变化。为了描述这类运动，我们使用位移、速度和加速度矢量。将这些矢量分解为相互垂直的分量是关键方法——这使我们能够在每个方向上单独应用一维运动方程。
Edexcel specifications introduce this concept early in mechanics, using projectiles as the primary context. Mastery of 2D motion underpins later topics such as circular motion and fields, making it a cornerstone of A‑Level Physics.
Edexcel 课程大纲在力学部分早期就引入这一概念，并以抛体作为主要情境。掌握二维运动是后续学习圆周运动和场等内容的基础，是 A‑Level 物理的基石。
2. Vectors and Components | 矢量与分量
Any vector quantity, such as initial velocity u, can be split into a horizontal component uₓ and a vertical component uy. When launched at an angle θ to the horizontal, the components are:
uₓ = u cosθ
uy = u sinθ
任何矢量，例如初速度 u，都可以分解为水平分量 uₓ 和垂直分量 uy。当以与水平方向成 θ 角发射时，分量为：
gnitude of the resultant velocity at any instant is then found by Pythagoras’ theorem: v = √vₓ² + vy², and its direction by tan φ = vy / vₓ.
这些公式假设角度是从水平方向测量。任意时刻的合速度大小可由勾股定理求得：v = √vₓ² + vy²，方向通过 tan φ = vy / vₓ 确定。
3. Independence of Perpendicular Motions | 垂直运动的独立性
A fundamental principle discovered by Galileo states that horizontal and vertical motions are independent of each other. The horizontal motion does not affect how fast an object falls, and the vertical motion does not alter its constant horizontal speed (ignoring air resistance).
伽利略发现的一个基本原理指出，水平和垂直运动是相互独立的。水平运动不影响物体下落的速度，而垂直运动也不改变其恒定的水平速率（忽略空气阻力）。
This means we can analyse an object’s flight by treating the constant‑velocity horizontal movement and the uniformly accelerated vertical movement as two separate, simultaneous happenings. The only shared quantity is time t.
这意味着我们可以将匀速的水平运动和匀加速的垂直运动视为两个独立且同时发生的过程来分析。唯一共享的物理量是时间 t。
4. Projectile Motion: Basics | 抛体运动基础
A projectile is any object that is given an initial velocity and then moves under the force of gravity alone. In Edexcel A‑Level Physics, air resistance is neglected unless stated otherwise, so the only acceleration is the free‑fall acceleration g directed vertically downwards.
抛体是指任何被赋予初速度后仅在重力作用下运动的物体。在 Edexcel A‑Level 物理中，除非另有说明，否则忽略空气阻力，因此唯一的加速度是垂直向下的自由落体加速度 g。
By choosing upward as positive, we set ay = –g (≈ – ng a constant horizontal velocity. These simple conditions lead to a parabolic trajectory.
取向上为正方向，则 ay = –g（约 –9.81 m s⁻²）。水平加速度 aₓ = 0，因此水平速度恒定。这些简单的条件导致轨迹为抛物线。
5. Horizontal Motion in Projectiles | 抛体运动的水平运动
: vₓ = uₓ = u cosθ. The horizontal displacement after time t is therefore straightforward:
x = u cosθ × t
由于 aₓ = 0，速度的水平分量永不改变：vₓ = uₓ = u cosθ。因此，经过时间 t 后的水平位移十分简洁：
This equation tells us that the horizontal range is entirely determined by the constant horizontal speed and the total time of flight. No horizontal force acts, so the projectile travels sideways at a steady pace as it rises and falls.
该公式表明，水平射程完全由恒定的水平速率和总飞行时间决定。因为没有水平力作用，抛体在上升和下落过程中以稳定的速度横向移动。
6. Vertical Motion under Gravity | 重力作用下的垂直运动
The vertical motion is governed by the uniform acceleration equations (SUVAT) with a = –g. If uy = u sinθ is the initial vertical component, the relevant equations are:
vy = u sinθ – g t
y = u sinθ t – ½ g t²
vy² = (u sinθ)² – 2 g y
垂直运动由匀加速方程（SUVAT）描述，其中 a = –g。若 uy = u sinθ 为初始垂直分量，则相关方程为：
At the peak of its flight, the vertical velocity momentarily becomes zero (vy ulate the time to reach maximum height and the maximum height itself. The symmetry of the upward and downward journeys holds only when launch and landing levels are equal.
在飞行最高点，垂直速度瞬时为零（vy = 0）。利用这一条件可计算到达最大高度的时间以及最大高度本身。仅当发射点与落地点高度相同时，上升与下降过程才具有对称性。
7. Trajectory and Parabolic Path | 轨迹与抛物线路径
Eliminating time t between x = u cosθ t and y = u sinθ t – ½ g t² yields the equation of the trajectory:
y = x tanθ – (g x²) / (2 u² cos²θ)
在 x = u cosθ t 与 y = u sinθ t – ½ g t² 之间消去时间 t，得到轨迹方程：
This is a quadratic relation between y and x, confirming the path is a parabola. The term (g / (2 u² cos²θ)) controls the curvature: a larger launch speed or a smaller g will make the trajectory flatter.
这是 y 与 x 之间的二次关系，证明路径是一条抛物线。项 (g / (2 u² cos²θ)) 控制着曲线的弯曲程度：更大的初速度或更小的 g 将使轨迹更平缓。
8. Time of Flight, Maximum Height and Range | 飞行时间、最大高度与射程
For a projectile launched and landing on the same horizontal level, three key quantities can be derived from the SUVAT equations:
- Time of flight: T = (2 u sinθ) / g
- Maximum height: H = (u² sin²θ) / (2 g)
- Horizontal range: R = (u² sin 2θ) / g
对于发射与落点在同一水平面上的抛体，可从 SUVAT 方程推导三个关键量：
- 飞行时间：T = (2 u sinθ) / g
- 最大高度：H = (u² sin²θ) / (2 g)
- 水平射程：R = (u² sin 2θ) / g
tary angles (e.g. 30° and 60°) give the same range but different flight times and heights.
射程公式揭示了一个关键结论：当初速度大小固定时，当 sin 2θ = 1 即 θ = 45° 时射程最大。互余的发射角（如 30° 和 60°）可获得相同射程，但飞行时间和最大高度不同。
| Angle θ | Time of flight (T) | Maximum height (H) | Range (R) |
|---|---|---|---|
| 30° | u/g | u²/(8g) | (√3 u²)/(2g) |
| 45° | (√2 u)/g | u²/(4g) | u²/g |
| 60° | (√3 u)/g | 3u²/(8g) | (√3 u²)/(2g) |
This table assumes u is the initial speed and g the gravitational field strength. The expressions confirm the symmetry between 30° and 60°.
该表假设 u 为初速度，g 为重力场强度。这些表达式证实了 30° 和 60° 之间的对称性。
9. Projectile Launched at an Angle | 斜抛运动
hed from the ground at an angle θ, both initial horizontal and vertical components are non‑ he projectile is launched horizontally (θ = 0) from a height, the analysis simplifies to uy = 0 and x = u t, y = –½ g t².
当抛体以角度 θ 从地面发射时，其初始水平和垂直分量均不为零。这是一般情况，涵盖了上述标准抛物线运动。如果抛体从某一高度水平发射（θ = 0），分析则简化为 uy = 0，x = u t，y = –½ g t²。
In many exam questions, objects are projected horizontally from cliffs or tables. The time of flight then depends only on the vertical fall, while the horizontal range is the product of the constant horizontal speed and that time.
在许多考试题中，物体从悬崖或桌面水平抛出。此时飞行时间仅取决于垂直下落距离，而水平射程等于恒定水平速率与该时间的乘积。
10. Effects of Air Resistance (Briefly) | 空气阻力的影响简述
In real life, air resistance cannot always be ignored. It acts opposite to the direction of motion and increases with speed. Consequently, the horizontal velocity decreases, the trajectory becomes asymmetric, and both the maximum height and range are reduced compared to ideal calculations.
在现实生活中，空气阻力往往不可忽略。其方向与运动方向相反，且随速度增大而增大。因此，水平速度会减小，轨迹变得不对称，与理想计算相比最大高度和射程都会减小。
Edexcel questions typically specify ‘neglecting air resistance’ or ask for a qualitative description of its effects. Recognising that the actual path is no longer a perfect parabola is often a required skill.
Edexcel 试题通常会说明“忽略空气阻力”，或要求定性地描述其影响。认识到实际路径不再是完美的抛物线常常是必备的技能。
11. Examples and Problem Solving | 实例与问题解决
maximum height is H = (20² × (sin30°)²)/(2×9.81) ≈ 5.1 m. The horizontal range is R = (20² × sin60°)/9.81 ≈ 35.3 m.
设想一个足球以 20 m s⁻¹ 的初速度沿 30° 角踢出。求其飞行时间：T = (2 × 20 × sin30°)/9.81 ≈ 2.04 s。最大高度为 H = (20² × (sin30°)²)/(2×9.81) ≈ 5.1 m。水平射程 R = (20² × sin60°)/9.81 ≈ 35.3 m。
When tackling problems, always split initial velocity into components, list known SUVAT variables for each direction, and choose the appropriate equation. Drawing diagrams and marking a sign convention (e.g. upward positive) prevents sign errors.
解题时，务必将初速度分解为分量，列出每个方向的已知 SUVAT 变量，并选择合适的方程。绘制示意图并标注正方向约定（如向上为正）可以避免符号错误。
12. Extension: Circular Motion as 2D Motion | 扩展：作为二维运动的圆周运动
Projectile motion is not the only important type of motion in a plane. In later units, students encounter circular motion, where an object moves along a circular path at constant speed. Although the speed is constant, the direction changes continuously, so the velocity vector is not constant – this implies a centripetal acceleration directed towards the centre.
抛体运动并非平面内唯一重要的运动类型。在后续单元中，学生会接触到圆周运动，物体以恒定速率沿圆轨道运动。尽管速率不变，但方向持续改变，因此速度矢量并非恒定——这意味着存在指向圆心的向心加速度。
The treatment of circular motion also relies on resolving vectors and understanding that perpendicular components can produce a resultant that turns the velocity without changing its magnitude. This reinforces the vector fundamentals covered in 2D translational motion.
圆周运动的分析同样依赖于矢量分解，并需要理解垂直分量如何产生一个只改变速度方向而不改变大小的合量。这进一步巩固了二维平移运动中所涉及的矢量基础。
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