📚 Accelerated Motion | 加速运动
Accelerated motion is a cornerstone of classical mechanics, describing how the velocity of an object changes with time under the influence of a net force. In CIE A-Level Physics, you are expected to analyse motion using graphs, equations, and vectors, with a strong focus on uniformly accelerated motion and its applications, including free fall and projectiles. A deep understanding of these concepts will not only help you solve numerical problems but also interpret real-world motion scenarios.
加速运动是经典力学的基石，描述物体在合力作用下速度如何随时间变化。在 CIE A-Level 物理中，你需要运用图像、方程和矢量来分析运动，重点掌握匀加速运动及其应用，包括自由落体和抛体运动。深入理解这些概念不仅能帮助你解决计算题，还能解释现实世界中的运动现象。
1. What Is Acceleration? | 什么是加速度？
Acceleration is defined as the rate of change of velocity with respect to time. It is a vector quantity, meaning it has both magnitude and direction. The SI unit of acceleration is metre per second squared (m s⁻²). If an object speeds up in the direction of motion, acceleration is positive; if it slows down, acceleration is negative (often called deceleration). Instantaneous acceleration is the gradient of a velocity–time graph at a specific instant, while average acceleration is total change in velocity divided by total time.
加速度定义为速度对时间的变化率。它是一个矢量，既有大小也有方向。加速度的国际单位是米每二次方秒（m s⁻²）。如果物体沿运动方向加速，加速度为正；如果减速，加速度为负（常称为减速度）。瞬时加速度是速度–时间图在某时刻的斜率，而平均加速度等于速度的总变化量除以总时间。
2. Scalar vs Vector Quantities in Motion | 运动中的标量与矢量
Distance and speed are scalar quantities; they only have magnitude. Displacement, velocity, and acceleration are vectors; they require both magnitude and direction. This distinction is crucial when dealing with motion in one or two dimensions. For example, a car travelling 100 m north undergoes a displacement of 100 m north, while the distance travelled is simply 100 m. In calculations, you must assign a positive direction and consistently use sign conventions for vectors.
路程和速率是标量，只有大小。位移、速度和加速度是矢量，既有大小又有方向。在处理一维或二维运动时，这一区别至关重要。例如，一辆汽车向北行驶 100 m，位移为向北 100 m，而路程仅为 100 m。在计算中，你必须规定正方向，并对矢量始终使用一致的符号规则。
3. Uniformly Accelerated Motion and the SUVAT Equations | 匀加速运动与 SUVAT 方程
When acceleration is constant, the motion is called uniformly accelerated. In this case, five key variables – displacement s, initial velocity u, final velocity v, acceleration a, and time t – are connected by a set of four kinematic equations often referred to as the SUVAT equations. These equations are derived from the definitions of velocity and acceleration, and they apply only when acceleration is constant.
当加速度恒定时，运动称为匀加速运动。此时，五个关键变量——位移 s、初速度 u、末速度 v、加速度 a 和时间 t——由一组四个运动学方程联系起来，常称为 SUVAT 方程。这些方程由速度和加速度的定义导出，仅适用于加速度恒定的情况。
The first equation links final velocity, initial velocity, acceleration and time:
v = u + at
方程一联系末速度、初速度、加速度和时间：
v = u + at
The second equation gives displacement as the average velocity multiplied by time:
s = ½(u + v)t
方程二将位移表示为平均速度乘以时间：
s = ½(u + v)t
The third equation expresses displacement in terms of initial velocity, acceleration and time:
s = ut + ½at²
方程三用初速度、加速度和时间表示位移：
s = ut + ½at²
The fourth equation relates final velocity, initial velocity, acceleration and displacement without involving time:
v² = u² + 2as
方程四联系末速度、初速度、加速度和位移而不涉及时间：
v² = u² + 2as
In problem solving, identify which three variables you know and which one you need to find, then select the equation that contains those four variables. Always ensure that the units are consistent – for instance, convert km h⁻¹ to m s⁻¹ when using SI units.
解题时，确定已知的三个变量和需要求出的变量，然后选择包含这四个变量的方程。务必保证单位一致——例如，使用国际单位制时应将 km h⁻¹ 转换为 m s⁻¹。
4. Interpreting Velocity–Time Graphs | 解读速度–时间图
A velocity–time graph is one of the most powerful tools in kinematics. The gradient of the graph represents acceleration: a straight sloping line indicates constant acceleration, while a curved line indicates changing acceleration. The area under the graph between two time values gives the displacement during that interval. If the graph crosses the time axis, the total area is the sum of the magnitudes of the areas above and below the axis, but displacement takes sign into account. For example, a triangle above the axis represents positive displacement, while one below represents negative displacement.
速度–时间图是运动学中最有力的工具之一。图线的斜率表示加速度：一条倾斜的直线表示加速度恒定，而曲线表示加速度在变化。图线下某段时间内的面积代表该段时间内的位移。如果图线穿过时间轴，总面积是轴上和轴下面积绝对值之和，但位移需考虑正负号。例如，轴上方的三角形代表正位移，轴下方的三角形代表负位移。
5. Displacement–Time Graphs and Their Slopes | 位移–时间图及其斜率
On a displacement–time graph, the gradient at any point gives the instantaneous velocity. A straight line indicates constant velocity, while a curve indicates acceleration or deceleration. If the gradient is increasing, the object is accelerating; if it is decreasing, the object is decelerating. Note that a horizontal line shows the object is at rest. The shape of the graph reveals much about the type of motion, and you should practise sketching graphs for different scenarios such as a ball thrown upwards or a car braking.
在位移–时间图上，任意点的斜率给出瞬时速度。直线表示匀速运动，曲线表示加速或减速。若斜率增大，物体在加速；若斜率减小，物体在减速。注意，水平线表示物体静止。图线的形状能揭示运动类型，你应当练习为不同情景绘制草图，例如上抛的小球或刹车的汽车。
6. Free Fall and Acceleration due to Gravity | 自由落体与重力加速度
Near the Earth’s surface, all objects in free fall experience a constant downward acceleration due to gravity, denoted g. The standard value is 9.81 m s⁻², though 9.8 m s⁻² or 10 m s⁻² may be used in exam questions. Since air resistance is negligible in ideal free fall, the motion can be analysed using the SUVAT equations with a = g (or a = –g, depending on your sign convention). For an object dropped from rest, u = 0, and the equations simplify considerably: v = gt, s = ½gt², and v² = 2gs.
在地球表面附近，所有自由落体都经历由重力引起的恒定向下加速度，记作 g。标准值为 9.81 m s⁻²，不过考题中可能使用 9.8 m s⁻² 或 10 m s⁻²。由于理想自由落体中空气阻力可忽略，可使用 SUVAT 方程分析，令 a = g（或 a = –g，取决于正方向规定）。对于从静止释放的物体，u = 0，方程大为简化：v = gt，s = ½gt²，v² = 2gs。
7. Projectile Motion: Resolving into Components | 抛体运动：分解为分量
A projectile is an object launched with an initial velocity and then allowed to move under gravity alone. The key to analysing projectile motion is to treat the horizontal and vertical motions independently. Horizontally, there is no acceleration (ignoring air resistance), so the horizontal component of velocity remains constant: v_x = u cos θ. Vertically, the motion is uniformly accelerated with a = –g (taking upwards as positive), giving v_y = u sin θ – gt. This separation allows you to use SUVAT equations separately in each direction.
抛体是以某一初速度抛出后仅在重力作用下运动的物体。分析抛体运动的关键是分别处理水平和竖直两个方向的运动。水平方向没有加速度（忽略空气阻力），因此水平速度分量保持不变：v_x = u cos θ。竖直方向是匀加速运动，a = –g（取向上为正），因此 v_y = u sin θ – gt。这种分离使你可以在每个方向上单独使用 SUVAT 方程。
8. Trajectory Equation and Range | 轨迹方程与射程
By eliminating time t from the horizontal displacement x = (u cos θ)t and the vertical displacement y = (u sin θ)t – ½gt², we obtain the trajectory equation:
y = x tan θ – (g x²) / (2 u² cos² θ)
This is a parabolic path. The maximum height H and the horizontal range R are given by:
H = (u² sin² θ) / (2g)
R = (u² sin 2θ) / g
从水平位移 x = (u cos θ)t 和竖直位移 y = (u sin θ)t – ½gt² 中消去时间 t，得到轨迹方程：
y = x tan θ – (g x²) / (2 u² cos² θ)
这是一条抛物线路径。最大高度 H 和水平射程 R 分别为：
H = (u² sin² θ) / (2g)
R = (u² sin 2θ) / g
The maximum range for a given launch speed is achieved at θ = 45°, a result often used in exam questions. Remember that these formulas assume launch and landing at the same height and zero air resistance.
对于给定的初速度，最大射程在 θ = 45° 时取得，这一结果常出现在考题中。记住，这些公式假设抛出点与落地点在同一高度，且空气阻力为零。
9. The Effect of Air Resistance and Terminal Velocity | 空气阻力的影响与终极速度
In real life, air resistance opposes motion and is velocity-dependent. As an object falls, air resistance increases until it balances the weight, after which the net force is zero and the object falls at a constant terminal velocity. This motion is no longer uniformly accelerated, so SUVAT equations cannot be used directly. Skydivers and raindrops are common examples. Understanding the shape of the velocity–time graph for a falling object with air resistance is a typical examination requirement.
在现实中，空气阻力阻碍运动且与速度有关。物体下落时，空气阻力不断增大，直到与重力平衡，此后合力为零，物体以恒定的终极速度下落。这种运动不再是匀加速运动，因此不能直接使用 SUVAT 方程。跳伞者和雨滴是常见的例子。理解存在空气阻力时下落物体的速度–时间图线形状是典型的考试要求。
10. Experimental Measurement of g | 通过实验测量 g
One standard method to determine g involves dropping a steel ball from a known height and using a trapdoor or light gates to measure the time of fall. By varying the height and recording the time, you can plot a graph of s against t² for the equation s = ½gt². The gradient of the straight line is ½g, from which g can be calculated. Alternatively, using a ticker-tape timer attached to a falling mass can provide a direct recording of displacement over equal time intervals, allowing calculation of acceleration.
测定 g 的一种标准方法是让钢球从已知高度下落，并使用活板门或光电门测量下落时间。通过改变高度并记录时间，你可以根据 s = ½gt² 绘制 s–t² 图。直线的斜率是 ½g，由此可计算出 g。另一种方法是，在重物下落时使用打点计时器，可以直接记录等时间间隔内的位移，从而计算加速度。
11. Common Misconceptions and Exam Tips | 常见误解与考试技巧
Many students mistakenly think that a positive acceleration always means speeding up; in fact, if the velocity is negative and acceleration is negative, the object speeds up in the negative direction. Also, at the highest point of a vertical throw, velocity is zero but acceleration is still g. In projectile questions, never mix horizontal and vertical components in the same SUVAT equation. Always draw a clear sign convention and label u, v, a, s, t for each direction. Practise switching between graphical, algebraic, and descriptive methods to deepen your understanding.
许多学生误以为正加速度总是意味着加速；实际上，如果速度为负且加速度也为负，物体是向负方向加速的。同样，在竖直上抛的最高点，速度为零但加速度仍为 g。在抛体问题中，切勿将水平和竖直分量混用在同一个 SUVAT 方程中。始终画出明确的符号规定，并为每个方向标出 u、v、a、s、t。多练习图形法、代数法和描述法之间的切换，以加深理解。
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