📚 Kinematics - Describing Motion | 运动学——描述运动
Kinematics is the branch of mechanics that describes how objects move without considering the forces that cause the motion. In CIE A-Level Physics, you need to define motion quantities precisely, interpret displacement-time, velocity-time and acceleration-time graphs, and apply the equations of uniformly accelerated motion to solve problems.
运动学是力学的一个分支，描述物体如何运动而不考虑引起运动的力。在 CIE A-Level 物理中，你需要准确定义运动量，解读位移-时间、速度-时间和加速度-时间图像，并应用匀加速运动方程解决问题。
1. Scalars and Vectors | 标量与矢量
In kinematics, all physical quantities are classified as scalars or vectors. A scalar has magnitude only, while a vector has both magnitude and direction. This distinction is essential because combining or resolving vector quantities requires direction as well as size.
在运动学中，所有物理量都分为标量和矢量。标量只有大小，而矢量既有大小又有方向。这一区别至关重要，因为合成或分解矢量时既需要方向也需要大小。
Common scalar quantities include distance, speed, mass, time and energy. Common vector quantities include displacement, velocity, acceleration and force. In CIE exams, you may be asked to state whether a given quantity is a scalar or a vector and to explain the difference.
常见的标量包括路程、速率、质量、时间和能量。常见的矢量包括位移、速度、加速度和力。在 CIE 考试中，你可能需要判断某一物理量是标量还是矢量，并解释两者的区别。
| Scalars 标量 | Vectors 矢量 |
|---|---|
| distance 路程 | displacement 位移 |
| speed 速率 | velocity 速度 |
| mass 质量 | acceleration 加速度 |
| time 时间 | force 力 |
2. Distance and Displacement | 路程与位移
Distance is the total length of the path travelled by an object. It is a scalar quantity, so it is always positive and depends on the actual route taken between two points.
路程是物体所经过路径的总长度。它是标量，因此总是正值，并且取决于两点之间实际经过的路线。
Displacement is the straight-line change in position from the initial point to the final point. It is a vector quantity, so it has both magnitude and direction. Displacement can be positive, negative or zero, and it does not depend on the path taken.
位移是从起点到终点的直线位置变化。它是矢量，因此既有大小又有方向。位移可以是正值、负值或零，并且与所经过的路径无关。
For example, if a runner completes one full lap of a circular track of radius r, the distance travelled is 2πr, but the displacement is zero because the final position is the same as the initial position. If the runner stops halfway around, the distance is πr, but the displacement is 2r in a specified direction across the circle.
例如，如果一名跑步者绕半径为 r 的圆形跑道跑完一整圈，路程为 2πr，但位移为零，因为最终位置与初始位置相同。如果跑步者停在半圈处，路程为 πr，但位移为 2r，方向为穿过圆心的指定方向。
3. Speed and Velocity | 速率与速度
Speed is the rate of change of distance. It is a scalar quantity and has magnitude only. Average speed is calculated by dividing the total distance travelled by the total time taken.
速率是路程的变化率。它是标量，只有大小。平均速率等于总路程除以总时间。
average speed = total distance ÷ total time
Velocity is the rate of change of displacement. It is a vector quantity, so it has both magnitude and direction. Average velocity is calculated by dividing the total displacement by the total time taken.
速度是位移的变化率。它是矢量，因此既有大小又有方向。平均速度等于总位移除以总时间。
average velocity = total displacement ÷ total time
A body moving at constant speed is not necessarily moving at constant velocity. If the direction of motion changes, the velocity changes even though the speed stays the same. Uniform velocity requires both constant speed and motion in a straight line.
以恒定速率运动的物体不一定以恒定速度运动。如果运动方向改变，即使速率不变，速度也会改变。匀速运动要求速率恒定且沿直线运动。
4. Acceleration | 加速度
Acceleration is the rate of change of velocity. It is a vector quantity, so direction matters. The average acceleration of an object is given by the change in velocity divided by the time taken.
加速度是速度的变化率。它是矢量，因此方向很重要。物体的平均加速度等于速度变化量除以所用时间。
a = Δv ÷ Δt
The SI unit of acceleration is metres per second squared, written m s⁻². If an object slows down, its acceleration is in the opposite direction to its velocity. This is often called deceleration or negative acceleration when the chosen positive direction is the direction of motion.
加速度的国际单位是米每二次方秒，写作 m s⁻²。如果物体减速，其加速度方向与速度方向相反。当选取运动方向为正方向时，这通常称为减速或负加速度。
Uniform acceleration means the velocity changes by equal amounts in equal time intervals. In many CIE kinematics problems, acceleration is constant, such as a car accelerating uniformly or a ball falling freely under gravity.
匀加速意味着在相等的时间间隔内速度的变化量相等。在许多 CIE 运动学问题中，加速度是恒定的，例如汽车匀加速运动或球在重力作用下自由下落。
5. Displacement-Time Graphs | 位移-时间图像
A displacement-time graph shows how displacement changes with time. The gradient of the graph at any point gives the instantaneous velocity. A straight line indicates constant velocity, while a curved line indicates changing velocity, meaning the object is accelerating or decelerating.
位移-时间图像展示位移随时间的变化。图像上任意一点的斜率表示瞬时速度。直线表示速度恒定，曲线表示速度在变化，即物体正在加速或减速。
A horizontal line on a displacement-time graph means the displacement is not changing, so the velocity is zero. A positive gradient means motion in the positive direction, and a negative gradient means motion in the negative direction.
位移-时间图像上的水平线表示位移不变化，因此速度为零。正斜率表示向正方向运动，负斜率表示向负方向运动。
The steepness of the line or curve shows the magnitude of the velocity. A steeper gradient means a higher speed. If the gradient changes from positive to negative, the object changes direction at that instant.
直线或曲线的陡峭程度表示速度的大小。斜率越陡，速率越高。如果斜率从正变为负，物体在该时刻改变运动方向。
6. Velocity-Time Graphs | 速度-时间图像
A velocity-time graph shows how velocity changes with time. The gradient of the graph at any point gives the instantaneous acceleration. A horizontal line represents constant velocity, a straight sloping line represents constant acceleration, and a curved line represents changing acceleration.
速度-时间图像展示速度随时间的变化。图像上任意一点的斜率表示瞬时加速度。水平线表示匀速运动，倾斜直线表示匀加速运动，曲线表示加速度在变化。
The area between the graph and the time axis represents displacement. Area above the time axis is positive displacement, and area below the time axis is negative displacement. To find total displacement, add the signed areas; to find total distance travelled, add the magnitudes of all areas.
图像与时间轴之间的面积表示位移。时间轴上方的面积为正位移，时间轴下方的面积为负位移。求总位移时，将带符号的面积相加；求总路程时，将所有面积的大小相加。
ine is a triangle, so the displacement is ½ × base × height = ½ × 10 s × 20 m s⁻¹ = 100 m.
例如，一个速度-时间图像在 10 s 内从 0 匀速增加到 20 m s⁻¹，表示匀加速运动。图线下的面积为三角形，因此位移为 ½ × 底 × 高 = ½ × 10 s × 20 m s⁻¹ = 100 m。
7. Acceleration-Time Graphs | 加速度-时间图像
An acceleration-time graph shows how acceleration changes with time. The area under the graph between two times gives the change in velocity during that interval. A horizontal line represents constant acceleration, and a line along the time axis represents zero acceleration, meaning constant velocity.
加速度-时间图像展示加速度随时间的变化。图像下方两个时间点之间的面积表示该时间间隔内的速度变化量。水平线表示加速度恒定，沿时间轴的直线表示加速度为零，即速度恒定。
ity is simply acceleration × time, which matches the area of the rectangle under the line.
对于匀加速运动的物体，加速度-时间图像是一条水平直线。速度的变化量等于加速度 × 时间，这与图像下方矩形的面积一致。
The gradient of an acceleration-time graph is called jerk, which is the rate of change of acceleration. Jerk is not usually required in CIE A-Level kinematics, but you should know that it exists and is not needed for standard motion problems.
加速度-时间图像的斜率称为急动度，即加速度的变化率。CIE A-Level 运动学通常不要求计算急动度，但你应该知道它的存在，标准运动问题中不需要使用。
8. Equations of Uniformly Accelerated Motion | 匀加速运动方程
For motion in a straight line with constant acceleration a, the following four equations link the five quantities s, u, v, a and t. In these equations, s is displacement, u is initial velocity, v is final velocity, a is constant acceleration, and t is time.
对于加速度 a 恒定的直线运动，以下四个方程联系了五个物理量 s、u、v、a 和 t。在这些方程中，s 是位移，u 是初速度，v 是末速度，a 是恒定加速度，t 是时间。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
These equations are only valid when acceleration is constant and the motion is in a straight line. Before using them, you must choose a positive direction. Quantities acting in the positive direction are positive, and quantities acting in the opposite direction are negative.
这些方程仅在加速度恒定且运动为直线运动时成立。使用前必须先规定正方向。沿正方向的物理量为正，沿反方向的物理量为负。
= 0 + (2.0)(5.0) = 10 m s⁻¹. Its displacement is s = ut + ½at² = 0 + ½(2.0)(5.0)² = 25 m.
例题：一辆汽车从静止开始以 2.0 m s⁻² 匀加速运动 5.0 s。它的末速度为 v = u + at = 0 + (2.0)(5.0) = 10 m s⁻¹。它的位移为 s = ut + ½at² = 0 + ½(2.0)(5.0)² = 25 m。
9. Free Fall and Projectile Motion | 自由落体与抛体运动
A body falling freely near the Earth' 81 m s⁻² unless the question specifies otherwise. Free fall acceleration is independent of mass in the absence of air resistance.
在地球表面附近自由下落的物体具有恒定的向下加速度，等于自由落体加速度 g。在 CIE 考试中，除非题目另有说明，取 g = 9.81 m s⁻²。在没有空气阻力的情况下，自由落体加速度与质量无关。
must apply the same sign convention to displacement, initial velocity and final velocity.
当取向上为正方向时，自由落体物体的加速度为 a = -9.81 m s⁻²。当取向下为正方向时，加速度为 a = +9.81 m s⁻²。你必须对位移、初速度和末速度使用相同的正方向规定。
In projectile motion, the horizontal and vertical components of motion are independent. If air resistance is negligible, the horizontal velocity remains constant, while the vertical motion is uniformly accelerated by gravity. The horizontal motion has zero acceleration, and the vertical motion has acceleration g downward.
在抛体运动中，水平方向和竖直方向的分运动相互独立。如果空气阻力可忽略，水平速度保持不变，而竖直方向以重力加速度 g 匀加速运动。水平方向加速度为零，竖直方向加速度为向下的 g。
At the highest point of a projectile, the vertical velocity is zero, but the horizontal velocity is still the same as at launch. The time to reach the highest point is found from the vertical component, and the total time of flight is twice this time for a projectile launched and landing at the same height.
在抛体的最高点，竖直速度为零，但水平速度仍与抛出时相同。到达最高点的时间由竖直分量求得，对于从同一高度抛出并落地的抛体，总飞行时间是该时间的两倍。
For a projectile launched from ground level with initial speed u at angle θ to the horizontal, the time of flight and horizontal range are:
对于从地面以初速度 u、与水平方向成 θ 角抛出的物体，飞行时间和水平射程为：
T = 2u sinθ ÷ g
R = u² sin(2θ) ÷ g
10. Practical Skills and Exam Tips | 实验技能与应试技巧
Common practical methods for studying motion include using a light gate connected to a data logger, using a motion sensor, using ticker tape with a vibrating marker, or analysing video frames. These methods measure displacement and time so that velocity and acceleration can be calculated.
研究运动的常见实验方法包括使用连接数据采集器的光门、运动传感器、带振动打点的纸带，或分析视频帧。这些方法测量位移和时间，从而计算出速度和加速度。
Δt. Alternatively, plot a velocity-time graph and find the gradient of the straight line.
实验测定加速度时，可以测量初速度、末速度和所用时间，然后使用 a = Δv ÷ Δt。或者绘制速度-时间图像，求出直线的斜率。
Exam tips: always define your positive direction before using suvat equations; convert quantities to SI units such as metres, seconds and metres per second; do not confuse displacement with distance; and in projectile problems, resolve the motion into horizontal and vertical components and work with them separately.
应试技巧：使用匀加速运动方程前先规定正方向；将物理量换算为国际单位制，如米、秒和米每秒；不要混淆位移与路程；在抛体问题中，将运动分解为水平和竖直分量并分别处理。
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