📚 AS Physics Unit 1 Formula Derivations (Jan 2020 Context) | AS物理单元1公式推导（2020年1月卷背景）
In the January 2020 AS Physics Unit 1 exam paper, many questions required not just recalling formulas but also understanding how they are derived. Being able to reproduce a derivation shows deep comprehension and helps avoid memorisation mistakes. This article walks you through the key derivations that underpin the mechanics, materials and waves topics tested in that paper, providing step‑by‑step reasoning in both English and Chinese.
在2020年1月的AS物理单元1试卷中，很多题目不仅要求记住公式，还要理解它们是如何推导出来的。能够自己推导公式体现了透彻的理解，也能避免死记硬背带来的错误。本文将带你逐步复习该试卷所涉及的力学、材料和波动部分的核心公式推导，并提供中英双语解释。
1. Deriving v = u + at | 推导 v = u + at
Acceleration is the rate of change of velocity. For uniform acceleration, the average acceleration a is (final velocity v minus initial velocity u) divided by time taken t. Rearranging the definition directly gives the first equation of motion.
加速度是速度的变化率。对于匀加速运动，平均加速度 a 等于（末速度 v 减初速度 u）除以所用时间 t。直接移项即可得到第一个运动学方程。
a = (v − u) / t → v = u + at
2. Deriving s = ut + ½ at² | 推导 s = ut + ½ at²
For uniform acceleration, the displacement s equals average velocity multiplied by time. The average velocity is (u + v)/2. Substituting v = u + at from the first equation yields s = ((u + u + at)/2) × t, which simplifies to s = ut + ½ at².
匀加速运动的位移 s 等于平均速度乘时间。平均速度为 (u + v)/2。将 v = u + at 代入，得 s = ((u + u + at)/2) × t，化简后得到 s = ut + ½ at²。
s = ( (u + v) / 2 ) t → s = ut + ½ at²
3. Deriving v² = u² + 2as | 推导 v² = u² + 2as
By eliminating time t from the first two equations, we obtain a relationship between initial velocity, final velocity, acceleration and displacement. Square v = u + at to get v² = u² + 2uat + a²t². Factor 2a from the last two terms: v² = u² + 2a (ut + ½ at²) = u² + 2as.
消去前两个方程中的时间 t，可以得出初速度、末速度、加速度和位移之间的关系。将 v = u + at 平方得 v² = u² + 2uat + a²t²。提取后两项的公因子 2a，得 v² = u² + 2a(ut + ½ at²) = u² + 2as。
v² = u² + 2as
4. Deriving the Principle of Conservation of Momentum | 推导动量守恒定律
Consider two bodies A and B interacting. According to Newton’s third law, the force FA on B is equal and opposite to FB on A. If the interaction lasts for time Δt, the impulses are FA on B Δt = −FB on A Δt. Impulse equals change in momentum, so ΔpB = −ΔpA. Therefore the total momentum change is zero, meaning the total momentum of the system remains constant provided no external force acts.
考虑两个相互作用的物体 A 和 B。根据牛顿第三定律，A 对 B 的作用力与 B 对 A 的作用力大小相等、方向相反。若作用时间为 Δt，则冲量 FA on B Δt = −FB on A Δt。冲量等于动量的变化量，因此 ΔpB = −ΔpA。系统总动量的变化为零，即在没有外力作用的条件下，系统总动量守恒。
Impulse = Δp → Δptotal = 0
5. Work Done and Kinetic Energy | 功与动能的关系
When a constant net force F acts on an object over a displacement s, the work done W = F s. Using Newton’s second law F = ma and the equation v² = u² + 2as, we can write as = (v² − u²)/2. Then W = ma × s = m × (v² − u²)/2 = ½ m v² − ½ m u². This shows that the net work done equals the change in kinetic energy.
当恒定的合力 F 作用在物体上并使其发生位移 s 时，所做的功 W = F s。利用牛顿第二定律 F = ma 和运动学方程 v² = u² + 2as，可得 as = (v² − u²)/2。于是 W = ma × s = m × (v² − u²)/2 = ½ m v² − ½ m u²。这表示合力所做的功等于动能的变化量。
W = F s = ½ m v² − ½ m u²
6. Derivation of Centripetal Acceleration a = v² / r | 向心加速度 a = v² / r 的推导
An object moving at constant speed v in a circle of radius r travels from point P to Q in a short time Δt. The change in velocity Δv points toward the centre. Since the velocity vectors form an isosceles triangle, and the displacement triangle is similar, we have |Δv| / v = chord PQ / r. For a small angle, chord PQ ≈ arc PQ = v Δt. Thus |Δv| / v = v Δt / r, giving acceleration a = |Δv| / Δt = v² / r, directed inward.
一物体以恒定速率 v 在半径为 r 的圆周上运动，在很短的 Δt 时间内从 P 点运动到 Q 点。速度的变化量 Δv 指向圆心。速度矢量构成等腰三角形，与位移三角形相似，因此 |Δv| / v = 弦长 PQ / r。当角度很小时，弦长 PQ ≈ 弧长 PQ = v Δt。于是 |Δv| / v = v Δt / r，从而向心加速度 a = |Δv| / Δt = v² / r，方向指向圆心。
a = v² / r (towards the centre)
7. Young Modulus Formula | 杨氏模量公式的推导
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 / L. The Young modulus E of a material is the ratio of stress to strain within the limit of proportionality. Substituting the definitions gives E = (F / A) / (ΔL / L) = F L / A ΔL. This formula is used extensively in the Jan 2020 paper to determine stiffness and extension of wires.
应力 (σ) 定义为单位横截面积上的力：σ = F / A。应变 (ε) 定义为单位原长上的伸长量：ε = ΔL / L。杨氏模量 E 是在比例极限内应力与应变的比值。代入定义式可得 E = (F / A) / (ΔL / L) = F L / A ΔL。这一公式在2020年1月试卷中被广泛用于计算金属丝的刚度和伸长量。
E = stress / strain = (F / A) / (ΔL / L) = F L / A ΔL
8. Deriving the Double‑Slit Interference Equation | 推导双缝干涉方程
For Young’s double‑slit experiment, constructive interference occurs when the path difference between waves from the two slits is an integer multiple of the wavelength. If d is the slit separation and θ the angle to the nth bright fringe, the path difference is d sin θ. Setting this equal to nλ gives nλ = d sin θ. For small angles, sin θ ≈ tan θ = x / D, leading to the fringe spacing formula Δx = λ D / d.
在杨氏双缝实验中，当来自两条狭缝的波的路程差等于波长的整数倍时，发生相长干涉。设狭缝间距为 d，第 n 级亮纹对应的角度为 θ，则路程差为 d sin θ。令其等于 nλ 即得 nλ = d sin θ。在小角度下，sin θ ≈ tan θ = x / D，进一步可推导出条纹间距公式 Δx = λ D / d。
nλ = d sin θ → Δx = λ D / d
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply