AS Physics Insert 2 Jan22 Formula Derivations | AS物理公式推导（2022年1月卷二）

📚 AS Physics Insert 2 Jan22 Formula Derivations | AS物理公式推导（2022年1月卷二）

The formula insert provided in the AS Physics examination (Paper 2, January 2022) contains a concise summary of essential relationships that govern mechanics, materials, waves and electricity. Relying solely on memorisation can be risky; understanding where these equations come from deepens your grasp of physical principles and makes it easier to apply them correctly under exam pressure. This article walks you through the derivations of the most critical AS Physics formulas, linking theory to practice.

AS物理考试（2022年1月卷二）提供的公式表集中概括了力学、材料、波和电学的基本关系。仅凭记忆有一定风险，理解这些方程的来源可以加深对物理原理的领悟，并帮助你在考试压力下正确应用它们。本文将带你逐步推导最重要的AS物理公式，将理论与实践联系起来。

1. Defining the Symbols and Sign Conventions | 定义符号与正方向约定

Before any derivation, we must clearly define the variables. For uniformly accelerated linear motion, u is initial velocity, v final velocity, a constant acceleration, t time interval, and s displacement. All vectors follow a chosen positive direction. A deceleration is simply a negative acceleration.

在进行任何推导之前，必须明确定义变量。对于匀加速直线运动，u 表示初速度，v 末速度，a 恒定加速度，t 时间间隔，s 位移。所有矢量都遵循选定的正方向。减速运动就是加速度取负值。

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2. Deriving v = u + at from the Definition of Acceleration | 由加速度定义推导 v = u + at

Acceleration is defined as the rate of change of velocity. For constant acceleration, a = (v – u) / t. Multiply both sides by t to obtain at = v – u. Rearranging gives the familiar first suvat equation.

加速度定义为速度的变化率。对于恒定加速度，a = (v – u) / t。两边同乘 t 得 at = v – u。整理后就得到熟悉的第一个匀加速运动方程。

v = u + at

This relation is linear: velocity changes by the same amount each second. If a = 2 m s⁻², then every second the velocity increases by 2 m s⁻¹.

该关系是线性的：每秒速度的变化量相同。如果 a = 2 m s⁻²，那么每秒速度增加 2 m s⁻¹。

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3. Deriving s = (u + v)t / 2 from a Velocity-Time Graph | 由速度-时间图推导 s = (u + v)t / 2

The area under a velocity-time graph equals the displacement. For constant acceleration, the graph is a straight line sloping from u to v. The area is a trapezium of parallel sides u and v and width t. Its area is the average of the parallel sides multiplied by the width.

速度-时间图的线下面积等于位移。对于恒定加速度，图线是一条从 u 斜升到 v 的直线。该面积是一个梯形，两平行边分别为 u 和 v，宽度为 t。其面积等于平行边的平均值乘以宽度。

s = (u + v) t / 2

This derivation reinforces the idea that average velocity for constant acceleration is simply (u+v)/2, which can only be used when acceleration is uniform.

这个推导强化了一个概念：匀加速运动中的平均速度就是 (u+v)/2，但仅当加速度恒定时才能使用。

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4. Substituting to Obtain s = ut + ½at² and v² = u² + 2as | 代入推导 s = ut + ½at² 和 v² = u² + 2as

Express v in s = (u+v)t/2 using v = u + at. Substitution gives s = (u + u + at)t/2 = (2u + at)t/2 = ut + ½at². To eliminate t, rearrange v = u + at to t = (v – u)/a, then substitute into s = (u+v)t/2. This yields s = (u+v)(v-u)/(2a) = (v² – u²)/(2a). Multiply both sides by 2a to obtain the timeless equation.

利用 v = u + at 将 s = (u+v)t/2 中的 v 表示出来。代入得 s = (u + u + at)t/2 = (2u + at)t/2 = ut + ½at²。为了消去 t，将 v = u + at 变形为 t = (v – u)/a，再代入 s = (u+v)t/2。得到 s = (u+v)(v-u)/(2a) = (v² – u²)/(2a)。两边同乘 2a 就得到了不含时间的方程。

s = ut + ½at²

v² = u² + 2as

These two equations complete the standard set of four suvat formulas. Each is useful in different scenarios, depending on which variable is unknown.

这两个方程与前面的共同组成了标准的四个匀加速运动公式。根据未知量的不同，每个方程在不同的情境中各有用途。

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5. Newton’s Second Law and the Impulse-Momentum Theorem | 牛顿第二定律与冲量-动量定理

Newton’s second law states that the net force is proportional to the rate of change of momentum. For constant mass, F = d(mv)/dt = m(dv/dt) = ma. The impulse experienced by an object equals the average force multiplied by the time interval, which is also the change in momentum.

牛顿第二定律指出，合力与动量的变化率成正比。对于质量不变的情况，F = d(mv)/dt = m(dv/dt) = ma。物体受到的冲量等于平均力乘以时间间隔，也等于动量的变化量。

F = ma

FΔt = Δp = mv – mu

Deriving impulse from F=ma: multiply both sides by Δt, recognising aΔt = Δv. Thus FΔt = mΔv = m(v-u). This links force, time and velocity change directly.

从 F=ma 推导冲量：两边同乘 Δt，注意到 aΔt = Δv。于是 FΔt = mΔv = m(v-u)。这直接将力、时间和速度变化联系了起来。

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6. Work Done, Kinetic Energy and the Work-Energy Principle | 做功、动能与功能原理

Work done by a constant force is W = F s cosθ. When the force acts in the direction of displacement, θ = 0° and cosθ = 1, so W = F s. Using F = ma and the suvat equation v² = u² + 2as, we can link work done to the change in kinetic energy.

恒力做的功为 W = F s cosθ。当力与位移同向时，θ = 0°，cosθ = 1，因此 W = F s。利用 F = ma 和运动学方程 v² = u² + 2as，可以将功与动能的变化联系起来。

Start with W = F s = (ma) s. From v² = u² + 2as, rearrange to as = (v² – u²)/2. Substitute: W = m × (v² – u²)/2 = ½mv² – ½mu². Thus the net work done on an object equals its change in kinetic energy.

从 W = F s = (ma) s 出发。由 v² = u² + 2as 整理得 as = (v² – u²)/2。代入：W = m × (v² – u²)/2 = ½mv² – ½mu²。因此，物体所受的净功等于它动能的变化。

W = F s

Eₖ = ½mv²

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7. Gravitational Potential Energy and the Principle of Conservation of Energy | 重力势能与能量守恒原理

Lifting an object of mass m through a vertical height h near the Earth’s surface requires work done against gravity. The force needed to lift it at constant speed equals its weight mg. Therefore the work done, which is stored as gravitational potential energy, is W = F h = mgh.

在地球表面附近，将质量为 m 的物体竖直举高 h，需要克服重力做功。以恒定速度举起它所需的力等于其重量 mg。因此所做的功（储存为重力势能）为 W = F h = mgh。

ΔEₚ = mgΔh

In the absence of resistive forces, the total mechanical energy (kinetic + potential) remains constant. Thus for a falling object, ½mv² = mgh if it starts from rest, allowing a direct derivation of impact speed: v = √(2gh).

在没有阻力的情况下，总机械能（动能+势能）保持不变。因此，对于从静止下落的物体，若初始高度为 h，则 ½mv² = mgh，可直接推导出落地速度 v = √(2gh)。

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8. Power, Efficiency and Their Link to Force and Velocity | 功率、效率及其与力和速度的关系

Power is the rate of doing work: P = W / t. Substituting W = F s gives P = F s / t = F v, provided the force is in the direction of motion. This formula is useful for vehicles moving at constant speed against resistive forces.

功率是做功的速率：P = W / t。代入 W = F s 得到 P = F s / t = F v，前提是力与运动方向相同。该公式对于匀速克服阻力运动的车辆非常有用。

P = W / t

P = F v

Efficiency is defined as useful output power (or energy) divided by input power (or energy). It is often expressed as a percentage: efficiency = (useful output / input) × 100%. It can never exceed 100% due to energy dissipated as heat, sound or other wasted forms.

效率定义为有用输出功率（或能量）与输入功率（或能量）之比。通常表示为百分比：效率 = (有用输出 / 输入) × 100%。由于能量会以热、声等形式散逸，效率永远不会超过100%。

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9. Hooke’s Law and Elastic Potential Energy | 胡克定律与弹性势能

For a spring that obeys Hooke’s Law, the applied force F is proportional to the extension x, so F = kx, where k is the spring constant. The work done to stretch the spring is the area under the force-extension graph, which is a triangle of base x and height kx.

对于服从胡克定律的弹簧，外力 F 与伸长量 x 成正比，即 F = kx，其中 k 为劲度系数。拉伸弹簧所做的功是力-伸长图下的面积，这个三角形底为 x、高为 kx。

F = kx

The stored elastic potential energy Eₑ = ½Fx = ½(kx)x = ½kx². The factor ½ arises because the average force during the stretch is half of the final force. This energy store may be released as kinetic energy in a subsequent recoil.

储存的弹性势能 Eₑ = ½Fx = ½(kx)x = ½kx²。因子 ½ 的出现是因为拉伸过程中的平均力是最大力的一半。这个能量储存可以在随后的回弹中以动能形式释放。

Eₑ = ½kx²

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10. Ohm’s Law, Resistivity, and Electrical Power | 欧姆定律、电阻率与电功率

Ohm’s law states that, at constant temperature, the current I through a conductor is directly proportional to the potential difference V across it. The constant of proportionality is the resistance R.

欧姆定律指出，在温度不变的条件下，通过导体的电流 I 与导体两端的电势差 V 成正比。比例常数就是电阻 R。

V = I R

Resistance depends on the conductor’s length L, cross-sectional area A, and a material property called resistivity ρ. Physically, R = ρL / A. Combining this with V=IR allows calculation of current in various circuit configurations.

电阻取决于导体的长度 L、横截面积 A 以及材料特性电阻率 ρ。物理关系为 R = ρL / A。将其与 V=IR 结合，可以计算各种电路配置中的电流。

From the definitions of potential difference (energy per unit charge) and current (charge per unit time), electric power P = V I. Substituting V = I R gives P = I²R, and substituting I = V / R gives P = V² / R.

根据电势差的定义（单位电荷的能量）和电流的定义（单位时间的电荷），电功率 P = V I。代入 V = I R 得 P = I²R，代入 I = V / R 得 P = V² / R。

P = V I

P = I²R = V² / R

These power formulas are essential for analysing energy transfer in resistors, and appreciate why high-voltage transmission reduces resistive losses.

这些功率公式对于分析电阻中的能量转换至关重要，并能理解为何高压输电可降低电阻损耗。

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11. Density, Pressure and the Wave Equation | 密度、压强与波动方程

Density ρ is mass per unit volume: ρ = m / V. Pressure p is force per unit area: p = F / A. In a fluid, the pressure at a depth h due to the weight of the fluid above is p = ρ g h. This is derived by considering the weight of a column of fluid of area A: weight = mg = ρVg = ρAh g, so pressure = weight / A = ρ g h.

密度 ρ 是单位体积的质量：ρ = m / V。压强 p 是单位面积上的力：p = F / A。在流体中，深度 h 处的压强由上方流体的重量引起，p = ρ g h。推导方法是考虑截面积为 A 的流体柱的重量：重量 = mg = ρVg = ρAh g，因此压强 = 重量 / A = ρ g h。

ρ = m / V

p = ρ g h

The wave speed v, frequency f and wavelength λ are connected by the simple relationship arising from the definition of speed as distance over time. A wave travels one wavelength in one period T, so v = λ / T = λ f. This universal wave equation holds for all transverse and longitudinal waves.

波速 v、频率 f 和波长 λ 由简单的定义关联起来：速度是距离除以时间。波在一个周期 T 内传播一个波长，因此 v = λ / T = λ f。这个普适的波动方程适用于所有横波和纵波。

v = f λ

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12. Summary of Derivative Logic and Exam Tips | 推导逻辑总结与应试技巧

Every formula on the AS insert is a condensed statement of a physical model derived from definitions, graphs or conservation laws. When tackling problems, identify which variables are given and which is unknown, then select the formula that relates them. Always check that the equation is homogeneous in units: dimensions on the left must match dimensions on the right.

AS公式表中的每一个公式都是物理模型的浓缩表达，由定义、图像或守恒定律推导而来。解题时，先辨认已知量和未知量，再选择联系它们的公式。始终要检查方程在单位上是否量纲一致：左边的量纲必须与右边相匹配。

Practise rewriting the derivations until the steps feel natural. This not only strengthens memory but also builds the analytical skills needed for the longer structured questions on Paper 2.

反复练习推导过程，直到步骤自然流畅。这不仅能强化记忆，还能培养卷二中较长的结构性题目所需的分析技能。

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