📚 IB WJEC Physics: Kinematics Essentials | IB WJEC 物理：运动学考点精讲
Kinematics forms the bedrock of classical mechanics, describing motion without reference to its causes. In both IB and WJEC syllabi, a firm grasp of vectors, uniformly accelerated motion, and graphical analysis is essential. This article systematically covers every key topic, from basic definitions to projectile motion, equipping you with the tools needed for exam success.
运动学是经典力学的基石，它描述物体的运动而不追究其成因。在 IB 和 WJEC 的考试大纲中，牢固掌握矢量、匀加速运动及图像分析至关重要。本文系统梳理了从基本定义到抛体运动的每个考点，助你掌握考试所需的全部工具。
1. Scalars and Vectors | 标量与矢量
A scalar is a physical quantity that only possesses magnitude (size). Distance, speed, mass, and time are typical scalars. A vector, on the other hand, has both magnitude and direction; displacement, velocity, acceleration, and force are vectors. Understanding this distinction is the first step in kinematics, as vector addition requires special rules, such as the tip-to-tail method or component resolution.
标量是只具有大小（量值）的物理量。路程、速率、质量和时间都是典型的标量。而矢量既有大小又有方向；位移、速度、加速度和力都是矢量。理解这一区别是学习运动学的第一步，因为矢量相加需要遵循特殊规则，例如三角形定则或分解法。
ed triangles, the cosine rule a² = b² + c² − 2bc cos θ and the sine rule are often needed.
在 IB 和 WJEC 考试中，你可能会被要求用比例图或直角三角学进行位移矢量的相加。对于非直角三角形，经常需要用到余弦定理 a² = b² + c² − 2bc cos θ 和正弦定理。
2. Displacement, Velocity and Acceleration | 位移、速度与加速度
Displacement (s) is the change in position of an object—it is a vector pointing from start to finish, with SI unit metre (m). Velocity (v) is the rate of change of displacement; thus, it too is a vector. Speed, by contrast, is the scalar rate of change of distance. Acceleration (a) is defined as the rate of change of velocity, meaning any change in either magnitude or direction constitutes acceleration.
位移（s）是物体位置的变化量——它是一个从起点指向终点的矢量，SI 单位为米（m）。速度（v）是位移的变化率；因此它也是一个矢量。相反，速率是路程的标量变化率。加速度（a）的定义是速度的变化率，这意味着速度的大小或方向发生任何变化都会产生加速度。
is given by vₐᵥ = Δs / Δt, while instantaneous velocity is the gradient of a displacement– In constant acceleration contexts, these definitions lead directly to the SUVAT equations.
平均速度由 vₐᵥ = Δs / Δt 给出，而瞬时速度是位移–时间图像的斜率。同样，平均加速度为 aₐᵥ = Δv / Δt。在加速度恒定的情况下，这些定义直接导出匀加速运动方程。
3. Uniformly Accelerated Motion – SUVAT Equations | 匀加速运动方程
When acceleration is constant, five key variables—displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t)—are linked by four equations. In the IB data booklet and WJEC formulae sheet they appear as:
当加速度恒定时，五个关键变量——位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）——由四个方程联系。在 IB 公式手册和 WJEC 公式表中，它们表述为：
v = u + at
s = ut + ½ at²
v² = u² + 2as
s = ½ (u + v) t
In solving problems, always list the known quantities and the variable you need, then select the equation that includes only one unknown. Pay attention to direction: assign a positive sign to one direction (usually the initial motion) and treat opposite vectors as negative.
解题时，先列出已知量和待求变量，然后选择一个只含一个未知量的方程。注意方向设定：给一个方向指定正号（通常是初始运动方向），并将相反方向的矢量视为负值。
⁻¹ to 30 m s⁻¹ over 80 m. Use v² = u² + 2as to find a: 30² = 10² + 2a(80) → a = 5.0 m s⁻².
例如，一辆汽车从 10 m s⁻¹ 加速到 30 m s⁻¹，行驶了 80 m。使用 v² = u² + 2as 求 a：30² = 10² + 2a(80) → a = 5.0 m s⁻²。
4. Displacement–Time Graphs | 位移–时间图像
A displacement–time (s–t) graph plots displacement on the vertical axis against time on the horizontal axis. The gradient at any point equals the instantaneous velocity. A straight line indicates constant velocity, while a curved line indicates changing velocity (acceleration). A horizontal line means the object is at rest.
位移–时间（s–t）图将位移置于纵轴，时间置于横轴。任意一点的斜率等于瞬时速度。一条直线表明速度恒定，而曲线则表明速度在变化（加速度）。水平线表示物体静止。
It is vital not to confuse the shape of the graph with the actual path of the object. A downward-sloping straight line, for instance, does not mean the object is moving downwards; it means the object is moving in the negative direction with constant speed, assuming displacement is defined along a straight line.
至关重要的一点是，不要将图像的形状与物体的实际路径混淆。例如，一条向下倾斜的直线并不意味着物体正在向下运动；它表示物体正以恒定速率沿负方向运动，前提是位移是沿直线定义的。
5. Velocity–Time Graphs | 速度–时间图像
The velocity–time (v–t) graph is arguably the most powerful kinematic graph. Its gradient gives the instantaneous acceleration, and the area under the curve (between the line and the time axis) gives the displacement. Areas above the time axis correspond to positive displacement, while areas below represent negative displacement.
速度–时间（v–t）图可以说是最强大的运动学图像。它的斜率给出瞬时加速度，而曲线下的面积（曲线与时间轴之间）给出位移。时间轴上方的面积对应正位移，下方的面积则代表负位移。
In IB and WJEC questions, you may be asked to sketch a v–t graph from a description of motion or to calculate displacement by counting squares or by dividing the area into simple geometric shapes (rectangles, triangles, trapezoids). Remember that a straight sloping line in a v–t graph indicates constant acceleration.
在 IB 和 WJEC 的试题中，你可能会被要求根据运动描述绘制 v–t 图，或者通过数格或划分简单几何图形（矩形、三角形、梯形）来计算位移。请记住，v–t 图中的一条倾斜直线表示加速度恒定。
6. Acceleration–Time Graphs | 加速度–时间图像
Acceleration–time (a–t) graphs are less common but still appear. A horizontal line indicates constant acceleration. The area under an a–t graph gives the change in velocity (Δv). If the graph line lies above the axis, the velocity increases; if below, it decreases. A zero line means uniform velocity.
加速度–时间（a–t）图不常见，但仍会出现。一条水平线表示加速度恒定。a–t 图下的面积给出速度的变化量（Δv）。若图线在轴上方，速度增加；在轴下方，速度减小。零线表示匀速运动。
When given a piecewise a– s (jerk = rate of change of acceleration), although jerk is not part of the core syllabus.
当给出分段 a–t 图时，可以通过将初始速度与总面积相加来求末速度。这在涉及加加速度（急动度）的问题中尤为有用，不过加加速度不在核心考纲范围内。
7. Free Fall Motion | 自由落体运动
otion under the influence of gravity alone, with negligible air resistance. Near the Earth' apply the SUVAT equations by setting a = g (or a = −g depending on your sign convention).
自由落体是仅在重力作用下、空气阻力可忽略不计的运动。在地球表面附近，所有物体都受到一个恒定的、向下的重力加速度 g ≈ 9.81 m s⁻²。在运动学问题中，可以根据你设定的正方向，设 a = g（或 a = −g），然后应用匀加速运动方程。
nwards. The time to reach maximum height is t = u/g, and the total time of flight (returning to the same level) is 2u/g. Symmetry is a powerful problem-solving tool here.
一个经典的考题涉及竖直上抛的物体。在最高点，其速度为零，但加速度仍为向下的 g。到达最高点所需的时间为 t = u/g，而飞行的总时间（落回同一水平面）为 2u/g。此处的对称性是极有力的解题工具。
8. Projectile Motion | 抛体运动
tal (uₓ = u cos θ) and vertical (uᵧ = u sin θ) components. The horizontal motion has constant velocity (aₓ = 0), while the vertical motion experiences constant acceleration (aᵧ = −g, taking upward as positive).
抛体运动是在恒定重力加速度下的二维运动。关键在于将初速度分解为水平分量（uₓ = u cos θ）和竖直分量（uᵧ = u sin θ）。水平方向上为匀速运动（aₓ = 0），而竖直方向上则具有恒定的加速度（aᵧ = −g，取向上为正）。
en R = uₓ t = (u² sin 2θ)/g, showing that the maximum range is achieved at θ = 45° in the absence of air resistance.
飞行的总时间完全由竖直运动决定：对于落回同一高度的抛体，t = (2u sin θ)/g。水平射程则为 R = uₓ t = (u² sin 2θ)/g，这表明在没有空气阻力的情况下，当 θ = 45° 时可获得最大射程。
9. Relative Motion | 相对运动
f reference. The relative velocity of object A with respect to B is given by vₐ_b = vₐ − vb . For example, if a car moves east at 20 m s⁻¹ and a truck moves east at 15 m s⁻¹, the car's velocity relative to the truck is 5 m s⁻¹ east. If they move in opposite directions, the magnitude of relative velocity is the sum of their speeds.
运动总是相对于参考系来测量的。物体 A 相对于 B 的相对速度为 vₐ_b = vₐ − vb 。例如，若一辆车以 20 m s⁻¹ 向东行驶，一辆卡车以 15 m s⁻¹ 向东行驶，则车相对于卡车的速度为 5 m s⁻¹ 向东。若它们反向行驶，相对速度的大小则为两者速率之和。
In river-crossing problems, a boat's velocity relative to the water combines with the water's velocity relative to the ground. The resultant velocity determines the actual path. Vector addition through a scale diagram or component resolution yields the drift and the time to cross.
在渡河问题中，船相对于水的速度与水流相对于地面的速度进行合成。合成的速度决定了实际路径。通过比例图或分量分解法进行矢量加法，可以求出漂移距离和渡河时间。
10. Common Pitfalls and Exam Tips | 常见错误与考试技巧
A frequent mistake is confusing distance and displacement. Distance is the total path length (scalar), while displacement is the straight-line separation from start to finish (vector). Similarly, average speed is total distance divided by time, whereas average velocity is displacement divided by time—these quantities are often different when direction changes.
一个常见错误是混淆路程和位移。路程是总路径长度（标量），而位移是从起点到终点的直线距离（矢量）。同样，平均速率是总路程除以时间，而平均速度是位移除以时间——当方向变化时，这两个量通常不同。
Always check your sign convention before substituting into SUVAT equations. Draw a clear vector diagram for relative motion and projectile problems. When reading graphs, carefully note the axes labels—many errors arise from misidentifying a v–t graph as an s–t graph. Finally, practise multi-step problems where you analyse motion in phases (e.g., acceleration, constant velocity, deceleration) because they are common in both IB and WJEC papers.
在代入匀加速运动方程前，务必检查你的正方向规定。对于相对运动和抛体问题，要画出清晰的矢量图。阅读图像时，仔细留意坐标轴标签——许多错误都源于将 v–t 图误认为 s–t 图。最后，要练习多阶段运动分析题（如加速、匀速、减速），因为这类题目在 IB 和 WJEC 试卷中都很常见。
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