📚 AS Physics Unit1: Deriving the Insert Formulas (June 2019) | AS物理Unit1 2019年6月插入页公式推导
The June 2019 insert for AS Physics Unit1 (Mechanics and Materials) provides a concise reference sheet of fundamental equations. While memorising these formulas is essential for the exam, understanding how they are derived from basic principles reinforces deep learning and helps you apply them correctly in unfamiliar contexts. This article walks through the logical steps behind each key equation, using only the language of AS-level mathematics and physics.
2019年6月AS物理Unit1（力学与材料）的插入页提供了一张简洁的基础公式表。虽然记住这些公式对考试至关重要，但理解它们是如何从基本原理推导出来的，可以加深学习并帮助你在陌生情境中正确应用它们。本文以纯AS水平数学和物理语言，逐步阐述每个关键公式背后的逻辑推导过程。
1. Acceleration and the First SUVAT Equation | 加速度与第一个运动学方程
Acceleration is defined as the rate of change of velocity. For uniform acceleration, a = (change in velocity) / (time taken). If an object moves from initial velocity u to final velocity v over time t, then a = (v – u) / t.
加速度定义为速度的变化率。对于匀加速运动，a = (速度变化量) / (所用时间)。如果一个物体在时间t内从初速度u变为末速度v，则 a = (v – u) / t。
Rearranging this definition immediately gives the first equation of motion: v = u + a t. This is the simplest link between velocity, acceleration and time, valid only when acceleration is constant.
重新整理这个定义可立即得到第一个运动方程：v = u + a t。这是速度、加速度和时间之间最简单的联系，仅在加速度恒定时成立。
2. Deriving the Second SUVAT Equation via the Velocity–Time Graph | 通过速度-时间图推导第二个运动学方程
Displacement s is equal to the area under a velocity–time graph. For uniformly accelerated motion, the graph is a straight line between u and v. The area of this trapezium is average velocity × time: s = ½ (u + v) t.
位移s等于速度-时间图线下方的面积。对于匀加速运动，图线是连接u和v的直线。这个梯形的面积等于平均速度乘以时间：s = ½ (u + v) t。
Substituting v = u + a t from Section 1 into this area expression yields s = ½ (u + u + a t) t, which simplifies to s = u t + ½ a t². This derivation shows how geometry and algebra combine to produce the most commonly used displacement formula.
将第1节中的v = u + a t代入该面积表达式得到 s = ½ (u + u + a t) t，化简后为 s = u t + ½ a t²。这个推导表明几何和代数如何结合，得出最常用的位移公式。
3. The Timeless SUVAT Equation | 不含时间的运动学方程
Sometimes a problem does not provide the time t. We can eliminate t from the first two equations. From v = u + a t we get t = (v – u) / a. Insert this into s = ½ (u + v) t: s = ½ (u + v) × (v – u) / a.
有些问题不提供时间t。我们可以从前两个方程中消去t。由v = u + a t得到 t = (v – u) / a。将其代入 s = ½ (u + v) t：s = ½ (u + v) × (v – u) / a。
Multiplying out, s = (v² – u²) / (2a). Rearranging gives v² = u² + 2 a s. This equation is independent of time and is extremely useful for linking initial and final speeds directly to displacement and acceleration.
乘开得到 s = (v² – u²) / (2a)。重新整理后为 v² = u² + 2 a s。这个方程与时间无关，在直接联系初末速度与位移和加速度时非常有用。
4. Newton’s Second Law from Momentum | 从动量出发的牛顿第二定律
Newton’s second law in its most fundamental form states that the resultant force is equal to the rate of change of momentum: F = Δp / Δt, where momentum p = m v.
牛顿第二定律最基本的形式为：合力等于动量的变化率：F = Δp / Δt，其中动量p = m v。
If the mass of an object remains constant, then Δp = m v – m u = m (v – u) = m Δv. Substituting into the force expression gives F = m Δv / Δt = m a. Thus F = m a is a special case of the momentum principle for constant mass.
如果物体的质量保持不变，那么Δp = m v – m u = m (v – u) = m Δv。代入力的定义式得到 F = m Δv / Δt = m a。因此，F = m a 是质量不变时动量原理的一个特例。
- The insert lists both F = Δp / Δt and F = m a, reminding candidates that the momentum form must be used when mass changes (e.g., rocket problems).
- 插入页同时列出了F = Δp / Δt和F = m a，提醒考生当质量变化时（如火箭问题）必须使用动量形式。
5. Impulse and Change in Momentum | 冲量与动量变化
Impulse is defined as the product of force and the time for which it acts: Impulse = F Δt. From the momentum form of Newton’s second law, F = Δp / Δt, multiplying both sides by Δt gives F Δt = Δp.
冲量定义为力与其作用时间的乘积：冲量 = F Δt。由牛顿第二定律的动量形式F = Δp / Δt，两边同乘以Δt得到 F Δt = Δp。
Therefore, impulse equals the change in momentum. This relation is particularly powerful for analysing collisions and impacts where forces vary rapidly; the area under a force–time graph directly gives the impulse delivered.
因此，冲量等于动量的变化量。这一关系在分析力迅速变化的碰撞和冲击时尤为有力；力-时间图下方的面积直接给出了所施加的冲量。
6. Work Done by a Constant Force | 恒力所做的功
Work is done when a force moves an object in the direction of the force. For a constant force F acting over a displacement s, work W = F s. If the force is at an angle θ to the displacement, only the component F cosθ does work, so W = F s cosθ.
当力使物体沿力的方向移动时，力就做了功。对于恒力F作用在位移s上，功W = F s。如果力与位移的夹角为θ，只有分量F cosθ做功，所以 W = F s cosθ。
This definition is consistent with energy transfer: one joule of work is done when a force of one newton moves an object one metre in the direction of the force.
这一定义与能量转移一致：当1牛顿的力使物体沿力的方向移动1米时，所做的功为1焦耳。
7. Kinetic Energy and the Work–Energy Principle | 动能与功能原理
To derive kinetic energy, consider a constant net force F acting on an object of mass m. The work done W = F s. Using F = m a and the timeless SUVAT equation v² = u² + 2 a s, we can replace a s: s = (v² – u²) / (2a).
为了推导动能，考虑恒定的合力F作用在质量为m的物体上。所做的功W = F s。利用F = m a和不含时间的运动学方程 v² = u² + 2 a s，我们可以代换掉 a s：s = (v² – u²) / (2a)。
Then W = m a × (v² – u²) / (2a) = ½ m v² – ½ m u². The terms ½ m v² and ½ m u² are defined as the final and initial kinetic energy. Hence net work done equals the change in kinetic energy, a result known as the work–energy principle.
于是 W = m a × (v² – u²) / (2a) = ½ m v² – ½ m u²。½ m v² 和 ½ m u² 分别被定义为末动能和初动能。因此，合力所做的功等于动能的变化量，这一结果被称为功能原理。
8. Gravitational Potential Energy and Conservation of Energy | 重力势能与能量守恒
When an object is lifted vertically at constant speed, the lifting force equals the weight m g. The work done in lifting is W = F s = m g Δh, where Δh is the change in height. This work is stored as gravitational potential energy: ΔEgrav = m g Δh.
当一个物体以恒定速度被竖直提升时，提升力等于重力m g。提升过程中所做的功为 W = F s = m g Δh，其中Δh是高度的变化量。这些功被储存为重力势能：ΔEgrav = m g Δh。
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred. In an isolated system, the total energy remains constant. Often, kinetic energy converts to gravitational potential energy and vice versa, with any losses accounted for as thermal energy due to friction or air resistance.
能量守恒定律指出，能量不能凭空产生或消失，只能转移。在一个孤立系统中，总能量保持不变。动能常常转化为重力势能，反之亦然，而任何损失则为摩擦或空气阻力产生的热能。
- Efficiency = ( useful energy output / total energy input ) × 100%
- 效率 = ( 有用能量输出 / 总能量输入 ) × 100%
9. Power as the Rate of Work | 功率作为做功的速率
Power is defined as the rate of doing work (or transferring energy): P = W / t (or P = ΔE / t). For a constant force moving an object at constant speed v in the direction of the force, work done in time t is W = F s. Since s = v t, then P = (F v t) / t = F v.
功率定义为做功（或转移能量）的速率：P = W / t（或 P = ΔE / t）。当一个恒力使物体以恒定速度v沿力的方向运动时，在时间t内做的功为 W = F s。由于 s = v t，那么 P = (F v t) / t = F v。
This derived formula P = F v is particularly useful in transport problems, where the driving force and velocity of a vehicle determine the instantaneous power output.
这个推导出的公式P = F v在交通工具问题中特别有用，此时驱动力和车辆的速度决定了瞬时功率输出。
10. Hooke’s Law and the Spring Constant | 胡克定律与弹簧常数
Hooke’s law states that, within the elastic limit, the extension Δx of a spring is directly proportional to the applied force: F = k Δx, where k is the spring constant. The negative sign often seen indicates restoring force; at AS level we use magnitudes.
胡克定律指出，在弹性限度内，弹簧的伸长量Δx与施加的力成正比：F = k Δx，其中k为弹簧常数。常见的负号表示回复力；在AS水平我们使用大小。
A graph of force against extension is a straight line through the origin, with gradient equal to the spring constant k. This linear relationship is the foundation for understanding elastic behaviour in materials.
力-伸长量图是一条过原点的直线，其斜率等于弹簧常数k。这种线性关系是理解材料弹性行为的基础。
11. Elastic Strain Energy | 弹性应变能
When a spring is stretched, work is done, and energy is stored as elastic potential energy (strain energy). The work done is the area under the force–extension graph. For a spring obeying Hooke’s law, the graph is linear, so the area is a triangle: W = ½ F Δx.
当弹簧被拉伸时，做了功，能量以弹性势能（应变能）的形式储存。所做的功等于力-伸长量图下方的面积。对于遵守胡克定律的弹簧，图线是线性的，因此面积是一个三角形：W = ½ F Δx。
Substituting F = k Δx into this expression gives two equivalent forms: E = ½ k (Δx)² and E = ½ F Δx. These are listed on the insert for quick reference, but the geometric derivation helps when working with non-linear materials.
将F = k Δx代入该表达式可得两种等价形式：E = ½ k (Δx)² 和 E = ½ F Δx。这些均列在插入页上以便快速参考，但几何推导在处理非线性材料时很有帮助。
12. Stress, Strain and Young Modulus | 应力、应变与杨氏模量
Stress (σ) is defined as force per unit cross-sectional area: σ = F / A. Strain (ε) is the extension per unit original length: ε = ΔL / L₀. Both are used to describe material properties independently of the specimen’s dimensions.
应力（σ）定义为单位横截面积上的力：σ = F / A。应变（ε）是单位原始长度的伸长量：ε = ΔL / L₀。两者用于描述材料本身的性质，而与试样的尺寸无关。
Young modulus E is the ratio of tensile stress to tensile strain: E = σ / ε. Substituting the definitions gives the practical form: E = (F / A) / (ΔL / L₀) = F L₀ / (A ΔL). This formula appears on the insert and is used to calculate the stiffness of a material in the linear region of its stress–strain curve.
杨氏模量E是拉伸应力与拉伸应变之比：E = σ / ε。代入定义式可得实用形式：E = (F / A) / (ΔL / L₀) = F L₀ / (A ΔL)。这一公式出现在插入页上，用于在应力-应变曲线的线性区域计算材料的刚度。
| Quantity | Formula | Units |
| Stress | σ = F / A | Pa (N m⁻²) |
| Strain | ε = ΔL / L₀ | (dimensionless) |
| Young modulus | E = σ / ε | Pa |
Understanding these derivations transforms the insert from a list of symbols into a logical story of mechanics and materials, building confidence for both calculation and explanation questions.
理解了这些推导过程，插入页便不再只是一堆符号，而是力学与材料的逻辑叙事，既增强了计算题的信心，也提升了回答解释题的能力。
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