Speed and Velocity | 速率与速度

📚 Speed and Velocity | 速率与速度

In A-Level Physics, speed and velocity are two distinct but closely related concepts. Speed is the rate at which an object covers distance and is a scalar quantity, meaning it has only magnitude. Velocity, however, is the rate of change of displacement and is a vector quantity, possessing both magnitude and direction. A thorough understanding of these ideas is essential for mastering kinematics, interpreting motion graphs, and applying equations of motion under constant acceleration, as required by the CIE syllabus.

在A-Level物理中，速率和速度是两个不同但又紧密关联的概念。速率是物体通过距离的快慢，属于标量，只有大小。而速度是位移的变化率，属于矢量，既有大小又有方向。透彻理解这些概念对于掌握运动学、解读运动图像以及应用匀加速运动方程至关重要，这也是CIE教学大纲的要求。

1. Scalar and Vector Quantities | 标量与矢量

All physical quantities in mechanics can be classified as either scalars or vectors. A scalar has only magnitude, for example mass, temperature, distance, and speed. A vector has both magnitude and direction, such as displacement, velocity, acceleration, and force. The distinction directly affects how we combine quantities: scalars add by ordinary arithmetic, whereas vectors must be added using vector rules that account for direction.

力学中的所有物理量都可以分为标量和矢量。标量只有大小，例如质量、温度、距离和速率。矢量既有大小又有方向，例如位移、速度、加速度和力。这一区分直接影响我们如何合成量：标量可用普通算术相加，而矢量必须按照矢量法则，考虑方向后再相加。

In CIE examinations, you are often required to state whether a given quantity is scalar or vector and to perform vector addition or resolution. Speed, as a scalar, is always non‑negative. Velocity, as a vector, can take positive or negative values depending on the chosen coordinate system.

在CIE考试中，常会要求你判断某个物理量是标量还是矢量，并进行矢量加法或分解。速率是标量，总是非负。速度是矢量，根据所选坐标方向，取值可为正或负。

Scalar 标量	Vector 矢量
Distance (距离)	Displacement (位移)
Speed (速率)	Velocity (速度)
Mass (质量)	Weight (重量)
Energy (能量)	Momentum (动量)

2. Defining Speed | 定义速率

Speed (v) is defined as the distance travelled per unit time. Because it uses the total length of the path taken by the object, speed does not tell us anything about the direction of motion. The SI unit of speed is metres per second (m s⁻¹), though everyday contexts also use km h⁻¹.

速率(v)定义为每单位时间内通过的距离。由于它使用的是物体经过的总路径长度，速率不提供任何关于运动方向的信息。速率的国际单位是米每秒(m s⁻¹)，日常生活中也会使用千米每小时(km h⁻¹)。

Mathematically, if an object moves with uniform speed:

数学上，若物体做匀速运动：

v = d / t

where d is the total distance and t is the time taken. Speed is always a positive number because distance can never be negative.

其中d是总距离，t是所用时间。速率永远为正数，因为距离不可能为负。

Even for non‑uniform motion, the average speed can be calculated using the total distance divided by the total time. This average speed provides no detail about variations during the journey.

即使是非匀速运动，也可以用总距离除以总时间计算平均速率。这个平均速率并不能反映行程中速率的变化细节。

3. Average and Instantaneous Speed | 平均速率与瞬时速率

Average speed is a useful overall measure, but it hides changes in motion. For example, a car that travels 120 km in 2 hours has an average speed of 60 km h⁻¹, yet it might have stopped at traffic lights or exceeded the speed limit at some points.

平均速率是一个有用的整体度量，但它掩盖了运动的变化。例如，一辆车用2小时行驶了120公里，平均速率为60 km h⁻¹，但它可能在红绿灯处停过或在某些路段超速。

Instantaneous speed is the speed of an object at a particular moment. When you look at a speedometer, you are seeing instantaneous speed. In physics, instantaneous speed is the magnitude of the instantaneous velocity, and it can be found from the gradient of a distance–time graph at a specific point.

瞬时速率是物体在某一特定时刻的速率。你看车速表时，看到的就是瞬时速率。在物理中，瞬时速率是瞬时速度的大小，可以从距离–时间图在某一点的斜率求得。

On a distance–time graph, the gradient Δd/Δt gives the speed. A steeper line indicates a higher speed; a horizontal line means the object is stationary (zero speed).

在距离–时间图上，斜率Δd/Δt给出速率。越陡的线表示速率越高；水平线表示物体静止（速率为零）。

4. Defining Velocity | 定义速度

Velocity (v) is the rate of change of displacement. Since displacement is a vector, velocity is also a vector. The SI unit is the same as for speed (m s⁻¹), but velocity includes a direction or a sign (+/−) according to a chosen positive direction.

速度(v)是位移的变化率。因为位移是矢量，所以速度也是矢量。国际单位与速率相同(m s⁻¹)，但速度包含方向，或根据选定的正方向带有正负符号。

For motion in a straight line with constant velocity:

对于匀速直线运动：

v = Δx / Δt

where Δx is the change in displacement. If an object returns to its starting point, the total displacement is zero, and thus the average velocity over the whole trip is zero—even though the average speed is not zero.

其中Δx是位移的变化量。如果物体回到起点，总位移为零，因此全程的平均速度为零——尽管平均速率不为零。

Velocity can be positive or negative, indicating the direction of motion. For instance, if east is taken as positive, a velocity of +5 m s⁻¹ means 5 m s⁻¹ east, and −5 m s⁻¹ means 5 m s⁻¹ west.

速度可以为正或负，表示运动方向。例如，若规定向东为正，速度+5 m s⁻¹表示向东以5 m s⁻¹运动，而−5 m s⁻¹表示向西以5 m s⁻¹运动。

5. Displacement, Distance and Sign Convention | 位移、距离与符号规定

The difference between distance and displacement is fundamental. Distance is the total length of the path, always increasing and never negative. Displacement is the straight‑line length from the initial to the final position, with a direction.

距离与位移的区别是根本性的。距离是路径的总长度，始终增加且永不为负。位移是从起点到终点的直线长度，并带有方向。

Consider a runner who goes 100 m east and then 40 m west. The total distance covered is 140 m, while the displacement is 60 m east. If east is positive, the displacement is +60 m.

设想一名运动员向东跑100米，然后向西跑40米。所跑的总距离是140米，而位移是向东60米。如果规定东为正，则位移为+60米。

Choosing a sign convention is essential when solving velocity problems. Typically, right or up is taken as positive. When objects fall, one may take downward as positive to simplify signs. Whatever choice is made must be used consistently for displacement, velocity and acceleration.

在解速度问题时，选择符号规定至关重要。通常取向右或向上为正。当物体下落时，可能取向下为正以简化符号。无论作何选择，位移、速度和加速度的符号都必须一致使用。

6. Speed–Time and Velocity–Time Graphs | 速率–时间图与速度–时间图

A speed–time graph always has non‑negative values on the vertical axis because speed is a scalar. The area under a speed–time graph represents distance travelled, and the gradient represents the magnitude of acceleration (but not its direction).

速率–时间图的纵轴始终为非负值，因为速率是标量。速率–时间图下的面积代表所通过的距离，斜率代表加速度的大小（但不表示方向）。

A velocity–time graph is far more powerful for analysing motion. On such a graph:

速度–时间图在分析运动时功能强大得多。在这样的图上：

The gradient at any point gives the instantaneous acceleration.

任一点的斜率给出瞬时加速度。
The area between the graph and the time axis represents displacement. Areas above the axis correspond to positive displacement, areas below to negative displacement.

图线与时间轴之间的面积代表位移。时间轴上方的面积对应正位移，下方的面积对应负位移。
A horizontal line means constant velocity; a sloping line indicates uniform acceleration.

水平线表示匀速；斜线表示匀加速。

When the graph line crosses the time axis, the velocity changes sign, indicating a reversal of direction. In contrast, a speed–time graph would simply show a decrease to zero and then an increase, with no sign information.

当图线穿过时间轴时，速度变号，表明运动方向反转。相比之下，速率–时间图只会显示降到零再增大，没有方向信息。

7. Displacement–Time Graphs | 位移–时间图

A displacement–time graph plots displacement (x) on the vertical axis against time (t) on the horizontal axis. The gradient of this graph gives the velocity.

位移–时间图以位移(x)为纵轴、时间(t)为横轴。该图的斜率给出速度。

A straight, sloping line represents constant velocity. A positive gradient means motion in the positive direction; a negative gradient means motion in the negative direction.

一条倾斜的直线表示匀速。正斜率意味着朝正方向运动；负斜率表示朝负方向运动。
A horizontal line indicates the object is stationary (zero velocity).

水平线表示物体静止（速度为零）。
A curved line signals changing velocity, i.e. acceleration. The instantaneous velocity is found by drawing a tangent to the curve.

曲线表示速度在变化，即存在加速度。瞬时速度可通过在曲线上作切线求得。

If you compare a distance–time graph with a displacement–time graph for the same journey, the key difference is that distance always increases, so the distance–time graph never slopes downwards. The gradient of a distance–time graph gives the speed, not the velocity.

如果将同一行程的距离–时间图和位移–时间图进行比较，关键区别在于距离总是增加，因此距离–时间图永远不会向下倾斜。距离–时间图的斜率给出的是速率，而不是速度。

8. Uniform Acceleration and SUVAT Equations | 匀加速运动与SUVAT方程

When an object moves with constant acceleration in a straight line, five kinematic quantities are involved: initial velocity u, final velocity v, acceleration a, displacement s, and time t. These are linked by the SUVAT equations, which are used extensively in CIE A-Level problems.

当物体在直线上做匀加速运动时，涉及五个运动学量：初速度u、末速度v、加速度a、位移s以及时间t。它们通过SUVAT方程联系在一起，在CIE A-Level题目中广泛使用。

The four standard equations (omitting one variable each time) are:

四个标准方程（每次省略一个变量）如下：

v = u + at (no s)

s = ut + ½at² (no v)

v² = u² + 2as (no t)

s = ½(u + v)t (no a)

These equations apply only when acceleration is constant. They are vector equations; therefore, direction must be accounted for by using a consistent sign convention. For example, if upward is positive, then an object thrown upwards will have a negative acceleration due to gravity (a = −9.81 m s⁻²).

这些方程仅适用于加速度恒定的情况。它们都是矢量方程，因此必须使用一致的符号规定来体现方向。例如，如果取向上为正，那么上抛物体的重力加速度为负(a = −9.81 m s⁻²)。

A typical CIE problem might ask you to calculate the stopping distance of a car given u, v=0 and a negative acceleration. By identifying the known quantities and choosing the appropriate SUVAT equation, you can solve for the unknown easily.

典型的CIE考题可能会给出u、v=0以及一个负加速度，让你计算汽车的刹车距离。通过确定已知量并选择合适的SUVAT方程，就能轻松求解未知量。

9. Free Fall and Vertical Motion | 自由落体与竖直运动

Free fall is a special case of uniformly accelerated motion where the only force acting is gravity. Near the Earth’s surface, the acceleration due to gravity, g, is approximately 9.81 m s⁻² downwards. When analysing free fall, we treat g as a vector depending on our sign convention.

自由落体是匀加速运动的一个特例，唯一作用力是重力。在地球表面附近，重力加速度g约为9.81 m s⁻²向下。在分析自由落体时，我们根据符号规定将g视为矢量。

If we take downward as positive, then a dropped object has u = 0, a = g = +9.81 m s⁻², and the velocity increases positively. An object thrown downwards also has a positive initial velocity. If upward is taken as positive, then an object thrown upwards has an initial positive velocity but a = −9.81 m s⁻²; at the highest point, v = 0 momentarily, and then the object accelerates downwards.

若取向下为正，则释放的物体有u = 0, a = g = +9.81 m s⁻²，速度正向增加。向下抛出的物体也有正初速度。如果取向上为正，上抛物体具有正初速度，但a = −9.81 m s⁻²；在最高点瞬时v = 0，随后物体向下加速。

Using the SUVAT equations, you can predict the time of flight, maximum height, and impact velocity. For example, for a ball thrown vertically upwards with speed u, the time to reach maximum height is t = u/g, and the maximum height is s = u²/(2g), treating magnitudes.

利用SUVAT方程，可以预测飞行时间、最大高度和撞击速度。例如，一个以速率u竖直上抛的小球，到达最高点的时间为t = u/g，最大高度为s = u²/(2g)，这里取g的大小。

Always remember that displacement, velocity and acceleration in free fall are vectors. A student who forgets the sign convention may incorrectly add speed and velocity, leading to mistakes in calculations of displacement.

始终记住，自由落体中的位移、速度和加速度都是矢量。忘记符号规定的学生可能会错误地将速率和速度相加，导致位移计算出错。

10. Vector Addition and Relative Velocity | 矢量加法与相对速度

Velocity is a vector, so when an object participates in two or more motions, the resultant velocity is found by vector addition. This is typically done using a scale diagram (triangle or parallelogram method) or by resolving components perpendicularly.

速度是矢量，因此当物体参与两个或多个运动时，合速度通过矢量加法求得。通常可用比例图示法（三角形或平行四边形法则）或通过垂直分解的方法完成。

For example, a boat crossing a river with a current has its own velocity relative to the water and is also carried by the water’s velocity relative to the ground. The resultant velocity relative to the ground is the vector sum:

例如，一艘渡河的船具有相对于水的速度，同时也被水流相对于地面的速度带动。相对于地面的合速度为矢量和：

v_bg = v_bw + v_wg

where v_bw is the velocity of the boat relative to water and v_wg is the velocity of the water relative to ground. The direction of the resultant determines the actual path and the time to cross the river.

其中v_bw是船相对于水的速度，v_wg是水相对于地面的速度。合速度的方向决定了实际路径和渡河时间。

Relative velocity problems also appear in one dimension. If two cars move along the same straight road, the velocity of car A relative to car B is v_AB = v_A − v_B. This helps in determining how fast they approach or separate.

相对速度问题也会出现在一维情况下。若两车在同一直道上行驶，A车相对于B车的速度为v_AB = v_A − v_B。这有助于判断它们接近或分离的快慢。

In vector addition, speed as a scalar cannot simply be added to velocity. If a girl walks at 3 m s⁻¹ across a bus that is moving at 10 m s⁻¹, her speed relative to the ground is not necessarily 13 m s⁻¹ but the magnitude of the vector sum, which depends on direction.

在矢量加法中，速率作为标量不能简单地与速度相加。如果一个女孩在公共汽车上以3 m s⁻¹横向走动，而车以10 m s⁻¹向前运动，她相对于地面的速率不一定就是13 m s⁻¹，而是矢量和的大小，这取决于方向。

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Speed and Velocity | 速率与速度