Acceleration-time Graphs | 加速度-时间图

📚 Acceleration-time Graphs | 加速度-时间图

An acceleration-time graph is one of the most useful tools in A-Level mechanics. It shows how an object’s acceleration changes with time, and its most important feature is that the area between the graph and the time axis gives the change in velocity. Many Edexcel questions ask you to move between velocity-time and acceleration-time graphs, so a clear understanding of slopes, areas and signs is essential.

加速度-时间图是 A-Level 力学中最有用的工具之一。它展示物体的加速度如何随时间变化，其最重要的特征是图线与时间轴之间的面积表示速度的变化量。许多 Edexcel 考题要求你在速度-时间图和加速度-时间图之间进行转换，因此清晰理解斜率、面积和正负号至关重要。

1. What is an Acceleration-time Graph? | 什么是加速度-时间图？

An acceleration-time graph, often written as an a-t graph, has time t on the horizontal axis and acceleration a on the vertical axis. Acceleration is a vector quantity, so its sign indicates direction. The SI unit is metres per second squared, written as m/s² or m s⁻².

加速度-时间图通常写作 a-t 图，横轴为时间 t，纵轴为加速度 a。加速度是矢量，因此它的正负表示方向。其国际单位是米每二次方秒，写作 m/s² 或 m s⁻²。

In kinematics, the a-t graph is closely linked to the velocity-time graph. The acceleration at any instant is the gradient of a v-t graph, while the area under an a-t graph gives the change in velocity between two times. This two-way relationship is central to Edexcel Mechanics questions.

在运动学中，a-t 图与速度-时间图密切相关。任一时刻的加速度是 v-t 图的斜率，而 a-t 图下的面积则表示两个时刻之间速度的变化量。这种双向关系是 Edexcel 力学考题的核心。

2. Key Features: Slope and Area | 关键特征：斜率与面积

There are two main features of an a-t graph: the gradient of the curve and the area under the curve. The gradient tells you the rate of change of acceleration, while the area tells you the change in velocity.

a-t 图有两个主要特征：曲线的斜率和曲线下的面积。斜率表示加速度的变化率，而面积表示速度的变化量。

For most A-Level work, the area is far more important. You will often be asked to calculate the change in velocity by finding the area under the graph. The slope is less commonly examined, but it is still useful to know that it represents the rate at which acceleration itself is changing.

在大多数 A-Level 学习中，面积重要得多。你经常需要通过求图线下的面积来计算速度变化量。斜率虽然较少考查，但了解它表示加速度本身的变化率仍然很有用。

3. Reading the Slope: Rate of Change of Acceleration | 读取斜率：加速度的变化率

The gradient of an acceleration-time graph is the rate of change of acceleration with respect to time. In more advanced mechanics this quantity is called jerk, and its unit is metres per second cubed, m/s³ or m s⁻³.

加速度-时间图的斜率是加速度随时间的变化率。在更高阶的力学中，这个量称为急动度，其单位是米每三次方秒，写作 m/s³ 或 m s⁻³。

jerk = Δa / Δt

If the a-t graph is a horizontal line, the slope is zero, so the jerk is zero and the acceleration is constant. If the graph slopes upwards, acceleration is increasing with time. If it slopes downwards, acceleration is decreasing.

如果 a-t 图是一条水平线，则斜率为零，因此急动度为零，加速度恒定。如果图线向上倾斜，加速度随时间增大。如果图线向下倾斜，加速度随时间减小。

Edexcel does not usually require you to calculate jerk, but recognising the slope prevents confusion between an a-t graph and a v-t graph.

Edexcel 通常不要求你计算急动度，但识别斜率有助于避免混淆 a-t 图与 v-t 图。

4. Area Under the Curve: Change in Velocity | 曲线下面积：速度变化量

The area between an acceleration-time graph and the time axis between two times t₁ and t₂ gives the change in velocity Δv over that interval. The formula is:

在时间 t₁ 到 t₂ 之间，加速度-时间图与时间轴之间的面积给出了该时间段内的速度变化量 Δv。公式为：

Δv = ∫ a dt from t₁ to t₂

At A-Level you do not need to use calculus; you simply calculate the area as a rectangle, triangle or trapezium. The units must work out: acceleration in m/s² multiplied by time in s gives m/s, which is a velocity.

在 A-Level 阶段你不需要使用微积分，只需将面积按矩形、三角形或梯形计算。单位必须吻合：加速度的单位 m/s² 乘以时间的单位 s 得到 m/s，即速度单位。

It is crucial to remember that the area gives the change in velocity, not the final velocity. If the initial velocity is u, the final velocity is v = u + Δv.

必须记住，面积给出的是速度变化量，而不是末速度。如果初速度为 u，则末速度为 v = u + Δv。

Areas below the time axis are counted as negative. A negative area means the velocity changes in the negative direction, or that a positive velocity decreases.

时间轴以下的面积计为负值。负面积表示速度沿负方向变化，或正方向速度减小。

5. Case 1: Constant Acceleration | 情形一：匀加速运动

When acceleration is constant, the a-t graph is a horizontal straight line. For an acceleration a acting for a time interval Δt, the area under the graph is a rectangle.

当加速度恒定时，a-t 图是一条水平直线。对于在时间间隔 Δt 内作用的加速度 a，图线下的面积是一个矩形。

Δv = a × Δt

For example, if an object starts at 2 m/s and accelerates at 3 m/s² for 5 s, the change in velocity is:

例如，一个物体以 2 m/s 的初速度运动，并以 3 m/s² 的加速度加速 5 s，则速度变化量为：

Δv = 3 m/s² × 5 s = 15 m/s

The final velocity is therefore v = 2 m/s + 15 m/s = 17 m/s. On a v-t graph, this motion would appear as a straight line with constant gradient 3 m/s².

因此末速度为 v = 2 m/s + 15 m/s = 17 m/s。在 v-t 图上，这一运动表现为一条斜率恒为 3 m/s² 的直线。

6. Case 2: Uniformly Increasing Acceleration | 情形二：均匀增大的加速度

If acceleration increases uniformly with time, the a-t graph is a straight line with a non-zero gradient. The area under the graph between two times is a trapezium, so the change in velocity is the average acceleration multiplied by the time interval.

如果加速度随时间均匀增大，则 a-t 图是一条斜率不为零的直线。在两个时刻之间，图线下的面积是一个梯形，因此速度变化量等于平均加速度乘以时间间隔。

Δv = ½(a₁ + a₂) × Δt

Here a₁ is the initial acceleration and a₂ is the final acceleration over the interval. If the acceleration starts from zero and increases uniformly to a value a at time T, then the area is a triangle:

其中 a₁ 是时间段内的初加速度，a₂ 是末加速度。如果加速度从零开始均匀增大，并在时刻 T 达到 a，则面积为三角形：

Δv = ½ × a × T

This corresponds to a v-t graph that is curved, with its gradient increasing over time. Many students incorrectly use a rectangular area here, so always check the shape of the graph.

这对应于一条曲线 v-t 图，其斜率随时间增大。许多学生在这里错误地使用矩形面积，因此务必检查图线的形状。

7. From a Velocity-time Graph to an Acceleration-time Graph | 从速度-时间图到加速度-时间图

To sketch an a-t graph from a velocity-time graph, you need to find the gradient of the v-t graph at each point. The gradient of a v-t graph is the instantaneous acceleration.

要从速度-时间图绘制 a-t 图，你需要找到 v-t 图在每个点的斜率。v-t 图的斜率就是瞬时加速度。

If the v-t graph is a straight line, the gradient is constant, so the a-t graph is a horizontal line. If the v-t graph has a positive constant gradient, the a-t graph is a horizontal line above the time axis at the value of that gradient.

如果 v-t 图是一条直线，则斜率为常数，因此 a-t 图是一条水平线。如果 v-t 图具有正的恒定斜率，则 a-t 图是位于时间轴上方、数值等于该斜率的水平线。

If the v-t graph is a horizontal line, the velocity is constant, so the acceleration is zero and the a-t graph lies on the time axis. If the v-t graph curves, its gradient changes, so the a-t graph will be a non-horizontal curve or line.

如果 v-t 图是一条水平线，则速度恒定，因此加速度为零，a-t 图位于时间轴上。如果 v-t 图是曲线，其斜率变化，因此 a-t 图将是一条非水平的曲线或直线。

8. From an Acceleration-time Graph to a Velocity-time Graph | 从加速度-时间图到速度-时间图

Going in the opposite direction, you can build a v-t graph from an a-t graph by adding areas step by step. Start with the initial velocity u, then add the area under the a-t graph for each time interval. The result is the velocity at the end of that interval.

反过来，你可以通过逐步累加面积，从 a-t 图构建 v-t 图。从初速度 u 开始，然后将每个时间段内 a-t 图下的面积加上去。所得结果就是该时间段结束时的速度。

For a piecewise-constant a-t graph, the v-t graph consists of straight-line segments. The gradient of each segment equals the constant acceleration in that interval.

对于分段恒定的 a-t 图，v-t 图由直线段组成。每一段的斜率等于该时间段内的恒定加速度。

For example, if a-t is 2 m/s² for 4 s, then 0 for 6 s, and then -2 m/s² for 4 s, the velocity changes by +8 m/s, then 0 m/s, then -8 m/s. If the initial velocity is zero, the final velocity is zero.

例如，如果 a-t 图在前 4 s 为 2 m/s²，接着 6 s 为 0，最后 4 s 为 -2 m/s²，则速度变化量分别为 +8 m/s、0 m/s 和 -8 m/s。如果初速度为零，末速度也为零。

9. Sign Conventions and Direction | 符号约定与方向

Acceleration is a vector, so on an a-t graph positive values mean acceleration in the positive direction and negative values mean acceleration in the negative direction. However, a negative acceleration does not always mean the object is slowing down.

加速度是矢量，因此在 a-t 图上，正值表示沿正方向的加速度，负值表示沿负方向的加速度。然而，负加速度并不总是表示物体在减速。

If velocity and acceleration have the same sign, the object speeds up. If they have opposite signs, the object slows down. For example, an object moving in the positive direction with acceleration -2 m/s² is decelerating.

如果速度与加速度同号，物体加速；如果二者异号，物体减速。例如，一个沿正方向运动的物体加速度为 -2 m/s²，它正在减速。

When calculating areas, areas below the time axis are negative. If the total area under an a-t graph is negative, the overall change in velocity is negative, meaning the velocity has decreased or become more negative.

在计算面积时，时间轴以下的面积为负。如果 a-t 图下的总面积为负，则速度的总变化量为负，意味着速度减小或变得更负。

10. Worked Example: Building v-t and a-t Graphs | 例题：构建 v-t 图与 a-t 图

A car starts from rest and moves in a straight line. From 0 to 4 s it accelerates uniformly to 8 m/s. It then travels at constant velocity from 4 s to 10 s. From 10 s to 14 s it brakes uniformly and comes to rest.

一辆汽车从静止开始沿直线运动。从 0 到 4 s，它匀加速到 8 m/s；然后从 4 s 到 10 s 做匀速运动；从 10 s 到 14 s 它均匀刹车并停下。

First find the accelerations: from 0 to 4 s, a = (8 – 0) / 4 = 2 m/s². From 4 to 10 s, a = 0 m/s². From 10 to 14 s, a = (0 – 8) / 4 = -2 m/s².

首先求加速度：0 到 4 s，a = (8 – 0) / 4 = 2 m/s²；4 到 10 s，a = 0 m/s²；10 到 14 s，a = (0 – 8) / 4 = -2 m/s²。

The a-t graph is therefore a horizontal line at 2 m/s² from 0 to 4 s, a horizontal line at 0 m/s² from 4 to 10 s, and a horizontal line at -2 m/s² from 10 to 14 s.

因此，a-t 图在 0 到 4 s 为 2 m/s² 的水平线，4 到 10 s 为 0 m/s² 的水平线，10 到 14 s 为 -2 m/s² 的水平线。

Check the areas: +2 × 4 = +8 m/s, 0 × 6 = 0 m/s, -2 × 4 = -8 m/s. The total change in velocity is 0 m/s, which matches the fact that the car starts and ends at rest.

检查面积：+2 × 4 = +8 m/s，0 × 6 = 0 m/s，-2 × 4 = -8 m/s。速度总变化量为 0 m/s，与汽车开始和结束时均静止的事实相符。

11. Common Misconceptions | 常见误区

One common mistake is reading the y-value of an a-t graph as if it were velocity. An a-t graph tells you acceleration, not velocity. A value of 0 m/s² means zero acceleration, which means constant velocity, not necessarily rest.

一个常见错误是把 a-t 图的纵坐标值当作速度来读。a-t 图给出的是加速度，不是速度。0 m/s² 表示加速度为零，即速度恒定，而不一定是静止。

Another common error is treating the area under an a-t graph as displacement. The area under an a-t graph is change in velocity. Displacement is the area under a velocity-time graph.

另一个常见错误是把 a-t 图下的面积当作位移。a-t 图下的面积是速度变化量，而位移是速度-时间图下的面积。

Students also forget to add the initial velocity. If the area gives Δv = 10 m/s and the initial velocity is 3 m/s, the final velocity is 13 m/s, not 10 m/s. Finally, always include the sign of the area; negative areas reduce the velocity.

学生还容易忘记加上初速度。如果面积为 Δv = 10 m/s，初速度为 3 m/s，则末速度为 13 m/s，而不是 10 m/s。最后，始终要计入面积的正负号；负面积会减小速度。

12. Exam Tips for Edexcel A-Level | Edexcel A-Level 考试技巧

When a question asks you to sketch an acceleration-time graph from a velocity-time graph, work out the gradient of each straight-line segment first. Label the axes clearly and use the correct units, m/s² on the vertical axis and s on the horizontal axis.

当题目要求你根据速度-时间图绘制加速度-时间图时，先求出每一段直线的斜率。清楚标注坐标轴，并在纵轴使用 m/s²、横轴使用 s 的单位。

If you need to calculate a change in velocity, identify whether the area is a rectangle, triangle or trapezium. Write down the area expression before substituting numbers. Then apply v = u + Δv, making sure you know the initial velocity.

如果需要计算速度变化量，先判断面积是矩形、三角形还是梯形。写出面积表达式后再代入数值。然后应用 v = u + Δv，确保已知初速度。

Always check the overall consistency: the total area under the a-t graph should equal v_final – v_initial. If it does not, you have likely made a sign error or missed an interval.

始终检查整体一致性：a-t 图下的总面积应等于 v_final – v_initial。若不相等，你很可能出现了正负号错误或遗漏了某个时间段。

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Acceleration-time Graphs | 加速度-时间图