A-Level Physics: Deriving Centripetal Acceleration (From June 2018 Paper 3 Mark Scheme) | A-Level 物理：向心加速度公式推导（源自2018年6月卷3评分方案）

📚 A-Level Physics: Deriving Centripetal Acceleration (From June 2018 Paper 3 Mark Scheme) | A-Level 物理：向心加速度公式推导（源自2018年6月卷3评分方案）

In A-Level Physics, centripetal acceleration is a foundational concept that explains why an object moving in a circle experiences continuous acceleration towards the centre, even when moving at constant speed. The derivation of a = v²/r is frequently examined, and the June 2018 Paper 3 mark scheme provides an ideal template for how examiners award marks for logical rigor and vector reasoning. This article breaks down the derivation step by step, mirrors the scoring points from that mark scheme, and equips you with the knowledge to reproduce it confidently under exam conditions.

在A-Level物理中，向心加速度是一个基础概念，它解释了为何物体做圆周运动时，即使速率恒定，也会持续受到指向圆心的加速度。a = v²/r 的推导是高频考点，而2018年6月试卷3的评分方案恰好展示了考官如何根据逻辑严谨性和矢量推理进行给分。本文将逐步拆解该推导过程，对照该评分方案的得分要点，帮助你掌握并能在考场上自信地呈现出来。

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1. Circular Motion Fundamentals | 圆周运动基础

Uniform circular motion describes an object travelling at constant speed v along a circular path of radius r. The time taken for one complete revolution is the period T, and the frequency f = 1/T. Although the speed is constant, the direction of the velocity vector changes each instant. According to Newton’s first law, any change in velocity (either in magnitude or direction) requires an acceleration. That acceleration is directed radially inward and is known as centripetal acceleration.

匀速圆周运动描述的是物体以恒定速率v沿半径为r的圆形路径运动。完成一整圈的时间称为周期T，频率f = 1/T。尽管速率恒定，速度矢量的方向却在每一瞬间变化。根据牛顿第一定律，速度的任何变化（无论大小还是方向）都需要加速度。该加速度指向圆心，被称为向心加速度。

To derive the magnitude of this acceleration, we must consider how the velocity vector changes over a very small time interval Δt. This is where vector subtraction and geometry come together.

为了推导这个加速度的大小，我们必须考虑速度矢量在极短时间间隔Δt内的变化。这正是矢量减法与几何图形结合之处。

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2. Visualising the Change in Velocity | 速度变化的矢量图示

Imagine an object moving from point A to point B on the circle, sweeping out a small angle Δθ. The velocity at A, v_A, is tangent to the circle at A; the velocity at B, v_B, is tangent at B. Both have magnitude v but differ in direction. To find the change in velocity Δv, place the two vectors tail-to-tail: Δv is the vector connecting the tip of v_A to the tip of v_B. This forms an isosceles triangle with two sides of length v and a base Δv.

设想一个物体从圆周上的A点运动到B点，扫过一个小角度Δθ。A点的速度v_A与圆在A点相切；B点的速度v_B与圆在B点相切。两者大小均为v，但方向不同。为了求出速度的变化量Δv，将这两个矢量尾对尾放置：Δv就是从v_A末端指向v_B末端的矢量。这样就构成了一个等腰三角形，两条边长为v，底边为Δv。

The angle between v_A and v_B is also Δθ because each velocity is perpendicular to its corresponding radius, so the angle between the radii equals the angle between the tangents. This geometric relationship is crucial for the next step.

v_A与v_B之间的夹角同样是Δθ，因为每条速度都与对应的半径垂直，因此半径之间的夹角等于切线之间的夹角。这一几何关系对下一步至关重要。

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3. Geometry of the Vector Triangle | 矢量三角形的几何关系

When Δt is extremely small, the angle Δθ becomes very small. In this limit, the chord length AB (straight-line distance) approximates the arc length Δs along the circle. The triangle formed by the two radii and the chord (with sides r, r, and chord length) is similar to the velocity triangle (with sides v, v, and Δv). This similarity arises because both are isosceles triangles with the same small angle Δθ at the apex.

当Δt极其微小时，角度Δθ变得非常小。在这个极限下，弦长AB（直线距离）近似等于沿圆周的弧长Δs。由两个半径与弦构成的三角形（边长为r、r和弦长）与速度三角形（边长为v、v和Δv）相似。相似的原因在于两者均为等腰三角形，且顶角同为小角度Δθ。

From the similarity, we can write the ratio of corresponding sides:

Δv / v ≈ chord / r

由相似关系，可得对应边的比例：

Δv / v ≈ 弦长 / r

For a very small Δθ, the chord length is essentially equal to the arc length Δs. Thus, we approximate:

Δv / v ≈ Δs / r

由于Δθ非常小，弦长实质上等于弧长Δs。因此我们近似为：

Δv / v ≈ Δs / r

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4. Relating Arc Length and Time | 弧长与时间的联系

The distance travelled along the arc is directly related to the constant speed: Δs = v Δt. Substituting this into the approximate ratio gives:

Δv / v ≈ (v Δt) / r

沿弧运动的距离与恒定速率直接相关：Δs = v Δt。将其代入上述近似比例：

Δv / v ≈ (v Δt) / r

Rearranging the terms, we obtain an expression for the rate of change of velocity:

Δv / Δt ≈ v² / r

重新整理各项，我们得到速度变化率的表达式：

Δv / Δt ≈ v² / r

Instantaneous acceleration is defined as the limit of Δv/Δt as Δt approaches zero. In this limit, all the approximations become exact, yielding the centripetal acceleration:

a = v² / r

瞬时加速度定义为当Δt趋近于零时Δv/Δt的极限。在极限情况下，所有近似均变为精确，从而得出向心加速度：

a = v² / r

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5. Expressing Acceleration in Terms of Angular Speed | 用角速度表示加速度

Angular speed ω is defined as the rate of change of the angle swept: ω = Δθ/Δt. Since the linear speed v is related to angular speed by v = ωr, we can substitute this into the centripetal acceleration formula:

a = (ωr)² / r = ω² r

角速度ω定义为扫过角度的变化率：ω = Δθ/Δt。由于线速度v与角速度满足v = ωr，将其代入向心加速度公式：

a = (ωr)² / r = ω² r

This alternative form is especially practical when dealing with problems stated in terms of period T (where ω = 2π / T) or frequency f (ω = 2πf). For example, a disc rotating at a constant angular speed has a centripetal acceleration that increases linearly with radius.

这一替代形式在处理以周期T（ω = 2π / T）或频率f（ω = 2πf）表述的问题时尤为实用。例如，一个匀速转动的圆盘上，向心加速度随半径线性增加。

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6. Centripetal Force from Newton’s Second Law | 由牛顿第二定律导出向心力

According to Newton’s second law, the net force acting on an object is F = m a. Therefore, the centripetal force required to keep an object in circular motion is:

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