📚 Uniform and Non-uniform Acceleration | 匀加速与非匀加速
Acceleration is the rate of change of velocity. When we study motion in A‑Level Physics, the way acceleration behaves determines which equations and graphical tools we can use. In some idealised situations, acceleration stays constant – uniform acceleration – making calculations beautifully simple. In the real world, however, forces change with position, speed or time, producing non‑uniform acceleration that requires a more careful approach.
加速度是速度的变化率。在A-Level物理中，加速度的变化方式决定了我们可以使用哪些方程和图形工具。在某些理想情况下，加速度保持恒定——即匀加速度——这使得计算非常简洁。然而在真实世界中，力会随着位置、速度或时间改变，产生非匀加速度，需要我们更细致地处理。
1. What is Acceleration? | 什么是加速度？
Δt, where Δv is the change in velocity and Δt is the time interval. The direction of acceleration points in the direction of the change in velocity, not necessarily in the direction of motion.
加速度是矢量，定义为单位时间内速度的变化量。数学上，平均加速度 aₐᵥ 写作 aₐᵥ = Δv / Δt，其中 Δv 是速度变化量，Δt 是时间间隔。加速度的方向指向速度变化的方向，而不一定指向运动方向。
taneous acceleration, a, is the limit of the average acceleration as Δt → 0. In a velocity– adient of the tangent. This distinction becomes crucial when acceleration is not constant.
瞬时加速度 a 是当 Δt → 0 时平均加速度的极限。在速度–时间图中，瞬时加速度由切线的斜率给出。当加速度不恒定时，这种区别就变得至关重要。
2. Uniform Acceleration Defined | 匀加速定义
Uniform acceleration means the acceleration of an object is constant in both magnitude and direction. Under this condition, the velocity changes by equal amounts every second, and the average acceleration equals the instantaneous acceleration at any point in the motion.
匀加速意味着物体的加速度在大小和方向上都是恒定的。在这种条件下，速度每秒钟的改变量均等，且平均加速度等于运动过程中任何一点的瞬时加速度。
In A‑Level problems, many situations are modelled with uniform acceleration: an object sliding down a frictionless inclined plane, a car accelerating steadily, or a freely falling body near the Earth's surface (ignoring air resistance). Although these are idealisations, they provide a powerful starting point for analysing motion.
在A-Level题目中，许多情况被建模为匀加速：物体沿无摩擦斜面下滑、汽车稳定加速，或者忽略空气阻力时近地表面的自由落体。尽管这些是理想化模型，但它们为分析运动提供了强有力的起点。
3. Equations of Motion for Uniform Acceleration | 匀加速运动方程
For straight‑line motion with uniform acceleration a, we use five standard quantities: initial velocity u, final velocity v, displacement s, acceleration a, and time t. Four kinematic equations, often remembered as 'suvat' equations, relate these quantities without including force or mass.
对于加速度 a 恒定的直线运动，我们使用五个标准物理量：初速度 u、末速度 v、位移 s、加速度 a 和时间 t。四个运动学方程（常称作 'suvat' 方程）联系这些量，而不涉及力或质量。
v = u + at s = ½(u + v)t s = ut + ½at² v² = u² + 2as
These equations are derived from the definitions of velocity and acceleration under the assumption that a is constant. They cannot be applied directly when acceleration changes. Selecting the right equation depends on which variables are known and which are required.
这些方程由速度和加速度的定义推导而来，其前提是 a 为常数。当加速度变化时，不能直接套用。选择正确的方程取决于已知哪些量、需求哪些量。
- Without s: v = u + at
- Without a: s = ½(u + v)t
- Without v: s = ut + ½at²
- Without t: v² = u² + 2as
- 不含 s：v = u + at
- 不含 a：s = ½(u + v)t
- 不含 v：s = ut + ½at²
- 不含 t：v² = u² + 2as
4. Graphical Representation of Uniform Acceleration | 匀加速运动的图形表示
When acceleration is constant, the graphs of motion take simple shapes. The velocity–time (v–t) graph is a straight line with slope equal to a. The displacement–time (s–t) graph is a parabola because s depends on t². The acceleration–time (a–t) graph is a horizontal line at a.
当加速度恒定时，运动图形呈现简单形状。速度–时间 (v–t) 图是一条直线，其斜率等于 a。位移–时间 (s–t) 图是抛物线，因为 s 依赖于 t²。加速度–时间 (a–t) 图是位于 a 处的一条水平线。
The area under a v– rectangle plus a triangle or a trapezium, which matches s = ½(u+v)t. The gradient of an s– gives instantaneous velocity; with uniform acceleration, velocity rises linearly, so the s–t curve becomes steeper with time.
v–t 图下的面积给出位移。对于匀加速，这块面积是一个矩形加一个三角形或一个梯形，这与 s = ½(u+v)t 吻合。s–t 图的斜率给出瞬时速度；由于速度线性增加，s–t 曲线随时间推移越来越陡。
5. Non‑uniform Acceleration Defined | 非匀加速定义
Non‑uniform acceleration occurs when the acceleration of an object changes in magnitude, direction, or both. The rate of change of velocity is not constant, meaning equal changes in velocity no longer take place in equal time intervals. Real‑life motions almost always involve non‑uniform acceleration due to varying resultant forces.
当物体加速度的大小、方向或两者均发生变化时，就存在非匀加速。速度的变化率不是恒定的，意味着在相等的时间间隔内不再发生相等的速度变化。由于存在变化的合力，实际运动几乎总是涉及非匀加速。
Examples include a skydiver during the initial part of the jump before terminal velocity, a pendulum bob whose acceleration is greatest at the extremes and zero at the equilibrium point, or a car braking with a force that depends on the friction conditions. Under non‑uniform acceleration, the 'suvat' equations are invalid.
例子包括跳伞者在到达终端速度之前的初始阶段、摆锤在极端位置加速度最大而在平衡点为零，或者汽车以依赖于摩擦条件的力进行制动。在非匀加速下，'suvat' 方程不再有效。
6. Describing Non‑uniform Acceleration | 描述非匀加速度
To describe non‑ a function of time, v(t), we can differentiate to find acceleration. Conversely, if we have acceleration as a function of time, a(t), the change in velocity over an interval is the area under the a–t graph.
为了描述非匀加速度，我们常使用瞬时加速度的概念 a = dv/dt。如果知道速度随时间变化的函数 v(t)，我们可以对其求导以得到加速度。反过来，如果加速度是时间函数 a(t)，某段时间内的速度变化量就是 a–t 图下的面积。
u will not need to perform calculus; instead, you will use the idea that Δv = area under a‑t graph and Δs = area under v‑t graph. The slope of a curved v‑t graph at a point gives the instantaneous acceleration, found by drawing a tangent.
在许多CIE问题中，你并不需要进行微积分运算；相反，你将使用这样的思想：Δv = a‑t 图下面积，Δs = v‑t 图下面积。曲形 v‑t 图上某一点的斜率给出瞬时加速度，可以通过画切线获得。
7. Graphical Analysis of Non‑uniform Motion | 非匀加速运动的图形分析
In non‑uniform acceleration, the v‑t graph is no longer a straight line. Its slope changes continuously, reflecting the changing acceleration. The s‑t graph is no longer a simple quadratic; its slope increases non‑uniformly if the object speeds up with varying acceleration.
在非匀加速运动中，v‑t 图不再是直线。其斜率不断变化，反映出加速度在改变。s‑t 图也不再是简单的二次曲线；如果物体以变化的加速度加速，其斜率非均匀地增加。
A curved v‑t graph must be interpreted carefully. The area under the curve still equals displacement, but it often must be estimated by counting squares or by using geometric approximations. An acceleration that decreases with time (such as a car running out of fuel) produces a v‑t graph whose slope gradually becomes shallower.
曲形 v‑t 图必须谨慎解读。曲线下的面积仍然等于位移，但通常需要通过数格子或使用几何近似来估算。随时间减小的加速度（如燃料耗尽的汽车）产生的 v‑t 图，其斜率逐渐变得平缓。
8. Instantaneous vs Average Acceleration | 瞬时加速度与平均加速度
− v₁) / (t₂ − t₁). Instantaneous acceleration is the value at a specific instant. For uniform acceleration, these two are identical. For non‑ acceleration is not generally halfway between the initial and final instantaneous values.
平均加速度在一个有限的时间间隔内计算：aₐᵥ = (v₂ − v₁) / (t₂ − t₁)。瞬时加速度是某一特定时刻的值。对于匀加速，这两者是相同的。对于非匀加速，它们差别很大，且平均加速度通常并非初始和最终瞬时值的中值。
uations, you may be asked to determine the average acceleration over a period where a body' but remember this does not describe how the acceleration changed at each moment. To understand the detailed motion, examine the shape of the v‑t or a‑t graph.
在考试情境中，你可能会被要求确定加速度变化期间的平均加速度。使用 a = Δv/Δt，但要记住这并不能描述加速度在每个时刻如何变化。要理解详细的运动，需分析 v‑t 或 a‑t 图的形状。
9. Determining Acceleration from a v‑t Graph | 从速度–时间图求加速度
For a straight‑line v‑ culate slope = (v₂ − v₁) / (t₂ − t₁). This gives the constant acceleration. For a curved v‑t graph, draw a tangent at the instant of interest. Construct a large right‑angled triangle on the tangent, and compute slope = Δv / Δt, using the triangle's vertical and horizontal lengths measured from the axes scales.
对于直线 v‑t 图，选取两个间隔较大的点并计算斜率 = (v₂ − v₁) / (t₂ − t₁)。这给出恒定的加速度。对于曲形 v‑t 图，在感兴趣的时刻画出切线。在切线上构造一个大的直角三角形，利用坐标轴标度读取垂直和水平长度，计算斜率 = Δv / Δt。
To obtain the change in velocity from an a‑t graph, whether uniform or non‑uniform, find the area between the graph and the time axis. A constant acceleration yields a rectangular area; a varying acceleration yields an area bounded by a curve. Counting squares or using simple shapes like trapeziums can provide accurate estimates.
无论匀加速还是非匀加速，要从 a‑t 图获得速度变化量，需找出图线与时间轴之间的面积。恒定加速度产生矩形面积；变化加速度产生由曲线围成的面积。数格子或使用梯形等简单形状可以提供准确的估算。
10. Examples: Free Fall and Falling with Air Resistance | 实例：自由落体与有空气阻力的下落
A classic example of uniform acceleration is free fall near the Earth' . Any object, regardless of mass, accelerates at g, and suvat equations describe its motion beautifully. A feather and a hammer dropped on the Moon demonstrate this perfectly.
匀加速的一个经典例子是近地表面无空气阻力时的自由落体，此时 a = g ≈ 9.81 m s⁻² 向下。无论质量如何，任何物体都以 g 加速，suvat 方程完美描述其运动。在月球上羽球和锤子同时落地的实验完美地证明了这一点。
When air resistance is significant, the acceleration is non‑uniform. As a skydiver falls, the air resistance increases with speed, reducing the resultant force and hence the acceleration from g to nearly zero at terminal velocity. The v‑t graph starts with slope g and gradually flattens, illustrating non‑uniform acceleration.
当空气阻力显著时，加速度为非匀加速。跳伞者下落时，空气阻力随速度增加，使得合力减小，加速度从 g 降至终端速度时几乎为零。v‑t 图起始斜率为 g，随后逐渐变平，体现了非匀加速的特征。
11. Experimental Methods | 实验方法
In the laboratory, uniform acceleration can be investigated using a dynamics trolley pulled by a constant force, or a glider on a linear air track. Measuring velocity at regular time intervals, for example with light gates or a motion sensor, allows you to plot a v–t graph. A straight line indicates constant acceleration; its slope gives a.
在实验室中，可以用恒力拉动的动力学小车，或线性气垫导轨上的滑块来研究匀加速。利用光门或运动传感器等时间间隔测量速度，可绘制 v–t 图。直线表明加速度恒定；其斜率给出 a。
Non‑uniform acceleration can be observed with a pendulum or a mass on a spring. A motion sensor records displacement as a function of time. By differentiating the s‑t data, you can obtain v‑t and a‑t graphs. The acceleration is greatest when the displacement is maximum, and zero at equilibrium, producing a sinusoidal variation.
非匀加速可以通过摆或弹簧振子观察。运动传感器记录位移随时间变化。通过对 s‑t 数据微分，可以获得 v‑t 与 a‑t 图。加速度在位移最大时最大，在平衡位置为零，呈现出正弦变化。
12. Summary and Exam Tips | 总结与考试技巧
mplified case where a = constant. The four suvat equations apply, and graphs are linear (v‑t) or quadratic (s‑t). Non‑uniform acceleration reflects real‑ reas on graphs. Always check whether acceleration is constant before choosing your method.
匀加速是 a = 常数的特殊情况。四个 suvat 方程适用，图形为直线 (v‑t) 或抛物线 (s‑t)。非匀加速反映真实世界中的力，需要对图形的斜率和面积进行分析。在选择方法之前，务必检查加速度是否恒定。
- Memorise the suvat equations and the conditions under which they are valid.
- For non‑uniform motions, use Δv = area under a–t, Δs = area under v–t.
- When drawing tangents to curves, use a sharp pencil, construct a large triangle and read the axes carefully.
- In free‑fall problems, g can be taken as 9.81 m s⁻² unless told otherwise.
- Distinguish clearly between instantaneous and average values.
- 熟记 suvat 方程及其适用条件。
- 对于非匀运动，使用 Δv = a–t 图下面积，Δs = v–t 图下面积。
- 画曲线切线时，使用尖锐铅笔，构造大的三角形并仔细读取坐标轴。
- 在自由落体问题中，除非另有说明，g 可取 9.81 m s⁻²。
- 明确区分瞬时值和平均值。
The interplay between uniform and non‑uniform acceleration deepens your understanding of Newton's laws and the kinematics of everyday objects. Practise interpreting graphs, as examiners often test your ability to extract a, v, and s from non‑linear plots.
匀加速与非匀加速之间的相互联系加深了你对牛顿定律和日常物体运动学的理解。要多练习解读图形，因为考官常测试你从非线性图中提取 a、v 和 s 的能力。
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