Displacement-Time Graph Analysis | 位移-时间图像解析

📚 Displacement-Time Graph Analysis | 位移-时间图像解析

Displacement-time graphs are among the most fundamental tools in A-Level Mechanics. They allow us to visualise the motion of an object along a straight line and extract key kinematic quantities such as velocity and average velocity directly from the graph.

位移-时间图像是 A-Level 力学中最基础的工具之一。它使我们能够直观地描述物体沿直线的运动，并直接从图像中提取速度、平均速度等关键运动学量。

1. The Axes and Basic Setup | 坐标轴与基本设置

On a displacement-time graph, the vertical axis (y-axis) represents displacement s, usually measured in metres (m). The horizontal axis (x-axis) represents time t, usually measured in seconds (s). Each point on the graph has coordinates (t, s), meaning at time t the object has displacement s.

在位移-时间图像中，纵轴（y 轴）表示位移 s，通常以米（m）为单位；横轴（x 轴）表示时间 t，通常以秒（s）为单位。图像上的每个点都有坐标 (t, s)，表示在时刻 t 物体的位移为 s。

It is essential to remember that displacement is a vector quantity: it has both magnitude and direction. On a straight line, we usually assign one direction as positive and the opposite as negative. Displacement measures the position of the object relative to its starting point, not the total distance travelled.

务必牢记：位移是矢量，具有大小和方向。在直线上，我们通常规定某一方向为正，相反方向为负。位移衡量的是物体相对于出发点的位置，而非实际走过的总路程。

2. Reading Key Points | 关键点的读取

Reading a displacement-time graph requires careful attention to the coordinates of points. The y-coordinate at any time t gives the displacement from the origin at that instant. Where the graph crosses the time axis (s = 0), the object is passing through its starting point.

读取位移-时间图像时需注意点的坐标。任何时刻 t 对应的 y 坐标即为该时刻物体相对于原点的位移。当图像与时间轴相交（即 s = 0）时，表示物体正经过出发点。

The starting point of the motion is the y-intercept, where t = 0. If the graph begins at the origin, the object started from the reference position. If the y-intercept is positive, the object began on the positive side of the reference point.

运动的起点是 t = 0 时的 y 截距。若图像从原点开始，说明物体从参考位置出发；若 y 截距为正，说明物体从参考点的正方向一侧出发。

3. Gradient as Velocity | 斜率即速度

The single most important feature of a displacement-time graph is its gradient. Since velocity is defined as the rate of change of displacement with respect to time, the gradient of the s-t graph gives the velocity.

位移-时间图像最重要的特征是其斜率。由于速度定义为位移随时间的变化率，因此 s-t 图像的斜率即为速度。

velocity = gradient = Δs / Δt = (s₂ − s₁) / (t₂ − t₁)

For a straight-line section, this gives the constant velocity during that time interval. A steeper line indicates a greater speed, while a shallower line indicates a slower speed.

对于直线段部分，该公式给出该时间段内的恒定速度。直线越陡表示速度越大，直线越缓表示速度越小。

4. Stationary and Uniform Motion | 静止与匀速运动

If the graph is a horizontal straight line, the gradient is zero, and therefore the velocity is zero: the object is stationary. The displacement remains constant, meaning the object is not changing its position.

若图像为水平直线，则斜率为零，速度为零：物体处于静止状态。位移保持不变，说明物体的位置没有变化。

If the graph is a straight line with a non-zero constant gradient, the object is moving with uniform (constant) velocity. The displacement changes by an equal amount in every equal time interval.

若图像是斜率非零的直线，则物体做匀速运动。在每个相等的时间间隔内，位移的变化量相等。

5. Curved Graphs and Non-Uniform Motion | 曲线与非匀速运动

In many real situations, velocity is not constant. A curved displacement-time graph indicates that the velocity is changing — the object is accelerating or decelerating. The gradient of the curve at any point is found by drawing a tangent at that point.

在许多实际情况中，速度并非恒定。弯曲的位移-时间图像表明速度在变化——物体正在加速或减速。曲线上任意一点的斜率需要通过在该点作切线来求得。

If the slope becomes steeper as time increases, the object is speeding up (accelerating). If the slope becomes less steep, the object is slowing down (decelerating).

若斜率随时间增大而变得更陡，说明物体在加速；若斜率变得平缓，说明物体在减速。

6. Positive and Negative Gradients | 正斜率与负斜率

A positive gradient means the object is moving in the positive direction (the direction we defined as positive). A negative gradient means the object is moving in the opposite (negative) direction.

斜率为正意味着物体沿正方向运动；斜率为负意味着物体沿相反（负）方向运动。

When a graph crosses the time axis, the displacement momentarily becomes zero as the object passes through the reference point, but this does not mean the object has stopped — the velocity at that instant is still the gradient of the graph at that point.

当图像穿越时间轴时，位移瞬时变为零，说明物体经过参考点，但这并不意味着物体已停止——该瞬时的速度仍是图像在该点的斜率。

7. Distance vs Displacement | 路程与位移的区别

One of the most common mistakes is confusing displacement with distance. On a displacement-time graph, the total distance travelled is the sum of the absolute changes in s over every interval. The net displacement is simply the final y-coordinate minus the initial y-coordinate.

最常见的错误之一是混淆位移与路程。在位移-时间图像上，总路程等于每一时间间隔内 |Δs| 的累加；而净位移则等于终点 y 坐标减去起点 y 坐标。

For example, if an object moves 5 m forward and then 3 m backward, the total distance is 8 m, but the net displacement is only 2 m. The graph would first rise by 5 units and then fall by 3 units.

例如，物体先前进 5 m，再后退 3 m，总路程为 8 m，但净位移仅为 2 m。图像将先上升 5 个单位，再下降 3 个单位。

8. Instantaneous Velocity with Tangents | 切线法求瞬时速度

Instantaneous velocity is the velocity of an object at a single instant of time. To find it from a curved s-t graph, draw a tangent line at the point of interest and calculate its gradient.

瞬时速度是物体在某一时刻的速度。要从弯曲的 s-t 图像中求出它，需在所关注的时刻作一条切线，并计算切线的斜率。

To draw a good tangent, use a ruler and aim to touch the curve at only one point. Then choose two well-separated points on the tangent line and compute: v = Δs / Δt. Well-separated points reduce the percentage error from reading the coordinates.

画切线时，用直尺使直线只与曲线接触于一点。然后在切线上选取两个相距较远的点，计算 v = Δs / Δt。取点相距越远，坐标读取带来的百分比误差越小。

9. Average Velocity with Secants | 割线法求平均速度

The average velocity over a time interval [t₁, t₂] is found by joining the two endpoints with a straight line (a secant). The gradient of this secant gives the average velocity over that interval.

[t₁, t₂] 时间间隔内的平均速度可通过连接两端点的直线（割线）来求得。该割线的斜率就是这一时间间隔内的平均速度。

v_avg = (s₂ − s₁) / (t₂ − t₁)

This is identical in form to the gradient formula, but applied to the secant rather than the tangent. It gives a coarse, overall picture of the motion, ignoring all fluctuations within the interval.

这与斜率公式形式相同，但应用于割线而非切线。它给出了运动的总览，忽略了区间内的所有波动细节。

10. Worked Example | 例题详解

Consider the following motion: A particle starts at s = 2 m, moves forward at a constant speed, reaching s = 8 m at t = 3 s. It then stays at rest for 2 seconds. Finally, it returns to the origin, reaching s = 0 at t = 8 s.

考虑以下运动：一质点从 s = 2 m 处出发，匀速前进，在 t = 3 s 时到达 s = 8 m；随后静止 2 秒；最后返回原点，在 t = 8 s 时到达 s = 0。

For the first interval, the gradient is (8 − 2) / (3 − 0) = 6 / 3 = 2 m/s. In the second interval, the gradient is 0, so the velocity is 0. In the third interval, the gradient is (0 − 8) / (8 − 5) = −8 / 3 ≈ −2.67 m/s.

第一个时间段内，斜率为 (8 − 2) / (3 − 0) = 6 / 3 = 2 m/s。第二个时间段内，斜率为 0，速度为 0。第三个时间段内，斜率为 (0 − 8) / (8 − 5) = −8 / 3 ≈ −2.67 m/s。

The negative sign indicates motion in the negative direction. The total distance travelled is |8 − 2| + 0 + |0 − 8| = 6 + 0 + 8 = 14 m. The net displacement is 0 − 2 = −2 m.

负号表示沿负方向运动。总路程为 |8 − 2| + 0 + |0 − 8| = 6 + 0 + 8 = 14 m。净位移为 0 − 2 = −2 m。

11. Common Exam Questions and Tips | 常见考题与应试技巧

Common exam questions based on displacement-time graphs include: reading the displacement at a given time; finding the time when the object is at a certain position; calculating velocity from a straight section; estimating instantaneous velocity from a tangent; and comparing average speed with average velocity.

基于位移-时间图像的常见考题包括：求某时刻的位移；求物体到达某位置的时间；计算直线段的速度；用切线估算瞬时速度；以及比较平均速率与平均速度。

Exam tips: always check the units; use a ruler for tangents; read the scale carefully; remember that the gradient gives velocity, and the area under a velocity-time graph gives displacement, but the area under an s-t graph has no useful physical meaning.

考试提示：始终检查单位；画切线时使用直尺；仔细读取刻度；牢记斜率给出的是速度——v-t 图像下方面积对应位移，但 s-t 图像下方面积没有实用的物理意义。

12. Summary | 要点总结

In summary, displacement-time graphs provide a compact and powerful representation of linear motion. The key ideas are: the y-axis is displacement; the gradient equals velocity; a horizontal line means stationary; straight lines mean uniform velocity; curves mean changing velocity; tangents give instantaneous velocity; and secants give average velocity.

总结：位移-时间图像为直线运动提供了简洁而有力的表示方法。核心要点包括：y 轴为位移；斜率等于速度；水平线表示静止；直线表示匀速；曲线表示变速；切线给出瞬时速度；割线给出平均速度。

Mastering these graph interpretation skills will not only help you in Mechanics questions but also build a strong foundation for more advanced topics such as the velocity-time graph and integration in kinematics.

掌握这些图像解读技巧不仅有助于解决力学题目，还将为更高级的专题——如速度-时间图像以及运动学中的积分应用——打下坚实的基础。

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Displacement-Time Graph Analysis | 位移-时间图像解析