📚 A-Level Physics: Uniform and Non-Uniform Acceleration | 匀变速与非匀变速运动
In A-Level Physics, the study of motion begins with kinematics, the branch of mechanics that describes how objects move without considering the forces that cause the motion. Two fundamental categories of motion are uniform acceleration (constant acceleration) and non-uniform acceleration (changing acceleration). Understanding the distinction between these two is essential for solving problems involving projectiles, free fall, and variable force scenarios.
在 A-Level 物理中，运动学是力学的一个重要分支，它描述物体如何运动，而不考虑导致运动的原因。匀变速运动（加速度恒定）与非匀变速运动（加速度变化）是运动的两个基本类别。理解两者之间的区别，对于解决抛体运动、自由落体以及变力场景中的问题至关重要。
1. Defining Uniform and Non-Uniform Acceleration | 匀变速与非匀变速运动的定义
Uniform acceleration occurs when an object's velocity changes at a constant rate over time. This means the acceleration vector has a constant magnitude and direction. A classic example is an object in free fall near the Earth's surface, where air resistance is neglected and the acceleration due to gravity, g, is approximately 9.81 m s⁻².
匀变速运动是指物体的速度随时间以恒定速率变化，即加速度矢量的大小和方向都保持不变。一个典型的例子是地球表面附近的自由落体运动（忽略空气阻力），重力加速度 g 约为 9.81 m s⁻²。
Non-uniform acceleration, on the other hand, occurs when the acceleration itself changes over time. This can be due to a changing net force or a changing mass. For example, a rocket burning fuel experiences decreasing mass and thus changing acceleration, even if the thrust is constant.
非匀变速运动则是指加速度本身随时间发生变化。这可能是由于合外力变化或质量变化引起的。例如，火箭在燃烧燃料时质量不断减小，即使推力恒定，其加速度也会发生变化。
a = dv/dt = d²s/dt²
Where a is acceleration, v is velocity, and s is displacement. For uniform acceleration, a is a constant; for non-uniform acceleration, a is a function of time, displacement, or velocity.
其中 a 是加速度，v 是速度，s 是位移。对于匀变速运动，a 为常数；对于非匀变速运动，a 是时间、位移或速度的函数。
2. The Four Kinematic Equations for Uniform Acceleration | 匀变速运动的四个运动学方程
When acceleration is constant, the following equations (often called the "suvat" equations) apply:
当加速度恒定时，以下方程（通常称为 suvat 方程）适用：
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
Here, u is the initial velocity, v is the final velocity, a is the constant acceleration, t is the time, and s is the displacement.
其中 u 为初速度，v 为末速度，a 为恒定加速度，t 为时间，s 为位移。
These equations are only valid when the acceleration is uniform. In CIE A-Level exams, you must always check whether the acceleration is constant before applying them. For example, a car accelerating along a straight road with a constant engine force and no resistive forces is a good approximation of uniform acceleration.
这些方程仅在加速度均匀时成立。在 CIE A-Level 考试中，在使用这些方程之前，务必检查加速度是否恒定。例如，一辆汽车在直路上以恒定发动机动力且无阻力加速，可以近似视为匀变速运动。
3. Displacement-Time Graphs | 位移-时间图像
Graphical analysis is a key skill in A-Level Physics. For uniform acceleration, the displacement-time graph is a parabola. The gradient of the s-t graph gives the velocity at that instant.
图像分析是 A-Level 物理的重要技能。对于匀变速运动，位移-时间图像是一条抛物线。s-t 图像的斜率表示该时刻的速度。
For non-uniform acceleration, the s-t graph is not a simple parabola. Its slope changes in a more complex way, and the curvature is not constant. To interpret such graphs, you must use the tangent at a point to find instantaneous velocity.
对于非匀变速运动，s-t 图像不是简单的抛物线，其斜率变化更为复杂，曲率不恒定。要解释此类图像，必须使用某点的切线来求瞬时速度。
- Uniform acceleration: s-t graph is a parabola; gradient increases linearly with time.
- 匀变速：s-t 图像为抛物线，斜率随时间线性增加。
- Non-uniform acceleration: s-t graph has changing curvature; gradient changes non-linearly.
- 非匀变速：s-t 图像曲率变化，斜率非线性变化。
4. Velocity-Time Graphs | 速度-时间图像
The v-t graph is more directly useful for analysing acceleration. For uniform acceleration, the v-t graph is a straight line with a constant slope equal to the acceleration a.
速度-时间图像更直接地用于分析加速度。对于匀变速运动，v-t 图像是一条直线，其斜率恒定且等于加速度 a。
For non-uniform acceleration, the v-t graph is a curve. The gradient at any point gives the instantaneous acceleration. The area under the v-t graph represents the displacement, regardless of whether the acceleration is uniform or not.
对于非匀变速运动，v-t 图像是一条曲线。任意一点的斜率表示瞬时加速度。v-t 图像下的面积代表位移，无论加速度是否均匀都成立。
A common exam question involves comparing two objects: one with uniform acceleration and one with non-uniform acceleration. You may be asked to determine which object travels further in a given time, which requires you to compare areas under the curves rather than final velocities.
常见的考题是比较两个物体：一个匀变速，一个非匀变速。题目可能要求你判断在给定时间内哪个物体运动得更远，这需要比较曲线下的面积而非末速度。
5. Acceleration-Time Graphs | 加速度-时间图像
An a-t graph provides the clearest visual distinction between uniform and non-uniform acceleration. For uniform acceleration, the a-t graph is a horizontal line. For non-uniform acceleration, the a-t graph is not horizontal; it may be a straight line with a slope or a curve.
加速度-时间图像能最清晰地显示匀变速与非匀变速的区别。匀变速运动的 a-t 图像是一条水平线；非匀变速运动的 a-t 图像不是水平线，可能是斜线或曲线。
The area under an a-t graph gives the change in velocity. This is true for both uniform and non-uniform acceleration, but calculating the area is simpler when the graph is a simple shape.
a-t 图像下的面积表示速度的变化量。这适用于匀变速和非匀变速运动，但当图形为简单形状时，计算面积更为容易。
| Type of Motion | a-t graph | v-t graph | s-t graph |
| Uniform acceleration | Horizontal line | Straight line | Parabola |
| Non-uniform acceleration | Curve or sloped line | Curve | Non-parabolic curve |
| 匀变速 | 水平线 | 直线 | 抛物线 |
| 非匀变速 | 曲线或斜线 | 曲线 | 非抛物线 |
6. Free Fall and Projectile Motion as Examples of Uniform Acceleration | 自由落体与抛体运动：匀变速的实例
In the absence of air resistance, an object moving under gravity alone experiences a uniform acceleration of about 9.81 m s⁻² directed towards the centre of the Earth. This is true for both vertical free fall and projectile motion (where the horizontal component of velocity remains constant, while the vertical component changes uniformly).
在没有空气阻力的情况下，物体仅在重力作用下运动，其加速度约为 9.81 m s⁻²，方向指向地心。这对于垂直自由落体和抛体运动都成立（抛体运动中水平速度分量保持不变，而垂直分量均匀变化）。
ical motion follows the suvat equations with a = g, while the horizontal motion has a = 0.
抛体运动是 CIE A-Level 的经典考点。关键是将水平方向和垂直方向的运动分开处理。垂直方向遵循 a = g 的 suvat 方程，而水平方向 a = 0。
Time of flight = 2u sinθ / g
Maximum height = u² sin²θ / (2g)
Range = u² sin2θ / g
However, if air resistance is significant, the acceleration becomes non-uniform because the drag force depends on velocity, so these equations no longer apply exactly.
然而，如果空气阻力显著，加速度将变为非匀变速，因为阻力取决于速度，所以上述方程不再完全适用。
7. Non-Uniform Acceleration in Real-World Contexts | 现实世界中的非匀变速运动
Non-uniform acceleration is far more common in real life than uniform acceleration. Examples include:
在现实生活中，非匀变速运动远比匀变速运动常见。例如：
- A car coasting to a stop with variable air resistance and rolling resistance.
- 一辆汽车因变化的空气阻力和滚动阻力而滑行停止。
- A mass oscillating on a spring, where the acceleration is proportional to displacement but continuously changing direction and magnitude.
- 弹簧上振动的物体，其加速度与位移成正比，但方向和大小不断变化。
- A rocket ascending through the atmosphere, where both thrust and mass change over time.
- 火箭穿过大气层上升，推力和质量都随时间变化。
- An object falling with air resistance, which eventually reaches terminal velocity when acceleration becomes zero.
- 物体在空气中下落，受到阻力作用，最终达到终端速度时加速度变为零。
For such cases, the suvat equations cannot be used. Instead, you may need to use calculus, numerical methods, or graphical analysis. In CIE A-Level, you are often expected to interpret v-t or a-t graphs rather than derive exact equations.
对于这些情况，不能使用 suvat 方程。你需要使用微积分、数值方法或图像分析。在 CIE A-Level 中，通常要求你解读 v-t 或 a-t 图像，而不是推导精确方程。
8. Using Calculus to Handle Non-Uniform Acceleration | 用微积分处理非匀变速运动Published by TutorHao | A-Level Physics Revision Series | aleveler.com
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