Example Responses for AQA A-Level Further Maths Paper 5 (Unit FM2) | AQA A-Level 进阶数学 Paper 5（FM2）范例作答解析

📚 Example Responses for AQA A-Level Further Maths Paper 5 (Unit FM2) | AQA A-Level 进阶数学 Paper 5（FM2）范例作答解析

In this article we look at how to write high-scoring example responses for AQA A-Level Further Maths Paper 5, also known as Unit FM2. We deconstruct model answers, highlight the steps that examiners reward, and point out common traps.

本文将解析 AQA A-Level 进阶数学第五卷（FM2 单元）的高分示例作答。我们一步步拆解标准答案，指出考官希望看到的得分步骤，并提醒你常见的陷阱。

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1. What FM2 Paper 5 Tests | FM2 第五卷考查什么

Unit FM2 is part of AQA’s Further Mechanics option. It tests your ability to apply advanced mechanical principles to moving particles, rigid bodies and systems. Typical topics include simple harmonic motion, circular motion, work and energy, impulse and collisions, and rigid-body equilibrium.

FM2 单元是 AQA“进阶力学”选项的一部分，考查你将高等力学原理应用于运动质点、刚体与系统的能力。典型考点包括简谐运动、圆周运动、功与能量、冲量与碰撞、刚体平衡等。

The exam is not just about getting the final number correct. Method marks are awarded for showing a logical chain of reasoning, including equations you cannot see directly from the question. A well-structured response often gains full marks even when arithmetic slips occur.

考试并不只要求得到最终数字正确，方法分也会授予那些展示出逻辑推理链的书写过程，包括不能直接从题目中看出的方程。即使最后计算有误，结构清晰的作答往往也能拿到满分。

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2. The Key Structure of a Model Response | 优秀作答的核心结构

A model response in FM2 should follow a consistent pattern. Examiners use this structure to locate marks quickly, so make it easy for them.

FM2 中的优秀作答应当遵循一致的结构。考官通常按此结构快速寻找得分点，所以请让他们的工作变得简单。

• Read the question and identify the governing principle, such as Newton’s second law, conservation of momentum or the conservation of energy.

  读题并识别主导原理，例如牛顿第二定律、动量守恒或能量守恒。

• Draw a clear diagram, define a positive direction and label every force or velocity with a symbol.

  画出清晰示意图，规定正方向，并用符号标出每一个力或速度。

• Write the governing equation in symbols before substituting any values. This is where most method marks are awarded.

  先以符号形式写出主导方程，然后再代入数值。这是大多数方法分的得分位置。

• Substitute the given values carefully, include units in the final answer, and state any limiting cases such as friction being at its maximum.

  细心代入已知数值，在最终答案中加上单位，并说明任何极限情况，例如摩擦力达到最大值。

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3. Example 1: Simple Harmonic Motion | 示例一：简谐运动

Consider a particle of mass 0.5 kg performing simple harmonic motion with amplitude 0.6 m and period 2 s. We are asked for its maximum speed and maximum acceleration.

设一个质量为 0.5 kg 的质点做简谐运动，振幅为 0.6 m，周期为 2 s。求它的最大速度和最大加速度。

The standard model response starts by evaluating the angular frequency.

标准作答首先要计算角频率。

ω = 2π/T = π rad/s

For SHM, the required formulae are v_max = ωA and a_max = ω²A. Substituting the values gives:

对于简谐运动，所需公式为 v_max = ωA 和 a_max = ω²A。代入数值得到：

v_max = π × 0.6 ≈ 1.88 m/s

a_max = π² × 0.6 ≈ 5.92 m/s²

The examiner awards method marks for quoting the standard formula, accuracy marks for substituting the correct numbers, and the final mark for giving units. In your response, always write the formula in symbols first, then show the arithmetic.

考官会因写出标准公式而给方法分，因代入正确数值而给准确分，因写出单位而给最后的一分。所以作答时务必先用符号写出公式，再展示运算过程。

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4. Example 2: Banked Circular Track | 示例二：倾斜圆周轨道

A car of mass 1000 kg travels around a banked circular track of radius 80 m with no friction assisting the motion. The track is inclined at angle θ, with tan θ = 0.4. Determine the speed at which the car can travel without tending to slide.

一辆质量为 1000 kg 的汽车沿半径为 80 m 的倾斜环形轨道行驶，轨道没有摩擦力帮助转弯。斜面倾角为 θ，且 tan θ = 0.4。求汽车不侧滑时能够行驶的速度。

The model response should begin by resolving forces vertically and horizontally. With no friction, the normal reaction N provides both the vertical support and the centripetal force.

标准作答先从竖直和水平方向分解力。在没有摩擦力时，法向反力 N 既提供竖直支持力，也提供向心力。

N cos θ = mg, N sin θ = mv²/r

Dividing the second equation by the first removes N and m, giving a neat formula.

将第二式除以第一式，可以消去 N 和 m，得到简洁的公式。

tan θ = v²/(rg)

Then solve for v.

然后求解 v。

v = √(80 × 9.8 × 0.4) ≈ 17.7 m/s

Notice that the mass is not used in the final calculation. A strong

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