📚 Displacement–time graphs | 位移-时间图
Displacement–time graphs are one of the most important tools for describing motion in kinematics. They show how the position of an object (measured from a defined origin) changes over time. Understanding these graphs is essential for solving problems in A-Level Physics and Mechanics, especially within the Edexcel specification.
位移-时间图是运动学中描述运动最重要的工具之一。它们展示物体位置（从定义的起点测量）如何随时间变化。理解这些图对于解决A-Level物理和力学问题至关重要，尤其是在Edexcel考试大纲中。
1. Introduction to Displacement–Time Graphs | 位移-时间图简介
A displacement–time graph plots displacement (usually denoted s or x) on the vertical axis against time (t) on the horizontal axis. The graph provides a visual representation of an object's motion, allowing us to determine velocity, direction of travel, and whether the object is stationary, moving uniformly, or accelerating.
位移-时间图将位移（通常用 s 或 x 表示）绘制在纵轴上，时间（t）绘制在横轴上。该图直观地表示了物体的运动，使我们能够确定速度、运动方向，以及物体是静止、匀速还是加速运动。
2. Axes and Variables | 坐标轴与变量
The vertical axis represents displacement from a reference point. Displacement is a vector quantity, which means it has both magnitude and direction. Positive displacement usually indicates movement to the right or upwards relative to the origin, while negative displacement indicates movement to the left or downwards. The horizontal axis represents time, always increasing to the right.
纵轴表示相对于参考点的位移。位移是一个矢量，既有大小又有方向。正位移通常表示相对于起点向右或向上运动，负位移表示向左或向下运动。横轴代表时间，始终向右递增。
3. Slope Represents Velocity | 斜率代表速度
The gradient (slope) of a displacement–time graph equals the velocity of the object. If the graph is a straight line, the velocity is constant. The steeper the line, the greater the speed. Mathematically, velocity is the rate of change of displacement with respect to time:
v = Δs / Δt
位移-时间图的斜率（梯度）等于物体的速度。如果图是一条直线，则速度恒定。直线越陡，速率越大。数学上，速度是位移随时间的变化率：
v = Δs / Δt
4. Constant Positive Velocity | 正匀速运动
When the graph is a straight line with a positive slope, the object moves with constant positive velocity. The displacement increases uniformly with time. This implies no acceleration and a steady movement away from the origin in the positive direction.
当图形是一条斜率为正的直线时，物体以恒定的正速度运动。位移随时间均匀增加。这意味着没有加速度，物体沿正方向稳定远离起点。
5. Zero Velocity (Stationary) | 零速度（静止）
A horizontal line on a displacement–time graph indicates that displacement does not change over time. The slope is zero, so velocity is zero. The object is stationary at a constant displacement, which could be positive, negative, or zero.
位移-时间图上的水平线表明位移不随时间变化。斜率为零，因此速度为零。物体静止在某一恒定位移处，该位移可以是正、负或零。
6. Constant Negative Velocity | 负匀速运动
A straight line with a negative slope indicates constant negative velocity. The displacement decreases at a steady rate as time progresses, meaning the object is moving back towards the origin or in the negative direction.
斜率为负的直线表示恒定的负速度。位移随时间稳定减少，意味着物体正向原点返回或沿负方向运动。
7. Curved Graphs – Changing Velocity | 曲线图——速度变化
If the displacement–time graph is curved, the slope is not constant; hence the velocity is changing. This indicates acceleration. A curve that becomes steeper over time represents increasing velocity (speeding up), while a curve that flattens represents decreasing velocity (slowing down).
如果位移-时间图是曲线，斜率不恒定，因此速度在变化，这表明存在加速度。随时间变陡的曲线表示速度增加（加速），而趋于平缓的曲线表示速度减小（减速）。
8. Instantaneous Velocity from Tangent | 从切线求瞬时速度
For a curved displacement–time graph, the instantaneous velocity at a particular time is found by drawing a tangent to the curve at that point and calculating its gradient. The tangent's slope gives the velocity at that instant, which can be approximated by Δs/Δt using two very close points on the tangent.
对于曲线位移-时间图，某一时刻的瞬时速度通过在该点处画切线并计算其斜率来求得。切线的斜率给出该瞬间的速度，可用切线上两个非常靠近的点通过 Δs/Δt 近似计算。
9. Turning Points and Maximum Displacement | 转折点与最大位移
A turning point occurs where the graph changes direction — from increasing to decreasing displacement or vice versa. At the exact maximum or minimum, the tangent is horizontal, meaning the instantaneous velocity is zero. This represents the moment the object momentarily stops before reversing direction.
转折点出现在图形改变方向的地方——位移从增加转为减小，或反之。在精确的最大值或最小值处，切线是水平的，意味着瞬时速度为零。这表示物体在反向运动前瞬间静止。
10. Common Mistakes to Avoid | 常犯错误
Students often misinterpret displacement–time graphs. Below is a table summarising key mistakes and the correct interpretations.
学生常误解位移-时间图。下表总结了主要错误与正确解释。
| Common Mistake | Correct Interpretation |
|---|---|
| Confusing displacement–time with distance–time | Displacement includes direction; distance is scalar. |
| Assuming a steep line always means high speed | Steepness indicates speed but also sign of velocity matters; steep negative slope means high speed in negative direction. |
| Finding area under displacement–time graph | Area under a displacement–time graph has no physical meaning; area under velocity–time graph gives displacement. |
| Using slope of a curve without tangents | Always draw tangent for instantaneous velocity on a curve; do not just use chord slope unless finding average velocity. |
中文对照：
| 常见错误 | 正确理解 |
|---|---|
| 混淆位移-时间图与路程-时间图 | 位移包含方向；路程是标量。 |
| 认为陡峭的线总是高速 | 陡峭程度表示速率，但速度的符号也很重要；陡峭的负斜率表示负方向高速。 |
| 求位移-时间图下的面积 | 位移-时间图下面积没有物理意义；速度-时间图下面积才是位移。 |
| 曲线上不使用切线就直接用斜率 | 要求曲线上的瞬时速度必须画切线；除非求平均速度，否则不要直接用割线斜率。 |
11. Worked Example | 例题解析
Example: A displacement– o t=8s, displacement stays at 20m; from t=8s to t=12s, displacement decreases uniformly from 20m to 0. Find the velocities in each stage.
例题：一个做直线运动的粒子的位移-时间图显示如下：t=0到5 s，位移从0均匀增加到20 m；5 s到8 s，位移保持在20 m；8 s到12 s，位移从20 m均匀减少到0。求各阶段的速度。
Solution: Stage 1 (0–5 s): slope = (20-0)/(5-0) = 4 m/s, positive constant velocity. Stage 2 (5–8 s): slope = 0, velocity = 0 (stationary). Stage 3 (8–12 s): slope = (0-20)/(12-8) = -20/4 = -5 m/s, constant negative velocity (moving back).
解答：阶段1（0-5 s）：斜率 = (20-0)/(5-0) = 4 m/s，正匀速。阶段2（5-8 s）：斜率 = 0，速度 = 0（静止）。阶段3（8-12 s）：斜率 = (0-20)/(12-8) = -20/4 = -5 m/s，负匀速（返回）。
This example illustrates how different sections of the graph correspond to different types of motion, and how the sign of velocity indicates direction.
这个例题展示了图的不同段落如何对应不同类型的运动，以及速度的符号如何表示方向。
12. Summary | 总结
Displacement– = constant velocity, curved line = changing velocity (acceleration). Tangent gives instantaneous velocity. Area under the graph is not meaningful in kinematics. Always pay attention to the sign of displacement and slope to determine direction.
位移-时间图提供了一种清晰的分析运动的方法。需牢记的要点：斜率 = 速度，水平线 = 静止，倾斜直线 = 匀速，曲线 = 速度变化（加速）。切线给出瞬时速度。图下面积在运动学中没有意义。务必注意位移和斜率的符号以确定方向。
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