📚 Displacement-time graphs | 位移-时间图
In CIE A-Level Physics, displacement-time graphs are one of the first tools used to describe motion. They show how displacement from a chosen origin changes with time, and their gradients reveal velocity. Mastering these graphs is essential for solving kinematics problems and for understanding more advanced motion graphs.
在 CIE A-Level 物理中，位移-时间图是描述运动的首批工具之一。它们显示相对于选定原点的位移如何随时间变化，其斜率反映速度。掌握这些图像对于解决运动学问题和理解更高级的运动图像至关重要。
1. What is Displacement? | 什么是位移？
Displacement is a vector quantity. It measures the straight-line distance from a chosen origin to an object's position, together with a direction. It is not the same as distance, which is a scalar and only gives total path length. In a displacement-time graph, the vertical axis shows displacement s from the origin, and the horizontal axis shows time t.
位移是矢量。它测量从选定原点到物体位置的直线距离，并带有方向。它不同于路程，路程是标量，仅给出总路径长度。在位移-时间图中，纵轴表示相对于原点的位移 s，横轴表示时间 t。
- Displacement can be positive, negative or zero, depending on which side of the origin the object lies.
- 位移可以是正值、负值或零，取决于物体位于原点的哪一侧。
- Distance is always non-negative and does not depend on direction, so a displacement-time graph does not directly show total distance travelled unless the direction changes are analysed.
- 路程总是非负的，且与方向无关，因此位移-时间图不直接显示总路程，除非对方向变化进行分析。
2. Axes and Basic Features | 坐标轴与基本特征
The vertical axis of a displacement-time graph has positive and negative displacement values. Positive values indicate one side of the origin, while negative values indicate the opposite side. The horizontal axis is time, and the graph is a continuous curve or line showing displacement at every instant. If the graph crosses the time axis, the object is passing through the origin at that instant.
位移-时间图的纵轴有正位移和负位移。正值表示原点的一侧，负值表示相反侧。横轴是时间，图像是显示每一瞬时位移的连续曲线或直线。如果图像穿过时间轴，物体在该瞬时正经过原点。
- The origin is chosen for convenience; it may be the starting point, but it does not have to be.
- 原点的选择是为了方便；它可以是起点，但不一定是。
- A continuous graph means displacement is defined at every moment, not just at a few recorded points.
- 连续图像意味着位移在每一时刻都有定义，而不仅仅是在少数记录点。
3. Gradient Equals Velocity | 斜率等于速度
The gradient (slope) of a displacement-time graph at any point equals the instantaneous velocity. This follows from the definition of velocity as the rate of change of displacement:
位移-时间图任意点的斜率等于瞬时速度。这源于速度定义为位移变化率：
v = Δs / Δt
If the gradient is positive, the velocity is positive. If the gradient is negative, the velocity is negative. If the gradient is zero, the object is momentarily at rest.
如果斜率为正，速度为正。如果斜率为负，速度为负。如果斜率为零，物体瞬时静止。
- Velocity is a vector, so both the size and direction of the gradient matter.
- 速度是矢量，因此斜率的大小和方向都很重要。
- The steeper the gradient, the greater the magnitude of the velocity.
- 斜率越陡，速度的大小越大。
4. Stationary Object: Horizontal Line | 静止物体：水平线
If an object is stationary, its displacement does not change over time. The graph is a horizontal straight line. Its gradient is zero, so the velocity is zero. The object may be at a positive, negative, or zero displacement, depending on its position relative to the origin.
如果物体静止，其位移不随时间变化。图像是一条水平直线。其斜率为零，因此速度为零。物体可能处于正、负或零位移，取决于其相对于原点的位置。
- ne at s = 5 m means the object remains 5 m away from the origin in the positive direction.
- 位于 s = 5 m 的水平线表示物体保持在正方向上距原点 5 m 处。
- A horizontal line at s = 0 m means the object stays at the origin.
- 位于 s = 0 m 的水平线表示物体停留在原点处。
- No matter where the horizontal line is, the object is not moving because displacement is constant.
- 无论水平线在哪里，物体都没有移动，因为位移保持不变。
5. Constant Velocity: Straight Line | 匀速运动：直线
For uniform motion, displacement changes at a constant rate. The graph is a straight line with a non-zero gradient. A positive gradient indicates movement in the positive direction; a negative gradient indicates movement in the negative direction. A steeper line represents a higher speed because the magnitude of the gradient is larger.
对于匀速运动，位移以恒定速率变化。图像是一条非零斜率的直线。正斜率表示沿正方向运动；负斜率表示沿负方向运动。线越陡代表速率越大，因为斜率的大小更大。
- 2 m s⁻¹, so the object moves at a constant velocity of 2 m s⁻¹ in the positive direction.
- 例如，从 (0 s, 0 m) 到 (10 s, 20 m) 的直线斜率为 20 m / 10 s = 2 m s⁻¹，因此物体以 2 m s⁻¹ 的恒定速度沿正方向运动。
- A straight line with gradient −3 m s⁻¹ represents motion at 3 m s⁻¹ in the negative direction, not slowing down.
- 斜率为 −3 m s⁻¹ 的直线表示以 3 m s⁻¹ 沿负方向运动，而不是减速。
6. Acceleration and Deceleration: Curved Lines | 加速与减速：曲线
If velocity is changing, the gradient is changing, so the graph is curved. If the curve becomes steeper over time, the object is accelerating in the direction of motion. If the curve becomes less steep, the object is decelerating. The curvature is linked to acceleration: when acceleration is positive, the gradient increases and the curve bends upward; when acceleration is negative, the gradient decreases and the curve bends downward.
如果速度在变化，斜率在变化，因此图像是曲线。如果曲线随时间变陡，物体沿运动方向加速。如果曲线变缓，物体减速。曲率与加速度有关：当加速度为正时，斜率增大，曲线向上弯曲；当加速度为负时，斜率减小，曲线向下弯曲。
- has a gradient that increases with time, showing constant positive acceleration from rest.
- 形如 y = x² 的位移-时间图，其斜率随时间增大，表示从静止开始的恒定正加速度。
- A graph with a positive gradient that becomes less steep, such as y = √x, shows positive velocity that is decreasing, so the object is decelerating while still moving in the positive direction.
- 正斜率逐渐变缓的图像，如 y = √x，表示正速度正在减小，因此物体在沿正方向运动的同时减速。
- A curve with a negative gradient becoming more negative shows acceleration in the negative direction, so the object is speeding up in the negative direction.
- 负斜率变得越来越负的曲线表示负方向的加速度，因此物体在负方向上加速。
7. Turning Points and Sign of Velocity | 转向点与速度符号
A turning point occurs where the graph has a local maximum or minimum. At that instant the gradient is zero, so velocity is zero. If the displacement reaches a maximum and then decreases, the object has stopped momentarily and reversed direction. The sign of the gradient before and after the turning point changes, showing a change in velocity direction.
转向点出现在图像具有局部最大值或最小值处。在该瞬时斜率为零，因此速度为零。如果位移达到最大值然后减小，物体瞬间停止并反转方向。转向点前后斜率的符号发生变化，表明速度方向改变。
- At a maximum point, the object moves in the positive direction before the turning point, stops for an instant, then moves in the negative direction afterwards.
- 在最大值点，物体在转向点之前沿正方向运动，瞬间停止，然后沿负方向运动。
- At a minimum point, the direction changes from negative to positive.
- 在最小值点，方向从负变为正。
- Turning points are important for finding when an object reverses direction, such as a ball thrown upwards reaching its highest point.
- 转向点对于确定物体何时反转方向很重要，例如向上抛出的球到达最高点。
8. Instantaneous and Average Velocity | 瞬时速度与平均速度
Instantaneous velocity is found by drawing a tangent to the curve at a specific time and calculating its gradient. Average velocity over a time interval is found by joining the two points on the graph with a straight chord and calculating its gradient:
瞬时速度通过在特定时刻作曲线的切线并计算其斜率来求得。某时间间隔内的平均速度通过用直线弦连接图上的两点并计算其斜率来求得：
v average = (s₂ − s₁) / (t₂ − t₁)
The tangent method is necessary when the graph is curved, because a chord gradient only gives the average velocity for the whole interval, not the velocity at one instant.
当图像为曲线时，切线法是必需的，因为弦的斜率只给出整个时间间隔的平均速度，而不是某一瞬时的速度。
- To draw a tangent accurately, place a ruler so it just touches the curve at the chosen point without cutting through it, then calculate rise over run.
- 为了准确画切线，将直尺放在刚好接触曲线上所选点而不穿过曲线，然后计算纵向变化与横向变化之比。
- Average speed is not the same as average velocity if direction changes; average speed uses total distance travelled, while average velocity uses net displacement.
- 如果方向改变，平均速率与平均速度不同；平均速率用总路程，而平均速度用净位移。
9. Common Misconceptions | 常见误区
Several misconceptions commonly arise when interpreting displacement-time graphs. Understanding these will help you avoid losing marks in exams. The key point is that the graph does not show the path taken by the object.
在解读位移-时间图时，常出现几个误区。理解这些有助于避免考试失分。关键在于：图像并不显示物体经过的实际路径。
- Misconception: The shape of the graph shows the actual route, like a hill or a track. Correction: It only shows how displacement from the origin changes with time; the actual path can be completely different.
- 误区：图像形状显示实际路线，如山坡或轨道。纠正：它只显示相对于原点的位移如何随时间变化；实际路径可能完全不同。
- Misconception: A steep line always means the object is moving uphill. Correction: Steepness relates to speed, not terrain or height.
- 误区：陡峭的线总是意味着物体在上坡。纠正：陡峭程度与速率有关，与地形或高度无关。
- Misconception: A negative gradient means the object is slowing down. Correction: Negative gradient indicates motion in the negative direction; the object may be speeding up if the gradient becomes more negative.
- 误区：负斜率意味着物体正在减速。纠正：负斜率表示沿负方向运动；如果斜率变得更负，物体可能在加速。
- Misconception: The area under a displacement-time graph gives displacement. Correction: For displacement-time graphs, the gradient gives velocity; area under the graph has no standard meaning in kinematics. Area is used for velocity-time graphs to find displacement.
- 误区：位移-时间图下方的面积给出位移。纠正：对于位移-时间图，斜率给出速度；图像下方的面积在运动学中没有标准意义。面积是用于速度-时间图来求位移的。
10. Worked Example | 例题解析
cribe the motion in each interval and calculate the relevant velocities and average speed.
一辆汽车沿直路行驶。位移-时间图有以下关键点：在 t = 0 s 时，s = 0 m；在 t = 2 s 时，s = 10 m；在 t = 5 s 时，s = 10 m；在 t = 8 s 时，s = 0 m。请描述每段时间的运动，并计算相关速度和平均速率。
- From 0 s to 2 s, the graph is a straight line from (0,0) to (2,10). Gradient:
- 从 0 s 到 2 s，图像是从 (0,0) 到 (2,10) 的直线。斜率为：
v = (10 − 0) / (2 − 0) = 5 m s⁻¹
The car moves at a constant velocity of 5 m s⁻¹ in the positive direction.
汽车以 5 m s⁻¹ 的恒定速度沿正方向运动。
- From 2 s to 5 s, the graph is a horizontal line at s = 10 m. Gradient = 0.
- 从 2 s 到 5 s，图像是位于 s = 10 m 的水平线。斜率为 0。
The car is stationary for 3 seconds at a displacement of 10 m from the origin.
汽车在距原点 10 m 处静止了 3 秒。
- From 5 s to 8 s, the graph is a straight line from (5,10) to (8,0). Gradient:
- 从 5 s 到 8 s，图像是从 (5,10) 到 (8,0) 的直线。斜率为：
v = (0 − 10) / (8 − 5) = −3.33 m s⁻¹
The car moves at a constant velocity of 3.33 m s⁻¹ in the negative direction, so it returns to the origin.
汽车以 3.33 m s⁻¹ 的恒定速度沿负方向运动，因此它返回原点。
- Total distance travelled = 10 m + 10 m = 20 m; total displacement = 0 m. Total time = 8 s.
- 总路程 = 10 m + 10 m = 20 m；总位移 = 0 m。总时间 = 8 s。
Average speed = total distance / total time = 20 m / 8 s = 2.5 m s⁻¹
Average velocity = total displacement / total time = 0 m / 8 s = 0 m s⁻¹
This example shows why average speed and average velocity are not the same when direction changes.
这个例子说明，当方向改变时，平均速率和平均速度为什么不同。
11. Displacement-time vs Velocity-time Graphs | 位移-时间图与速度-时间图的对比
It is easy to confuse displacement-time graphs with velocity-time graphs. In a displacement-time graph, the gradient gives velocity. In a velocity-time graph, the gradient gives acceleration and the area under the graph gives displacement. The table below summarises the key differences.
很容易混淆位移-时间图与速度-时间图。在位移-时间图中，斜率给出速度。在速度-时间图中，斜率给出加速度，图像下方的面积给出位移。下表总结了主要区别。
| Feature | 特征 | Displacement-time graph | 位移-时间图 | Velocity-time graph | 速度-时间图 |
|---|---|---|
| Vertical axis | 纵轴 | Displacement s | 位移 s | Velocity v | 速度 v |
| Gradient gives | 斜率给出 | Velocity | 速度 | Acceleration | 加速度 |
| Area under graph | 图像下方面积 | No standard kinematic meaning | 无标准运动学意义 | Displacement | 位移 |
| Horizontal line means | 水平线表示 | Object is stationary | 物体静止 | Constant velocity | 匀速运动 |
| Straight sloping line means | 倾斜直线表示 | Constant velocity | 恒定速度 | Constant acceleration | 恒定加速度 |
12. Exam Tips | 考试技巧
When answering exam questions on displacement-time graphs, keep these practical tips in mind. They will help you interpret graphs accurately and avoid common errors.
在回答有关位移-时间图的考试题目时，请记住这些实用技巧。它们
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