📚 Velocity–time graphs | 速度-时间图像
Velocity–time graphs are one of the most powerful tools in kinematics. They allow you to visualise how an object's velocity changes over time and to extract acceleration, displacement and the nature of motion directly from the graph. For Edexcel A-Level Physics, you must be able to sketch, interpret and calculate quantities from these graphs, including cases where the motion involves changes in direction and non‑constant acceleration.
速度–时间图像是运动学中最强大的工具之一。它可以让你直观地看到物体速度随时间的变化，并直接从图中提取加速度、位移以及运动的性质。在 Edexcel A-Level 物理中，你必须能够绘制、解读这类图像并进行相关计算，包括运动方向改变和非匀加速度的情况。
1. What is a Velocity–time Graph? | 什么是速度–时间图像？
A velocity–time graph plots instantaneous velocity on the vertical axis and time on the horizontal axis. Velocity is a vector, so the sign (positive or negative) tells you the direction of motion relative to a chosen coordinate system. The graph can consist of straight lines, curves or a combination of both, each revealing different types of motion.
速度–时间图像将瞬时速度标在纵轴上，将时间标在横轴上。速度是矢量，因此正负号表示相对于选定坐标系的运动方向。图像可以由直线、曲线或两者的组合构成，每种形态都揭示了不同类型的运动。
The standard units on a velocity–time graph are metres per second (m s⁻¹) for velocity and seconds (s) for time. You will often see labels such as v / m s⁻¹ and t / s, which follow the convention of dividing the quantity by its unit.
速度–时间图像的标准单位是速度用米每秒 (m s⁻¹)，时间用秒 (s)。你经常会看到诸如 v / m s⁻¹ 和 t / s 这样的标注，这遵循了物理量除以其单位的惯例。
2. Key Features of Velocity–time Graphs | 速度–时间图像的关键特征
Three fundamental features make velocity–time graphs incredibly informative:
三个基本特征使得速度–时间图像具有极高的信息量：
- Slope (gradient) – acceleration: The gradient of a v–t graph at any instant gives the instantaneous acceleration. A straight line indicates constant acceleration, while a curved line shows changing acceleration.
- 斜率（梯度）– 加速度：在任一时刻，v–t图像的斜率给出瞬时加速度。直线表示加速度恒定，曲线则表示加速度在变化。
- Area under the graph – displacement: The area between the graph line and the time axis represents displacement. Areas above the t‑axis are positive displacements; areas below the axis are negative.
- 图像下的面积 – 位移：图线与时间轴之间的面积代表位移。t轴以上的面积为正位移，轴以下的面积为负位移。
- Intercept – initial velocity: he : The value where the line crosses the velocity axis (t = 0) is the initial velocity, often labelled u in the suvat equations.
- 截距 – 初速度：图线与速度轴（t = 0）的交点即为初速度，在 suvat 方程中通常记为 u。
3. Slope Equals Acceleration | 斜率等于加速度
Acceleration is defined as the rate of change of velocity, so on a velocity–time graph it is given by the gradient. For a straight line between two points (t₁, v₁) and (t₂, v₂), the average acceleration is:
加速度定义为速度的变化率，因此在速度–时间图上它由斜率表示。对于两点 (t₁, v₁) 和 (t₂, v₂) 之间的直线，平均加速度为：
a = (v₂ − v₁) ÷ (t₂ − t₁) = Δv ÷ Δt
If the line slopes upwards as time increases, the gradient is positive and the object accelerates in the chosen positive direction. If the line slopes downwards, the gradient is negative, indicating deceleration (or acceleration in the opposite direction).
如果图线随时间上升，斜率为正，物体在选定的正方向上加速。如果图线下降，斜率为负，表示减速（或反方向加速）。
When the graph is a curve, you must draw a tangent to the point of interest to find the instantaneous acceleration. The steeper the tangent, the greater the magnitude of the acceleration.
当图像为曲线时，必须在感兴趣的点画切线以求出瞬时加速度。切线越陡，加速度的大小就越大。
4. Area Under the Graph Equals Displacement | 图像下的面积等于位移
Displacement is the integral of velocity with respect to time, so on a v–t graph it corresponds to the area bound by the graph line and the time axis over a given interval. For simple shapes, you can use geometry:
位移是速度对时间的积分，因此在 v–t 图上，它对应于给定区间内图线与时间轴所围成的面积。对于简单图形，你可以使用几何方法：
- Rectangle: area = v × Δt (for constant velocity)
- 矩形：面积 = v × Δt （匀速运动）
- Triangle: area = ½ × base × height (for constant acceleration from rest)
- 三角形：面积 = ½ × 底 × 高 （由静止开始的匀加速运动）
- Trapezium: area = ½ × (u + v) × t (for constant acceleration with non‑zero initial velocity)
- 梯形：面积 = ½ × (u + v) × t （初速不为零的匀加速运动）
A crucial point is that areas below the time axis must be treated as negative displacement. If the graph crosses the axis, you should calculate the total area above and below separately and then add them algebraically to obtain the net displacement. The total distance travelled, however, is the sum of the absolute values of the areas.
关键的一点是，时间轴以下的面积必须视为负位移。若图像穿过时间轴，应分别计算轴以上和以下的面积，然后代数相加得到净位移。而总路程则是各面积绝对值之和。
5. Interpreting Constant Velocity | 解读匀速运动
al Δt is simply the area of a rectangle: Δs = v × Δt. No net force acts on the object (Newton's First Law).
当速度恒定时，图像是一条水平直线。斜率为零，因此加速度为零。在时间间隔 Δt 内的位移就是矩形面积：Δs = v × Δt。物体不受净外力（牛顿第一定律）。
On an exam question, you might be asked to find the distance travelled by reading the velocity from the vertical axis and multiplying by the duration. Be careful with signs: if the line lies in the negative velocity region, the object moves backwards at constant speed.
在考试题目中，你可能需要从纵轴读取速度并乘以持续时间来求出运动距离。注意符号：若直线处于负速度区域，物体以恒定速率向后运动。
6. Interpreting Constant Acceleration | 解读匀加速运动
A straight line with a non‑zero gradient represents constant acceleration. If the line starts at u and ends at v over a time t, the area under the line is a trapezium, giving the familiar suvat equation:
一条具有非零斜率的直线代表匀加速运动。如果直线在时间t内从u变化到v，图线下的面积是一个梯形，由此可得出熟悉的 suvat 方程：
s = ½ (u + v) t
t + ½ a t² and v² = u² + 2 a s, all of which link directly to the geometry of the velocity–time graph.
该方程很容易从图中推导出来。同样，斜率给出加速度 a = (v − u) / t。你也可以将这些关系组合得到 s = ut + ½ a t² 和 v² = u² + 2 a s，所有这些都直接与速度–时间图的几何结构相关联。
When the line has a negative gradient, it indicates deceleration (or negative acceleration). The area is still displacement, but the velocity at the end may become zero or negative, meaning the object has reversed direction.
当图线斜率为负时，表示减速（或负加速度）。面积仍然是位移，但末速度可能变为零或负值，意味着物体已反向运动。
7. Changing Acceleration and Curved Graphs | 变加速度与曲线图
If the velocity–time graph is a curve, the acceleration is not constant. The instantaneous acceleration at any point is the gradient of the tangent at that point. As the curve becomes steeper, the magnitude of acceleration increases.
如果速度–时间图是曲线，加速度则不是恒定的。任一点的瞬时加速度是该点切线的斜率。曲线越陡，加速度的大小就越大。
the slope becomes more gradual. You may be asked to sketch the corresponding acceleration–time graph or to describe how the resultant force changes, since F = m a.
若曲线的斜率随时间变得越来越陡，则加速度在增大；若斜率变得越来越平缓，则加速度在减小。题目可能要求你画出相应的加速度–时间图像或描述合力的变化，因为 F = m a。
To find the displacement from a curved v–t graph, you cannot rely on simple geometric shapes. Instead, you estimate the area using methods such as counting squares or applying the trapezium rule, which are discussed in the next section.
要从曲线的 v–t 图求位移，不能依赖于简单的几何图形，而需要采用数格子或梯形法则等方法进行估算，这将在下一节讨论。
8. Calculating Displacement: Counting Squares and Formulas | 计算位移：数格子和公式
When a velocity–time graph is non‑linear, Edexcel may provide a printed grid. Each square represents a fixed amount of displacement. To estimate the area:
当速度–时间图是非线性时，Edexcel 可能会提供一张印刷的网格图。每个格子代表一个固定的位移量。估算面积的方法是：
- Count complete squares that lie entirely under the curve.
- 数出完全位于曲线下方的完整格子。
- Estimate the fraction of partially covered squares and add them to the total.
- 估算部分覆盖的格子所占比例，并加入总数。
- l number of squares by the area value of one square (e.g. 1 square = 1 m s⁻¹ × 1 s = 1 m).
- 将总格子数乘以每个格子的面积值（例如 1 格 = 1 m s⁻¹ × 1 s = 1 m）。
For sections that approximate a straight line, break the area into triangles and trapezia. You can also use the average velocity method if the acceleration is nearly constant over short intervals.
对于那些近似直线的部分，可将面积拆分为三角形和梯形。若加速度在短时间间隔内近似恒定，也可使用平均速度的方法。
9. Average Speed vs Average Velocity | 平均速率与平均速度
This distinction is often tested alongside velocity–time graphs. Average velocity is a vector: it equals total displacement divided by total time. Average speed is a scalar: it equals total distance divided by total time.
这个区别经常与速度–时间图一起考查。平均速度是矢量：它等于总位移除以总时间。平均速率是标量：它等于总路程除以总时间。
On a v–t graph, total displacement is the algebraic sum of areas, while total distance is the sum of absolute areas. Therefore, whenever the graph crosses the time axis, average speed will be greater than the magnitude of average velocity.
在 v–t 图上，总位移是面积的代数和，而总路程是各面积绝对值之和。因此，每当图像穿越时间轴时，平均速率将大于平均速度的大小。
For example, if a ball is thrown upwards and returns to the thrower's hand, the total displacement is zero, so average velocity is zero. But the total distance is twice the maximum height, giving a non‑zero average speed.
例如，若将一个球向上抛起并接住它，总位移为零，因此平均速度为零。但总路程是最大高度的两倍，从而得到一个非零的平均速率。
10. Converting Velocity–time to Displacement–time Graphs | 将速度–时间图转换为位移–时间图
Edexcel may ask you to sketch a displacement–time graph using information from a velocity–time graph. The key principle is that velocity is the gradient of the displacement–time graph. Therefore:
Edexcel 可能要求你利用速度–时间图的信息绘制位移–时间图。核心原理是速度等于位移–时间图的斜率。因此：
- Constant positive velocity → straight line with constant positive gradient on s–t graph.
- 恒定的正速度 → s–t 图上为恒定正斜率的直线。
- Constant acceleration (linear v–t) → parabolic curve on s–t graph, with increasing gradient if acceleration is positive.
- 匀加速（线性 v–t）→ s–t 图为抛物线，若加速度为正，则斜率逐渐增大。
- Velocity passing through zero → turning point (maximum or minimum) on s–t graph.
- 速度经过零点 → s–t 图上的转折点（极大值或极小值）。
To sketch accurately, calculate the displacement at several key times (by finding areas under the v–t graph) and then plot these points on the s–t axes. Connect the points with the correct curvature guided by the velocity values.
要精确绘图，可先计算若干关键时刻的位移（求 v–t 图下面积），然后将这些点标在 s–t 坐标系上，并根据速度值用正确的曲率连接各点。
11. Common Mistakes and Pitfalls | 常见错误与陷阱
Many candidates lose marks by confusing velocity–time and displacement–time graphs. Always check the label on the vertical axis. A horizontal line on a v–t graph means constant velocity, but on an s–t graph it means the object is stationary.
许多考生因混淆速度–时间图与位移–时间图而失分。务必检查纵轴的标注。v–t 图上的水平线表示匀速运动，而在 s–t 图上水平线表示物体静止。
Another common error is forgetting that area below the t‑axis is negative displacement. When calculating total distance, treat all areas as positive. Also, ensure you use the correct units: acceleration is in m s⁻², and area units come from multiplying velocity units (m s⁻¹) by time units (s), giving metres (m).
另一个常见错误是忘记 t 轴以下的面积代表负位移。计算总路程时，要将所有面积视为正值。同样，确保使用正确单位：加速度单位为 m s⁻²，面积单位由速度单位 (m s⁻¹) 乘以时间单位 (s) 得到米 (m)。
Students also sometimes read the intercept incorrectly or fail to recognise that a sudden change in slope means an abrupt change in acceleration, implying a jerk or a sudden force, which may be physically unrealistic in some contexts.
学生有时还会读错截距，或未能意识到斜率的突变意味着加速度的骤然变化，这隐含着 jerk（急动度）或突然的力，在某些情境中这在物理上可能是不现实的。
12. Exam-style Tips | 考试技巧
In Edexcel exams, velocity–time graphs often appear in multi‑stage motion scenarios, such as a car accelerating, cruising and braking. Start by labelling key values on the axes and breaking the graph into linear segments. This allows you to apply simple geometry for each segment.
在 Edexcel 考试中，速度–时间图常出现在多阶段运动的场景中，比如汽车加速、匀速行驶和刹车。首先应在坐标轴上标出关键数值，并将图像分割为若干线段，这样你就可以对每一段应用简单的几何方法。
Always state the formula you are using before substituting numbers. For example: 'acceleration = gradient = (12 m s⁻¹ − 4 m s⁻¹) ÷ (8 s − 2 s) = 1.33 m s⁻²' hat the area represents, and check whether the question asks for distance or displacement.
在代入数字之前，务必先列出你所使用的公式。例如：“加速度 = 斜率 = (12 m s⁻¹ − 4 m s⁻¹) ÷ (8 s − 2 s) = 1.33 m s⁻²”。要表现出你理解面积所代表的物理量，并看清题目要求的是路程还是位移。
When estimating areas under curves, explain your method clearly. If you use the trapezium rule, give an example calculation for the first two strips. Finally, pay attention to significant figures and ensure your final answer matches the precision of the data provided.
在估算曲线下面积时，要清晰地解释你的方法。若使用梯形法则，可给出前两个区间的计算示例。最后，注意有效数字，并确保最终答案与所给数据的精度相匹配。
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