📚 Deriving Key Equations from the PH02/INS Physics Insert | PH02/INS物理插入页关键公式推导
The PH02/INS Data, Formulae and Relationships insert provided in International AS Physics examinations contains essential equations for mechanics, materials, waves, and electricity. Understanding how these formulae are derived not only reinforces theoretical concepts but also improves the ability to apply them in novel situations. This article guides you through the step-by-step derivation of several core formulae from the insert.
国际AS物理考试中提供的PH02/INS数据、公式与关系插入页涵盖了力学、材料、波和电学等核心方程。理解这些公式的推导过程不仅能巩固理论知识，还能提升在新情境中应用它们的能力。本文将逐步推导插入页中若干关键公式。
1. Derivation of the SUVAT Equations | 匀加速运动方程推导
u) / t, where u is the initial velocity, v is the final velocity, and t is the time taken.
对于以恒定加速度 a 运动的物体，加速度定义为速度的变化率：a = (v - u) / t，其中 u 为初速度，v 为末速度，t 为所用时间。
a = (v - u) / t
Rearranging this definition immediately gives the first SUVAT equation.
重新整理这个定义式，直接得到第一个匀加速运动方程。
v = u + at
Since the acceleration is constant, the velocity– al velocities: vₐᵥ = (u + v)/2. Displacement s is the average velocity multiplied by time.
因为加速度恒定，速度-时间图为一条直线，平均速度为初速度与末速度的算术平均值：vₐᵥ = (u + v)/2。位移 s 等于平均速度乘以时间。
s = ½ (u + v) t
Substituting v = u + at into this expression for s gives the second equation.
将 v = u + at 代入上述位移表达式，得到第二个方程。
s = ut + ½ at²
/a. Substituting into the displacement equation and simplifying yields the third equation.
另一种方法是消去时间 t。由 v = u + at 得到 t = (v - u)/a，代入位移公式并化简，得到第三个方程。
v² = u² + 2as
These three equations form the basis of constant-acceleration kinematics and are widely used in projectile and linear motion problems.
这三个方程构成了匀加速运动学的基础，广泛应用于抛体运动和直线运动问题中。
2. Newton's Second Law and the Impulse–Momentum Theorem | 牛顿第二定律与冲量-动量定理
Newton' its momentum: F = Δp / Δt. For an object of constant mass m, momentum p = mv, so Δp = mΔv.
牛顿第二定律指出，作用于物体上的合力等于其动量的变化率：F = Δp / Δt。对于质量 m 恒定的物体，动量 p = mv，因此 Δp = mΔv。
F = (mv - mu) / Δt
ies to F = m (v - u)/Δt. Recognising (v - u)/Δt as acceleration a gives the familiar form.
如果质量不变，可简化为 F = m (v - u)/Δt。注意到 (v - u)/Δt 正是加速度 a，于是得到常见形式。
F = ma
Rearranging the momentum version gives the impulse– The product F Δt is the impulse, which equals the change in momentum. This is particularly useful when forces act over short time intervals.
重新整理动量版公式可得冲量-动量关系：F Δt = mv - mu。乘积 F Δt 为冲量，等于动量的变化量。这在力作用时间很短的场景中特别有用。
3. Elastic Potential Energy Stored in a Spring | 弹簧中储存的弹性势能
For a spring obeying Hooke' is the spring constant. The work done in stretching the spring from 0 to an extension x is the area under the force–extension graph, which is a triangle.
对于遵循胡克定律的弹簧，施加的力 F 与伸长量 x 成正比：F = kx，其中 k 为劲度系数。将弹簧从 0 拉伸至伸长 x 所做的功等于力-伸长图下的面积，该图形为三角形。
W = ½ F x
Substituting F = kx gives the elastic potential energy Eₑ stored in the spring.
代入 F = kx，即得到弹簧中储存的弹性势能 Eₑ。
Eₑ = ½ k x²
This derivation can also be approached by integrating the force over distance: E = ∫₀ˣ applies to energy stored in a stretched wire as long as the elastic limit is not exceeded.
这一推导也可通过对距离积分实现：E = ∫₀ˣ kx dx = ½ kx²。只要不超过弹性极限，该原理同样适用于拉伸金属丝储存的能量。
4. Defining and Deriving Young's Modulus | 杨氏模量的定义与推导
Young's modulus E describes the stiffness of a material and is defined as the ratio of tensile stress to tensile strain, provided the material obeys Hooke's law.
杨氏模量 E 描述材料的刚度，定义为拉伸应力与拉伸应变之比，前提是材料服从胡克定律。
Tensile strain is the fractional extension: ε = ΔL / L₀, where L₀ is the original length.
拉伸应力是单位横截面积上的力：σ = F / A。拉伸应变是长度的相对变化率：ε = ΔL / L₀，其中 L₀ 为原长。
σ = F / A ， ε = ΔL / L₀
Hence Young's modulus is given by:
因此杨氏模量由下式给出：
E = σ / ε = (F / A) / (ΔL / L₀) = (F L₀) / (A ΔL)
The unit of E is the pascal (Pa), the same as that of stress because strain is dimensionless. This formula allows the determination of the stiffness of wires and is fundamental in structural engineering.
E 的单位为帕斯卡（Pa），与应力单位相同，因为应变无量纲。这一公式可用于测定金属丝的刚度，是结构工程中的基础。
5. Snell's Law and the Critical Angle | 斯涅尔定律与临界角
When light passes from one transparent medium to another, Snell' ction: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media.
当光从一种透明介质进入另一种时，斯涅尔定律描述了入射角和折射角之间的关系：n₁ sin θ₁ = n₂ sin θ₂，其中 n₁ 和 n₂ 分别为两种介质的折射率。
n₁ sin θ₁ = n₂ sin θ₂
action becomes 90°. The incident angle at which this happens is the critical angle θc. Setting θ₂ = 90° and taking n₁ = n (denser medium), n₂ = 1 (air or vacuum), we get n sin θc = 1 × sin 90°.
当光从折射率较高的介质射向折射率较低的介质，且折射角变为 90° 时，发生全内反射。此时的入射角即为临界角 θc。设 θ₂ = 90°，n₁ = n（光密介质），n₂ = 1（空气或真空），则有 n sin θc = 1 × sin 90°。
n sin θc = 1
s principle explains the operation of optical fibres and prisms in cameras and binoculars.
因此，临界角公式为 sin θc = 1 / n。这一原理解释了光纤以及相机和双筒望远镜中棱镜的工作原理。
6. Derivation of the Thin Lens Formula | 薄透镜公式的推导
Consider a thin converging lens with focal length f. An object of height h₀ is placed at a distance u from the lens, and a real image of height h₁ is formed at distance v on the opposite side. By tracing two rays from the top of the object, we can set up similar triangles.
考虑焦距为 f 的薄凸透镜。将高度为 h₀ 的物体置于透镜一侧距离 u 处，在另一侧距离 v 处形成一个高度为 h₁ 的实像。通过从物体顶端画出两条光线，可构建相似三角形。
A second pair of similar triangles involving the focal point gives h₁ / h₀ = (v - f) / f.
利用物体和像构成的第一组相似三角形，放大率 m = h₁ / h₀ = v / u。第二组包含焦点的相似三角形给出 h₁ / h₀ = (v -
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