Accelerated Motion in A-Level Physics | A-Level 物理中的加速运动

📚 Accelerated Motion in A-Level Physics | A-Level 物理中的加速运动

In CIE A-Level Physics, accelerated motion is one of the foundational topics in mechanics. It connects the definitions of displacement, velocity and acceleration with graphical interpretation and the equations of uniformly accelerated motion. These tools are used to analyse objects falling under gravity, projectiles, and vehicles in multi-stage journeys.

在 CIE A-Level 物理中，加速运动是力学的基础主题之一。它将位移、速度和加速度的定义与图像解读以及匀加速运动方程联系起来。这些工具用来分析在重力作用下的落体、抛体以及多阶段运动的交通工具。

1. Motion Vocabulary and Acceleration | 运动术语与加速度

Displacement s is a vector quantity that describes an object’s change in position in a specified direction. Distance, by contrast, is a scalar and has no direction.

位移 s 是描述物体在指定方向上位置变化的矢量。而路程是标量，没有方向。

Velocity v is the rate of change of displacement. Acceleration a is the rate of change of velocity with respect to time.

速度 v 是位移的变化率。加速度 a 是速度随时间的变化率。

a = Δv / Δt

The SI unit of acceleration is m s⁻². Because velocity is a vector, acceleration is also a vector; it can involve a change in speed, a change in direction, or both.

加速度的 SI 单位是 m s⁻²。由于速度是矢量，加速度也是矢量；它可以涉及速率变化、方向变化，或两者兼有。

For uniform acceleration, the average acceleration equals the instantaneous acceleration, so a = (v − u) / t can be used with u as initial velocity and v as final velocity.

对于匀加速度，平均加速度等于瞬时加速度，因此可以用 a = (v − u) / t，其中 u 为初速度，v 为末速度。

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2. Velocity–Time Graphs: Gradient and Area | 速度–时间图：斜率与面积

A velocity–time graph is one of the most useful tools in CIE kinematics. The gradient at any point gives acceleration, and the area between the graph and the time axis gives displacement.

速度–时间图是 CIE 运动学中最有用的工具之一。任意一点的斜率给出加速度，图像与时间轴之间的面积给出位移。

If the graph is a straight line with positive gradient, the acceleration is constant and positive. A straight line with negative gradient indicates constant acceleration in the opposite direction.

如果图像是正斜率的直线，加速度恒定且为正。负斜率直线表示相反方向的恒定加速度。

The area under a velocity–time graph must be calculated with sign. Area above the time axis is positive displacement; area below the axis is negative displacement and reduces the net displacement.

计算速度–时间图下方面积时必须考虑正负号。时间轴上方面积为正位移；轴下方面积为负位移，会减少净位移。

For uniform acceleration from u to v over time t, the area is a trapezium, giving s = ½(u + v)t.

对于从 u 到 v、历时 t 的匀加速运动，面积是梯形，由此得 s = ½(u + v)t。

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3. Displacement–Time Graphs: Curvature and Velocity | 位移–时间图：曲率与速度

On a displacement–time graph, the gradient gives velocity. A straight line means constant velocity, while a curve means changing velocity, hence acceleration is present.

在位移–时间图上，斜率给出速度。直线表示速度恒定，曲线表示速度在变化，因此存在加速度。

If the curvature is concave upwards, the gradient is increasing, so the object is accelerating. If it is concave downwards, the gradient is decreasing, indicating acceleration in the opposite direction to motion or deceleration.

如果曲线向上凹，斜率在增加，物体在加速。如果曲线向下凹，斜率在减小，表明加速度与运动方向相反或减速。

It is important not to confuse the shape of a displacement–time graph with the path of the object; a parabolic curve does not mean the object moves in a parabola.

不要将位移–时间图的形状与物体的轨迹混淆；抛物线形曲线并不意味着物体沿抛物线路径运动。

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4. The SUVAT Equations of Uniformly Accelerated Motion | 匀加速运动的 SUVAT 方程

For motion in a straight line with constant acceleration, CIE expects candidates to recall and use the four SUVAT equations. The symbols are s = displacement, u = initial velocity, v = final velocity, a = constant acceleration and t = time.

对于加速度恒定的直线运动，CIE 要求考生记住并使用四个 SUVAT 方程。符号分别为：s 位移、u 初速度、v 末速度、a 恒定加速度、t 时间。

v = u + at

s = ut + ½at²

s = ½(u + v)t

v² = u² + 2as

These equations can only be used when acceleration is constant. If acceleration changes or the motion has distinct stages, each stage must be analysed separately.

这些方程只在加速度恒定时才能使用。如果加速度变化或运动分为不同阶段，则必须分段分析。

Always identify the known quantities and the unknown before selecting the equation that avoids the variable you do not need.

在选择方程前，先确定已知量和未知量，选择不包含不需要变量的方程。

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5. Deriving the SUVAT Equations Graphically | 用图像推导 SUVAT 方程

The four equations can be derived from the definition of acceleration and the area under a velocity–time graph. The first follows directly from a = (v − u) / t, rearranged as v = u + at.

四个方程可以由加速度定义和速度–时间图下方面积推导出来。第一个直接来自 a = (v − u) / t，重新整理为 v = u + at。

The displacement equation s = ½(u + v)t comes from the trapezium area under the velocity–time line.

位移方程 s = ½(u + v)t 来自速度–时间直线下的梯形面积。

Substituting v = u + at into s = ½(u + v)t gives s = ut + ½at². Eliminating t between v = u + at and s = ½(u + v)t gives v² = u² + 2as.

将 v = u + at 代入 s = ½(u + v)t 得到 s = ut + ½at²。在 v = u + at 与 s = ½(u + v)t 之间消去 t 得到 v² = u² + 2as。

Understanding these derivations helps when a question is set in unfamiliar notation or when one equation appears to fail because time is unknown.

理解这些推导有助于应对使用不熟悉符号的题目，或在时间未知看似无法用方程时提供思路。

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6. Free Fall and the Acceleration due to Gravity | 自由落体与重力加速度

An object in free fall moves under gravity alone with negligible air resistance. Near the Earth’s surface, the acceleration of free fall g is approximately 9.81 m s⁻² downwards.

物体在自由落体时仅在重力作用下运动，空气阻力可忽略。在地球表面附近，自由落体加速度 g 约为 9.81 m s⁻²，方向向下。

If the upward direction is chosen as positive, the acceleration of a falling object is a = −g = −9.81 m s⁻². The same sign convention must be applied to u, v and s.

如果规定向上为正，则下落物体的加速度 a = −g = −9.81 m s⁻²。同样的符号约定必须应用于 u、v 和 s。

An object thrown vertically upwards with speed u reaches maximum height when v = 0. Its maximum height is h = u² / (2

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