📚 Uniform and Non-uniform Acceleration | 匀加速与非匀加速运动
Acceleration is one of the core ideas in CIE A-Level kinematics. This article explains the difference between uniform acceleration, where the acceleration stays constant, and non-uniform acceleration, where it changes with time. It also covers the equations, graphs, and exam skills needed to solve problems confidently.
加速度是 CIE A-Level 运动学中的核心概念之一。本文解释匀加速运动（加速度保持恒定）与非匀加速运动（加速度随时间变化）之间的区别，并涵盖解题所需的方程、图像和考试技巧。
1. Defining Acceleration | 加速度的定义
Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, so both its magnitude and direction matter. In symbols, average acceleration is given by a = Δv ÷ Δt, where Δv is the change in velocity and Δt is the time interval. The SI unit is metre per second squared, written m s⁻².
加速度是速度随时间的变化率。它是一个矢量，因此大小和方向都很重要。用符号表示，平均加速度为 a = Δv ÷ Δt，其中 Δv 是速度的变化量，Δt 是时间间隔。国际单位是米每二次方秒，写作 m s⁻²。
In CIE A-Level problems, positive and negative signs indicate direction. A negative acceleration is often called deceleration or retardation when the velocity decreases in the chosen positive direction.
在 CIE A-Level 题目中，正负号表示方向。当速度沿选定的正方向减小时，负加速度通常称为减速或 retardation。
2. Uniform Acceleration: Key Features | 匀加速运动的关键特征
Uniform acceleration means the acceleration is constant: neither its magnitude nor its direction changes. The velocity-time graph is therefore a straight line with a constant gradient.
匀加速运动意味着加速度恒定：大小和方向都不改变。因此，速度-时间图是一条具有恒定斜率的直线。
With constant acceleration, the velocity changes by equal amounts in equal time intervals. This does not mean the object moves at constant speed; it means the rate of speeding up or slowing down is steady.
在加速度恒定的情况下，速度在相等的时间间隔内变化相同的量。这并不意味着物体做匀速运动；它表示加速或减速的快慢是稳定的。
3. The SUVAT Equations | 匀加速运动方程（SUVAT）
For motion in a straight line with uniform acceleration, four equations link displacement s, initial velocity u, final velocity v, acceleration a and time t:
对于匀加速直线运动，四个方程将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来：
v = u + at
s = ½(u + v)t
s = ut + ½at²
v² = u² + 2as
These equations are only valid when acceleration is constant. Each question should list known quantities and choose the equation that omits the unknown variable.
这些方程仅在加速度恒定时有效。解题时应列出已知量，并选择不含未知量的方程。
The SUVAT equations are derived from the definitions of velocity and acceleration, assuming a straight line and constant a.
SUVAT 方程由速度和加速度的定义推导而来，假设直线运动且 a 恒定。
4. Velocity-Time Graphs for Uniform Acceleration | 匀加速运动的速度-时间图
For uniform acceleration, the velocity-time graph is a straight line. The gradient of the line equals the acceleration a = Δv ÷ Δt. The area between the line and the time axis equals the displacement.
对于匀加速运动，速度-时间图是一条直线。直线的斜率等于加速度 a = Δv ÷ Δt。直线与时间轴之间的面积等于位移。
If the line slopes upward, acceleration is positive in the chosen direction. If it slopes downward, acceleration is negative, which may represent slowing down or speeding up in the opposite direction.
如果直线向上倾斜，加速度在选定方向为正。如果向下倾斜，加速度为负，可能表示减速或沿相反方向加速。
For example, a line from (0, 2 m s⁻¹) to (4 s, 10 m s⁻¹) has gradient (10 − 2) ÷ 4 = 2 m s⁻². The area is ½(2 + 10) × 4 = 24 m.
例如，从 (0, 2 m s⁻¹) 到 (4 s, 10 m s⁻¹) 的直线斜率为 (10 − 2) ÷ 4 = 2 m s⁻²。面积为 ½(2 + 10) × 4 = 24 m。
5. Non-uniform Acceleration: Changing Acceleration | 非匀加速运动：变化的加速度
Non-uniform acceleration occurs when the acceleration changes with time, either in magnitude, direction, or both. The velocity-time graph is then curved, not a straight line.
当加速度随时间变化时（大小、方向或两者都变化），就出现非匀加速运动。此时速度-时间图是曲线，而不是直线。
In such motion, the instantaneous acceleration at any moment is the gradient of the tangent to the velocity-time curve at that point.
在这种运动中，任一时刻的瞬时加速度是该点速度-时间曲线的切线斜率。
Many real motions are non-uniform, such as a car in traffic or a rocket whose mass decreases as fuel burns.
许多真实运动是非匀加速的，例如交通中的汽车，或质量随燃料燃烧而减小的火箭。
6. Velocity-Time Graphs for Non-uniform Motion | 非匀加速运动的速度-时间图
A curved velocity-time graph tells us how acceleration changes. Where the curve becomes steeper, the magnitude of acceleration increases. Where it flattens, the magnitude of acceleration decreases toward zero.
弯曲的速度-时间图可以告诉我们加速度如何变化。曲线越陡，加速度的大小越大；曲线越平缓，加速度的大小就越小并趋近于零。
The gradient of a tangent gives instantaneous acceleration, while the area under the curve still gives total displacement. Areas may need to be estimated by counting squares or using geometric approximations if the curve is irregular.
切线斜率给出瞬时加速度，而曲线下面积仍然给出总位移。如果曲线不规则，面积可能需要通过数格或几何近似来估算。
7. Acceleration-Time Graphs | 加速度-时间图
An acceleration-time graph is another useful tool. For uniform acceleration, it is a horizontal line at the constant value a. For non-uniform acceleration, it is a curve or a changing line.
加速度-时间图是另一个有用的工具。对于匀加速运动，它是一条位于恒定值 a 处的水平线。对于非匀加速运动，它是一条曲线或变化的线。
The area under an acceleration-time graph between two times gives the change in velocity Δv. This follows from a = Δv ÷ Δt, so Δv = a × Δt.
加速度-时间图在两个时刻之间的面积给出速度变化量 Δv。这由 a = Δv ÷ Δt 得出，因此 Δv = a × Δt。
A horizontal acceleration-time graph therefore produces the straight-line velocity-time graph characteristic of uniform acceleration.
因此，水平的加速度-时间图会产生匀加速运动特有的直线速度-时间图。
8. Free Fall as Uniform Acceleration | 自由落体作为匀加速运动
Near the Earth’s surface, when air resistance is negligible, all objects fall with the same uniform acceleration due to gravity, g = 9.81 m s⁻² downward. This is an important example of uniform acceleration.
在地球表面附近，当空气阻力可以忽略时，所有物体都以相同的重力加速度 g = 9.81 m s⁻² 竖直下落。这是匀加速运动的一个重要例子。
For a falling object released from rest, u = 0, a = g, and the SUVAT equations become v = gt and s = ½gt². If upward is chosen as positive, then a = -g.
对于从静止释放的下落物体，u = 0，a = g，SUVAT 方程变为 v = gt 和 s = ½gt²。如果选向上为正，则 a = -g。
Projectile motion also involves uniform acceleration vertically (g) and constant horizontal velocity, assuming no air resistance.
抛体运动也涉及竖直方向的匀加速运动（g）和水平方向的匀速运动，假设没有空气阻力。
9. Terminal Velocity and Non-uniform Acceleration | 终速与非匀加速运动
When a skydiver or falling sphere moves through air, air resistance increases with speed. The resultant force and acceleration decrease as speed increases, so the motion is non-uniform.
当跳伞者或下落球体穿过空气时，空气阻力随速度增大而增大。合力与加速度随速度增大而减小，因此运动是非匀加速的。
Eventually, air resistance equals the weight. The resultant force is zero, so the acceleration becomes zero and the object falls at a constant terminal velocity.
最终，空气阻力等于重力。合力为零，因此加速度变为零，物体以恒定的终速下落。
On a velocity-time graph, the gradient is initially steep, then becomes shallower, and finally reaches zero at terminal velocity. This curve is a classic example of non-uniform acceleration.
在速度-时间图上，斜率起初很陡，随后变缓，最终在终速时变为零。这条曲线是非匀加速运动的典型例子。
10. Calculating Non-uniform Acceleration from Graphs | 从图像计算非匀加速运动
If a velocity-time graph is curved, we cannot use SUVAT equations directly. Instead, draw a tangent at the required time and calculate its gradient. Use a large tangent triangle to reduce uncertainty.
如果速度-时间图是曲线，我们不能直接使用 SUVAT 方程。而应在所需时刻画切线，并计算其斜率。使用较大的切三角形可以减小误差。
For example, if a tangent at t = 3 s passes through (3 s, 5 m s⁻¹) and (5 s, 11 m s⁻¹), the instantaneous acceleration is (11 − 5) ÷ (5 − 3) = 3 m s⁻².
例如，如果 t = 3 s 处的切线经过 (3 s, 5 m s⁻¹) 和 (5 s, 11 m s⁻¹)，则瞬时加速度为 (11 − 5) ÷ (5 − 3) = 3 m s⁻²。
To estimate distance travelled, count squares under the curve and multiply by the value represented by one square, or split the area into approximate trapeziums.
要估算运动距离，可以数曲线下方的方格并乘以每一格所代表的数值，或将面积分割为近似梯形。
11. Worked Example: Comparing the Two Types | 例题：两种运动对比
A train starts from rest and accelerates uniformly at 2 m s⁻² for 10 s. Its final velocity is v = u + at = 0 + 2 × 10 = 20 m s⁻¹, and its displacement is s = ½at² = ½ × 2 × 10² = 100 m.
一列火车从静止开始，以 2 m s⁻² 匀加速运动 10 s。它的末速度为 v = u + at = 0 + 2 × 10 = 20 m s⁻¹，位移为 s = ½at² = ½ × 2 × 10² = 100 m。
If instead the train’s acceleration decreases from 2 m s⁻² to 0 m s⁻² over the same 10 s, the motion is non-uniform. The final velocity and distance cannot be found by simple SUVAT equations alone; they must be obtained from a graph or by integration if calculus is used.
如果火车的加速度在同样的 10 s 内从 2 m s⁻² 减小到 0 m s⁻²，则运动是非匀加速的。末速度和距离不能仅靠简单的 SUVAT 方程求出；必须通过图像或积分（如果使用微积分）获得。
This contrast shows why identifying whether acceleration is constant is the first step in any CIE kinematics problem.
这一对比说明，判断加速度是否恒定是解决任何 CIE 运动学问题的第一步。
12. Common Misconceptions and Exam Tips | 常见误区与考试技巧
A common mistake is using SUVAT equations when acceleration is not constant. Always check the question for a changing force, changing mass or a curved v-t graph. If acceleration varies, use graphical methods.
一个常见错误是在加速度不恒定时使用 SUVAT 方程。务必检查题目是否存在变化的力、变化的质量或弯曲的 v-t 图。如果加速度变化，应使用图像法。
Another mistake is confusing velocity and acceleration. A positive acceleration does not always mean speeding up if the velocity is negative. It means the velocity is becoming more positive.
另一个错误是混淆速度和加速度。如果速度为负，正加速度并不总是意味着加速。它表示速度正在变得更正。
In CIE exams, show your working clearly, state the positive direction, and include units. For graph questions, draw a ruler line for tangents and label the triangle used.
在 CIE 考试中，要清晰展示解题过程，说明正方向，并标注单位。对于图像题，应画出切线的直尺线，并标出所使用的三角形。
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