A-Level Physics Unit 5 Jun22: Essential Formula Derivations | A-Level 物理 单元5 2022年6月 核心公式推导

📚 A-Level Physics Unit 5 Jun22: Essential Formula Derivations | A-Level 物理 单元5 2022年6月 核心公式推导

Unit 5 of A-Level Physics covers a wide range of topics including nuclear physics, thermal physics, simple harmonic motion, and gravitational fields. Understanding the derivations behind key formulas is essential for mastering the subject and tackling exam questions effectively. This article revisits the most important derivations commonly tested in the June 2022 question paper, providing step-by-step explanations to reinforce your conceptual understanding.

A-Level物理单元5涵盖核物理、热物理、简谐运动与引力场等广泛领域。掌握核心公式的推导过程，不仅有助于深入理解物理概念，也能在考试中游刃有余。本文围绕2022年6月试卷中常见的推导题，逐步解析重要的公式由来，帮助大家巩固知识。

1. Deriving the Radioactive Decay Law | 推导放射性衰变定律

Radioactive decay is a random process where the rate of decay is directly proportional to the number of undecayed nuclei present. If N is the number of radioactive nuclei at time t, the decay constant λ gives the probability of decay per unit time. The activity can be written as:

放射性衰变是一个随机过程，衰变速率与现存未衰变原子核的数目成正比。设N为t时刻的放射性核数目，衰变常数λ表示单位时间的衰变概率，则活度可表示为：

dN/dt = –λN

By separating the variables, we obtain dN/N = –λ dt. Integrating both sides from N₀ at t = 0 to N at time t gives ∫(1/N) dN = –λ∫ dt, leading to ln(N/N₀) = –λt. Exponentiation yields the exponential decay law:

分离变量得 dN/N = –λ dt。对两边积分，从t=0时的N₀ 到时刻t的N，得到 ln(N/N₀) = –λt。取指数即得指数衰变定律：

N = N₀ e^–λt

This derivation shows that the number of undecayed nuclei falls exponentially with time, a direct consequence of the constant decay probability per nucleus.

这一推导表明，由于每个核的衰变概率恒定，未衰变核数随时间呈指数衰减。

2. Half-Life and the Decay Constant | 半衰期与衰变常数

The half-life t_½ is the time taken for the number of radioactive nuclei to reduce to half its initial value. Setting N = N₀/2 in the decay law N = N₀ e^–λt gives ½ = e^–λt_½. Taking natural logarithms yields –ln2 = –λt_½, so:

半衰期t_½ 是放射性核数目减半所需的时间。在衰变定律中令N = N₀/2，得到½ = e^–λt_½。取自然对数得 –ln2 = –λt_½，因此：

t_½ = ln2 / λ

This important relation tells us that a larger decay constant means a shorter half-life, as the substance decays more rapidly. It is frequently used to determine λ from experimental half-life data.

这一重要关系表明，衰变常数越大，半衰期越短，物质衰变得越快。实验中常通过半衰期数据来确定λ。

3. Kinetic Theory of Gases: Pressure Formula | 气体分子运动论的压强公式

Consider an ideal gas of N molecules, each of mass m, confined in a cubic box of side L. A molecule moving with velocity v_x along the x-axis collides elastically with a wall, reversing its x-component. The change in momentum is 2mv_x, and the time between collisions with the same wall is 2L/v_x. The force on the wall from one molecule averages to mv_x²/L. Summing over all molecules and using the mean square speed ⟨c²⟩ = ⟨v_x²⟩ + ⟨v_y²⟩ + ⟨v_z²⟩ = 3⟨v_x²⟩, the total force F = (N/L) × (1/3) m⟨c²⟩. Since pressure p = F/L² and volume V = L³, we obtain:

考虑一个包含N个分子（质量均为m）的理想气体，密闭在边长为L的立方盒中。一个分子以速度v_x沿x轴运动，与器壁发生弹性碰撞，动量改变量为2mv_xPublished by TutorHao | A-Level Physics Revision Series | aleveler.com

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