Steady Speed, Changing Velocity | 恒定速率，变化的速度

📚 Steady Speed, Changing Velocity | 恒定速率，变化的速度

In everyday language, “speed” and “velocity” are often used as if they were the same thing. However, in CIE A-Level Physics they have precise and different meanings. This article examines a counter-intuitive idea: an object can travel at a steady speed while its velocity is changing. The key example is uniform circular motion, which appears throughout mechanics, gravitation and charged particle motion.

在日常语言中，”速率”和”速度”经常被当作同一个概念。但在 CIE A-Level 物理中，它们有精确且不同的含义。本文探讨一个违反直觉的观点：物体可以以恒定速率运动，但速度却在变化。最典型的例子就是匀速圆周运动，它贯穿力学、万有引力和带电粒子运动等内容。

1. Speed vs Velocity: The Key Distinction | 速率与速度：关键区别

Speed is a scalar quantity, meaning it has magnitude only. Velocity is a vector quantity, meaning it has both magnitude and direction. In physics, a vector changes whenever its magnitude changes, its direction changes, or both change. Therefore, a car going around a roundabout at a constant speed still has a changing velocity because its direction is continuously changing.

速率是标量，只有大小。速度是矢量，既有大小又有方向。在物理中，只要矢量的大小改变、方向改变或两者都改变，矢量就发生了变化。因此，汽车以恒定速率绕环岛行驶时，速度仍然在变化，因为它的方向在不断改变。

This distinction is assessed regularly in CIE exams. Students often lose marks by writing “velocity is constant” when they mean “speed is constant” in circular motion questions.

这一区别在 CIE 考试中经常考查。在圆周运动题目中，学生常因把”速度恒定”写成”速率恒定”而丢分。

2. Uniform Circular Motion: A Visual Overview | 匀速圆周运动：直观概述

Uniform circular motion describes an object moving along a circular path at a constant speed. The word “uniform” refers to the constant speed, not the constant velocity. At any instant, the object’s velocity is directed along the tangent to the circle.

匀速圆周运动描述物体沿圆形路径以恒定速率运动。”匀速”指的是恒定速率，而不是恒定速度。在任意瞬间，物体的速度方向都沿着圆周的切线。

Imagine a stone tied to a string and whirled in a horizontal circle. The stone covers equal arc lengths in equal time intervals, so its speed is steady. Yet its direction of travel is different at every point, so its velocity vector is different at every point.

想象一块系在绳子上的石头在水平面内旋转。石头在相等时间间隔内经过相等的弧长，因此它的速率恒定。然而它在每一点的运动方向都不同，所以它在每一点的速度矢量都不同。

3. Why Velocity Changes at Steady Speed | 为什么恒定速率下速度仍在变化

To compare velocities at two points on a circle, draw the two velocity vectors as arrows. Although both arrows have the same length, they point in different directions. Subtracting the initial velocity from the final velocity gives the change in velocity:

为了比较圆上两点的速度，将两个速度矢量画成箭头。虽然两个箭头长度相同，但它们指向不同方向。用末速度减去初速度得到速度变化量：

Δv = v₂ − v₁

Since v₂ and v₁ are not the same vector, Δv is not zero. This non-zero change in velocity means the object is accelerating, even though its speed is unchanged.

由于 v₂ 和 v₁ 不是相同的矢量，Δv 不为零。速度变化量不为零意味着物体正在加速，尽管它的速率没有改变。

4. Direction of the Change in Velocity | 速度变化的方向

For small angular displacements, the vector triangle shows that the change in velocity points towards the centre of the circle. If an object moves from point P to point Q through a small angle Δθ, the velocity vectors at P and Q are separated by the same angle Δθ. Their difference, Δv, is approximately perpendicular to the original velocity and points radially inward.

对于小角度位移，矢量三角形表明速度变化量指向圆心。若物体从 P 点经过小角度 Δθ 运动到 Q 点，P 点和 Q 点的速度矢量也相隔同样的角度 Δθ。它们的差值 Δv 近似垂直于初速度，并沿径向指向圆心。

As Δθ becomes very small, this inward direction becomes exact. This is why the acceleration in uniform circular motion is always directed towards the centre: it is called centripetal acceleration.

当 Δθ 变得非常小时，这个指向圆心的方向变得精确。这就是为什么匀速圆周运动中的加速度始终指向圆心：它被称为向心加速度。

5. Centripetal Acceleration | 向心加速度

Acceleration is the rate of change of velocity. Since the velocity change is directed towards the centre, the acceleration is also directed towards the centre. The centripetal acceleration has magnitude:

加速度是速度的变化率。由于速度变化量

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