📚 Graphs describing motion | 描述运动的图像
In IB Physics, the ability to interpret and sketch motion graphs is fundamental to understanding kinematics. Displacement–time, velocity–time and acceleration–time graphs provide powerful visual tools for analysing how objects move, allowing you to extract instantaneous and average quantities, recognise types of motion, and solve problems without necessarily using SUVAT equations. This article systematically covers the skills required, common pitfalls, and examination tips to help you master graphs describing motion.
在 IB 物理中，解读和绘制运动图像是理解运动学的基础。位移–时间图、速度–时间图和加速度–时间图为分析物体运动提供了有力的可视化工具，使你可以提取瞬时量和平均量、识别运动类型，并在不依赖 SUVAT 方程的情况下解决问题。本文系统地介绍了所需技能、常见错误与考试技巧，帮助你掌握描述运动的图像。
1. Displacement–Time Graphs (s–t Graphs) | 位移–时间图
A displacement–time graph plots displacement s on the vertical axis against time t on the horizontal axis. Each point on the curve tells you where the object is (relative to the origin) at a given instant.
位移–时间图将位移 s 放在纵轴、时间 t 放在横轴。图线上每一点都给出了物体在某一时刻相对于原点的位置。
On an s–t graph, the slope of the line (gradient) equals the object's velocity. A straight, sloping line indicates uniform velocity: the steeper the line, the greater the speed. A horizontal line means the object is stationary — its displacement is not changing.
在 s–t 图中，图线的斜率（梯度）等于物体的速度。一条笔直的斜线表示匀速运动：线越陡，速率越大。水平线则意味着物体静止，位移不发生变化。
A curved line signals changing velocity. If the curve becomes steeper with time, the object is speeding up; if it flattens out, the object is slowing down. To find the instantaneous velocity at a point on a curve, draw a tangent and calculate its slope.
曲线意味着速度在变化。如果曲线随时间变得越来越陡，说明物体在加速；如果曲线趋于平缓，则物体在减速。要找出曲线上某一点的瞬时速度，可作该点的切线并计算其斜率。
v = Δs / Δt (for uniform sections) | vinst = slope of tangent
v = Δs / Δt （匀速段） | v瞬时 = 切线斜率
2. Velocity–Time Graphs (v–t Graphs) | 速度–时间图
A velocity–time graph has velocity v on the vertical axis and time t on the horizontal axis. This graph gives a direct picture of how an object's velocity changes, and it is arguably the most versatile of the three motion graphs.
速度–时间图以速度 v 为纵轴、时间 t 为横轴。它直观地给出了物体速度随时间的变化情况，可以说是三种运动图中用途最广的一种。
The slope of a v–t graph gives the acceleration. A straight, sloping line represents constant acceleration; a horizontal line indicates constant velocity (zero acceleration). A curved line means the acceleration is changing with time.
v–t 图的斜率代表加速度。一条笔直的斜线表示匀加速度；水平线表示匀速运动（加速度为零）；曲线表示加速度随时间变化。
Critically, the area between the graph line and the time axis (taking sign into account) represents the change in displacement (Δs). An area above the t-axis contributes a positive displacement, while an area below gives a negative displacement.
至关重要的是，图线与时间轴之间的面积（需考虑正负号）代表位移的变化量 Δs。位于 t 轴上方的面积贡献正向位移，下方的面积贡献负向位移。
a = Δv / Δt | Δs = area under v–t graph
a = Δv / Δt | Δs = v–t 图下面积
3. Acceleration–Time Graphs (a–t Graphs) | 加速度–时间图
An acceleration–time graph plots acceleration a against time t. Although less common in simple problems, it is essential for understanding how the net force (and hence acceleration) varies, and it links directly to the area under the curve.
加速度–时间图将加速度 a 作为纵轴、时间 t 作为横轴。尽管在简单问题中较少出现，但它在理解合力（以及加速度）如何变化方面至关重要，并且与曲线下面积直接相关。
The slope of an a–t graph has no standard name in IB Physics, but it represents the rate of change of acceleration (sometimes called “jerk”). You will not be asked to calculate it, but you should be aware that it exists.
a–t 图的斜率在 IB 物理中没有标准名称，它表示加速度的变化率（有时称为“加加速度”）。虽然不要求计算，但你应知道它的存在。
The key feature is the area under the a– nstant positive acceleration over an interval Δt gives a rectangular area, so Δv = a × Δt.
关键特征是 a–t 图下的面积：它等于速度的变化量 Δv。例如，在时间间隔 Δt 内加速度恒为正值，图线为一条水平线，形成的矩形面积即为 Δv = a × Δt。
When an a–t graph crosses the time axis, the velocity can still be positive; the sign of the area tells you whether the velocity is increasing or decreasing, but the object's direction depends on the sign of v, not a.
当 a–t 图穿过时间轴时，速度仍可能为正；面积的符号表示速度是在增加还是减少，但物体的运动方向取决于 v 的符号，而非 a 的符号。
4. Interpreting Slope: Velocity and Acceleration | 斜率解读：速度与加速度
Gradients are the central concept linking motion graphs. On an s–t graph, the gradient at any point or over any interval tells you the velocity. On a v–t graph, the gradient gives the acceleration.
斜率是连接运动图像的核心概念。在 s–t 图上，任意一点或某段时间内的梯度都给出速度；在 v–t 图上，梯度则给出加速度。
For a straight line on an s–t graph, the gradient is constant, and you can simply use v = (s₂ − s₁) / (t₂ − t₁). For a curve, you must draw a tangent at the point of interest, form a large right‑angled triangle, and calculate the ratio of the vertical change (Δs) to the horizontal change (Δt).
对于 s–t 图上的直线，梯度恒定，可以直接使用 v = (s₂ − s₁) / (t₂ − t₁)。对于曲线，则须在目标点画切线，构建一个足够大的直角三角形，计算纵向变化量 Δs 与横向变化量 Δt 的比值。
Similarly, on a v–t graph, a constant slope means uniform acceleration: a = (v − u) / t, which is the defining equation of constant acceleration. When the v–t line is curved, the instantaneous acceleration is found from the slope of the tangent.
类似地，在 v–t 图上，恒定的斜率代表匀加速度：a = (v − u) / t，这正是匀加速运动的定义式。当 v–t 图线为曲线时，瞬时加速度需通过切线斜率求得。
gradient of s–t = velocity | gradient of v–t = acceleration
s–t 斜率 = 速度 | v–t 斜率 = 加速度
5. Area Under Graphs: Displacement from v–t | 图下方面积：从速度–时间图求位移
The area enclosed between the v–t curve and the time axis represents the change in displacement. This principle is a direct consequence of the definition of velocity as the rate of change of displacement.
v–t 图与时间轴之间所包围的面积代表位移的变化量。这一原理直接源于速度的定义——即位移的变化率。
When the v–t graph consists of straight lines and simple shapes, you can calculate the area using geometry: rectangles for constant velocity, triangles and trapeziums for uniformly accelerated motion. If the graph lies partly below the axis, treat that area as negative.
当 v–t 图由直线和简单几何形状构成时，可用几何方法计算面积：匀速对应矩形，匀加速对应三角形或梯形。若图线部分位于时间轴下方，则该部分面积取负值。
For irregular v–t curves, you may be asked to count squares on a grid, estimate the area of one square, and multiply by the number of squares (counting those under the baseline as negative). IB exam papers often provide a grid for this purpose.
对于不规则的 v–t 曲线，考题可能要求你在坐标格上数格：先估算一个格子的面积大小，再乘以格子数（轴下方的格子计为负值）。IB 试卷常为此提供网格背景。
6. Uniform Motion and Constant Acceleration | 匀速直线运动和匀变速直线运动
Uniform motion (constant velocity) produces the simplest graphs: a straight, sloping line on an s–t graph; a horizontal straight line on a v–t graph; and a zero line (along the t‑axis) on an a–t graph.
匀速直线运动产生最简单的图像：s–t 图是一条倾斜的直线；v–t 图是一条水平直线；a–t 图是一条与时间轴重合的零线。
For constant acceleration, the v–t graph is a straight sloping line, the s–t graph is a parabola (curved), and the a– at², and v² = u² + 2as describe the exact mathematical relationships behind these graphs.
对于匀加速运动，v–t 图是一条倾斜的直线，s–t 图是一条抛物线（曲线），a–t 图则是一条位于恒定加速度值的水平线。SUVAT 方程 v = u + at, s = ut + ½ at² 和 v² = u² + 2as 描述了这些图像背后的精确数学关系。
When solving problems, it is often quicker to sketch a v–t graph and extract information from its gradient and area than to plug numbers into SUVAT formulas. Always label known values such as initial velocity u, final velocity v, and time intervals on your sketch.
在解题时，通常会比代入 SUVAT 公式更快地画出一幅 v–t 草图，通过斜率和面积提取所需信息。做草图时，务必标注已知量如初速度 u、末速度 v 和时间间隔。
7. Non‑uniform Motion and Curved Graphs | 非匀变速运动与曲线图
Real objects often experience changing acceleration, such as a car gradually pressing the accelerator or decelerating due to air resistance. In such cases, the v–t graph becomes a smooth curve, and the a–t graph is no longer horizontal.
真实物体常经历变化的加速度，例如汽车逐渐踩下油门或因空气阻力减速。此时，v–t 图成为平滑的曲线，a–t 图也不再是水平线。
To analyse a curved v–t graph, you may need to find the instantaneous acceleration at several points using tangents. The changing area under the curve still gives the displacement, but you may need integration or a counting‑squares method.
分析弯曲的 v–t 图时，可能需要在多个点作切线求瞬时加速度。曲线下的面积仍然代表位移，但计算时可能需要积分或数格子的方法。
On an s–t graph, non‑uniform motion produces a curve that changes its steepness in a non‑linear way. The direction of curvature (concave up or down) indicates whether the acceleration is positive or negative relative to the direction of motion.
在 s–t 图上，非匀变速运动产生一条非线性变化的曲线。曲线的弯曲方向（上凹或下凹）表明加速度相对于运动方向是正还是负。
8. Motion under Gravity and Symmetrical Graphs | 重力作用下的运动与对称图像
When an object is thrown vertically upwards and returns to the ground, its motion under constant gravitational acceleration produces characteristic graphs. Taking the upward direction as positive, the velocity–time graph is a straight line sloping downwards, crossing the time axis at the highest point.
当物体竖直上抛并落回地面时，它在恒定重力加速度作用下的运动会产生特征图像。若以向上为正方向，速度–时间图是一条向下倾斜的直线，并在最高点处穿过时间轴。
The displacement–time graph is a parabola that opens downwards, with its vertex at the maximum height. The upward and downward halves of the journey are symmetric in time if air resistance is negligible, so the time to rise equals the time to fall.
位移–时间图是一条开口向下的抛物线，其顶点对应最大高度。在忽略空气阻力的情况下，上升和下降段在时间上是对称的，因此上升时间等于下落时间。
The acceleration– = 0. Students often mistakenly think a = 0 at the peak; the graph clearly shows it is not.
加速度–时间图在整个飞行过程中都是一条位于 a = −g (g ≈ 9.8 m s⁻²) 的水平线，即使在最高点 v = 0 时也是如此。学生常误以为最高点加速度为零，而图像清楚地表明并非如此。
9. Key Skills: Sketching and Transforming Graphs | 关键技能：草图绘制与图像转换
IB exam questions frequently ask you to sketch a v–t graph from a given s–t graph, or an a–t graph from a v–t description. You must understand the “derivative” relationship: v is the slope of s, and a is the slope of v.
IB 考试常要求根据给定的 s–t 图画出 v–t 图，或根据 v–t 描述画出 a–t 图。你必须理解其中的“导数”关系：v 是 s 的斜率，a 是 v 的斜率。
When sketching, start by marking any known times where the motion changes (e.g., when the slope of s–t is zero, v–t must cross the axis). Then consider the sign and trend of the slope: increasing positive slope means increasing positive velocity, shown as a rising line on the v–t graph.
绘制草图时，先标出运动性质发生变化的已知时刻（例如，s–t 图斜率为零时，v–t 图必穿过时间轴）。然后考虑斜率的符号和变化趋势：斜率正且在增大，意味着速度为正且增大，在 v–t 图上表现为上升的线。
The table below summarises the link between the three graphs in terms of gradient and area, which is invaluable for graph‑transformation tasks.
下表从梯度和面积的角度总结了三种图之间的联系，对图像转换任务极有帮助。
| Graph / 图 | Slope gives / 斜率得出 | Area gives / 面积得出 |
|---|---|---|
| s–t | Velocity (v) / 速度 | — |
| v–t | Acceleration (a) / 加速度 | Displacement change (Δs) / 位移变化 |
| a–t | — | Velocity change (Δv) / 速度变化 |
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students confuse the slope of an s–t graph with the area under a v–t graph. Keep the principle clear: slope on s–t is velocity; area under v–t is displacement — they describe the same physical quantity but come from different operations.
许多学生混淆了 s–t 图的斜率与 v–t 图下的面积。务必清晰区分：s–t 图的斜率是速度；v–t 图下的面积是位移——两者描述的是同一个物理量，但来自不同的操作。
A frequent error is neglecting the sign of area in a v–t graph when the graph crosses the time axis. Always split the area into above‑axis and below‑axis parts, assign signs, and add them algebraically to find the net displacement. The total distance, however, is the sum of the absolute values of the areas.
一个常见错误是当 v–t 图穿过时间轴时忽略了面积的符号。务必将面积分为轴上方和轴下方两部分，赋予正负号，再代数相加求净位移。而总路程则是各面积绝对值之和。
In sketching problems, students often draw a v–t graph that implies an instantaneous jump in velocity, which would require infinite acceleration. In IB Physics, unless a collision is specified, velocity changes must be continuous; your graph should not have vertical segments except in idealised impulse scenarios.
在绘图题中，学生常常画出含有速度突变的 v–t 图，而这意味着无限大的加速度。在 IB 物理中，除非明确涉及碰撞，否则速度变化必须是连续的；除理想化的冲量情形外，你画的图不应出现竖直段。
A final tip: always label axes with quantities and units, and make the scale clear. In data‑based questions, use a full‑size triangle when finding a gradient from a graph — a triangle that covers more than half the drawn line to minimise uncertainty.
最后一条建议：始终在坐标轴上标注物理量和单位，并清晰标明刻度。在处理数据题时，求图线斜率应使用覆盖大半条线的直角三角形，以减小不确定度。
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