📚 Formula Derivations for OxfordAQA International A-Level Physics A2 Unit 4 | 牛津AQA国际A-Level物理A2第4单元公式推导
Unit 4 of the OxfordAQA International A-Level Physics course brings together further mechanics, fields, and nuclear physics. A deep understanding of the key formulas often requires working through their derivations. This article presents step-by-step derivations for the most important relationships, pairing each English explanation with its Chinese equivalent to support bilingual learners.
牛津AQA国际A-Level物理第4单元整合了进阶力学、场和核物理。透彻理解关键公式往往需要推导其来历。本文逐步展示最重要关系的推导，每个英文解释都配有对应的中文，为双语学习者提供支持。
1. Deriving the SUVAT Equations | 推导匀加速运动学方程
For motion with constant acceleration a, the definition a = (v – u) / t can be rearranged directly to give the first equation.
对于恒定加速度 a 的运动，定义式 a = (v – u) / t 可直接整理出第一个方程。
v = u + at
The area under a velocity-time graph represents displacement s. Since the graph is a straight line, the area is a trapezium: s = ½(u + v)t. Substituting the expression for v yields the second equation.
速度-时间图下的面积代表位移 s。该图是一条直线，面积是梯形面积：s = ½(u + v)t。代入 v 的表达式得到第二个方程。
s = ut + ½at²
By eliminating t using t = (v – u)/a and substituting into s = ½(u+v)t, we obtain the third useful link between velocity and displacement.
通过 t = (v – u)/a 消去 t 并代入 s = ½(u+v)t，我们得到速度与位移之间第三个有用的联系。
v² = u² + 2as
2. Independence of Horizontal and Vertical Motion in Projectiles | 抛体运动的水平与竖直独立性
In projectile motion, the horizontal and vertical components are independent. Horizontally, there is no acceleration (ignoring air resistance), so the horizontal displacement is simply x = uₓt, where uₓ = u cos θ.
在抛体运动中，水平和竖直分量相互独立。水平方向没有加速度（忽略空气阻力），因此水平位移为 x = uₓt，其中 uₓ = u cos θ。
Vertically, the motion is under constant downward acceleration g. Applying the SUVAT equation with initial vertical velocity uᵧ = u sin θ gives the vertical displacement.
竖直方向受恒定向下的加速度 g 影响。使用初始竖直速度 uᵧ = u sin θ 的匀加速方程，可得竖直位移。
y = (u sin θ)t – ½gt²
The time of flight is found by setting y = 0 (return to the original height), yielding t = (2u sin θ)/g. The range R is then obtained by substituting this time into the horizontal equation.
令 y = 0（返回原高度）可求出飞行时间 t = (2u sin θ)/g，再将此时间代入水平方程即得水平射程 R。
R = (u² sin 2θ) / g
3. Deriving Centripetal Acceleration | 推导向心加速度
Consider an object moving at constant speed v in a circle of radius r. In a short time Δt, its velocity vector turns through a small angle Δθ. The change in velocity Δv points toward the centre.
考虑物体以恒定速率 v 在半径为 r 的圆周上运动。在很短的 Δt 内，速度矢量转过小角度 Δθ。速度变化量 Δv 指向圆心。
From geometry, the magnitude of Δv is approximately v Δθ, and Δθ = v Δt / r. Therefore the magnitude of acceleration a = Δv/Δt = v * (v/r) = v²/r.
由几何关系，|Δv| ≈ v Δθ，且 Δθ = v Δt / r。因此加速度大小 a = Δv/Δt = v · (v/r) = v²/r。
a = v² / r and a = r ω²
Since v = rω, we can also write a = r ω². This acceleration is always directed radially inward.
由于 v = rω，也可写成 a = r ω²。该加速度始终沿径向指向圆心。
4. Centripetal Force and Angular Speed Relationships | 向心力与角速度关系推导
Applying Newton’s second law to circular motion gives the centripetal force F = ma = mv² / r. Using v = rω, we obtain two equivalent forms.
将牛顿第二定律应用于圆周运动，得向心力 F = ma = mv² / r。利用 v = rω 可得到两种等价形式。
F = mv² / r = mr ω²
For a given force, the angular speed ω can be related to the period T: since ω = 2π / T, we can rewrite the force as F = mr (4π² / T²). This is particularly useful when analysing planetary orbits or rotating systems.
对于给定向心力，角速度 ω 可与周期 T 关系：由于 ω = 2π / T，可将力改写为 F = mr (4π² / T²)。这在分析行星轨道或旋转系统时特别有用。
5. Electric Field Strength and Potential Gradient | 电场强度与电势梯度
A uniform electric field between two parallel plates separated by distance d and potential difference V has field strength E = V / d. This is derived from the work done per unit charge.
两块相距 d、电势差为 V 的平行板间的匀强电场，其场强为 E = V / d。这由单位电荷所做的功推导而来。
For a general field, the electric field strength is the negative gradient of potential: E = – dV / dr. In a radial field due to a point charge Q, E = Q / (4πε₀ r²), and the potential V = Q / (4πε₀ r). The derivative of V with respect to r yields – Q / (4πε₀ r²), confirming the relationship.
在一般电场中，场强是电势的负梯度：E = – dV / dr。在点电荷 Q 的径向电场中，E = Q / (4πε₀ r²)，而电势 V = Q / (4πε₀ r)。对 r 求导得 – Q / (4πε₀ r²)，验证了关系。
E = – dV / dr
6. Motion of a Charged Particle in a Uniform Electric Field | 带电粒子在匀强电场中的运动
When a charge q enters a uniform electric field E at right angles, the force F = qE causes a constant acceleration a = qE / m in the direction of the field. The path is parabolic, analogous to projectile motion.
当电荷 q 垂直射入匀强电场 E 时，力 F = qE 产生沿场方向的恒定加速度 a = qE / m。路径为抛物线，类似于抛体运动。
If the particle’s initial velocity is u horizontally, the vertical deflection y after travelling a plate length L is found from y = ½ a t², where t = L / u.
若粒子初速度为水平方向 u，穿越板长 L 后的竖直偏转量 y 由 y = ½ a t² 给出，其中 t = L / u。
y = (qE L²) / (2 m u²)
This derivation is often used in cathode-ray tube questions.
这一推导常用于阴极射线管问题。
7. Magnetic Force on a Moving Charge | 运动电荷在磁场中的力
The force experienced by a charge q moving with velocity v in a magnetic field B is given by F = q v B sinθ. The direction is perpendicular to both v and B, as given by Fleming’s left‑hand rule.
以速度 v 在磁场 B 中运动的电荷 q 所受的力为 F = q v B sinθ。其方向垂直于 v 和 B，由弗莱明左手定则判定。
When the particle moves perpendicular to the field (θ = 90°), the force simplifies to F = q v B. This force acts as a centripetal force, causing circular motion.
当粒子运动方向垂直于磁场时 (θ = 90°)，力简化为 F = q v B。该力充当向心力，导致圆周运动。
F = q v B (for θ = 90°)
8. Radius of Circular Path in a Magnetic Field | 磁场中圆周运动半径推导
Equating the magnetic force to the centripetal force gives q v B = m v² / r. Solving for r yields the cyclotron radius.
令磁场力等于向心力：q v B = m v² / r。解出 r 即得回旋半径。
r = m v / (q B)
This expression shows that the radius is directly proportional to momentum mv and inversely proportional to charge and magnetic field. The angular speed (cyclotron frequency) ω = v / r = q B / m is independent of speed.
该式表明半径与动量 mv 成正比，与电荷和磁场成反比。角速度（回旋频率）ω = v / r = q B / m 与速率无关。
9. Deriving a = – ω² x for Simple Harmonic Motion | 简谐运动 a = – ω² x 的推导
Simple harmonic motion (SHM) can be modelled as the projection of uniform circular motion onto a diameter. The displacement from equilibrium is x = A cos(ωt) or A sin(ωt).
简谐运动可以看作匀速圆周运动在直径上的投影。离开平衡位置的位移为 x = A cos(ωt) 或 A sin(ωt)。
Differentiating twice with respect to time gives the acceleration. For x = A cos(ωt), velocity v = – A ω sin(ωt), and acceleration a = – A ω² cos(ωt) = – ω² x.
对时间两次求导可得加速度。对于 x = A cos(ωt)，速度 v = – A ω sin(ωt)，加速度 a = – A ω² cos(ωt) = – ω² x。
a = – ω² x
The defining equation shows that acceleration is proportional to displacement and directed towards the equilibrium position.
该定义式表明加速度与位移成正比且指向平衡位置。
10. Gravitational Potential and Escape Velocity | 引力势与逃逸速度推导
The work done per unit mass moving from infinity to a point in a gravitational field gives the gravitational potential V. For a point mass M, the force is F = G M m / r², so the work done is the integral of – F dr from infinity to r.
将单位质量从无穷远移入引力场某点做的功等于引力势 V。对于质点 M，力为 F = G M m / r²，因此做功为从无穷远到 r 对 -F dr 的积分。
V = – G M / r
Escape velocity vₑ is the minimum speed needed to reach infinity with zero final kinetic energy. Using energy conservation: ½ m vₑ² + ( – G M m / R ) = 0, where R is the planet’s radius.
逃逸速度 vₑ 是刚好能到达无穷远处且动能归零的最小速率。由能量守恒：½ m vₑ² + ( – G M m / R ) = 0，其中 R 为行星半径。
vₑ = √(2 G M / R)
11. Mass-Energy Equivalence and Binding Energy | 质能等价与结合能推导
Einstein’s mass-energy relation E = m c² states that mass and energy are interchangeable. In nuclear reactions, the mass defect Δm is the difference between the total mass of separate nucleons and the mass of the nucleus.
爱因斯坦质能关系 E = m c² 表明质量与能量可以相互转化。在核反应中，质量亏损 Δm 是单个核子总质量与原子核质量之差。
E = m c²
The binding energy is then the energy equivalent of the mass defect: B.E. = Δm c². It represents the energy required to separate the nucleus into its individual protons and neutrons.
结合能即质量亏损的能量当量：B.E. = Δm c²。它代表将原子核分离成独立质子和中子所需的能量。
For practical calculations, 1 u of mass is equivalent to 931.5 MeV. Thus binding energy per nucleon can be compared for different nuclei.
实际计算中，1 u 质量相当于 931.5 MeV。因此可以比较不同原子核的每个核子结合能。
12. Radioactive Decay Law and Half-Life | 放射性衰变定律与半衰期推导
The activity A is proportional to the number of undecayed nuclei N: A = λ N, where λ is the decay constant. This leads to the exponential decay law via dN/dt = – λ N.
放射性活度 A 与未衰变核的数目 N 成正比：A = λ N，其中 λ 为衰变常量。这通过微分方程 dN/dt = – λ N 导出指数衰变律。
N = N₀ e^(-λ t)
The half-life T½ is the time when N = N₀ / 2. Substituting gives N₀ / 2 = N₀ e^(-λ T½), so e^(-λ T½) = ½, and taking natural logs yields T½ = ln 2 / λ.
半衰期 T½ 是 N = N₀ / 2 的时间。代入得 N₀ / 2 = N₀ e^(-λ T½)，故 e^(-λ T½) = ½，取自然对数得 T½ = ln 2 / λ。
T½ = ln 2 / λ
These relationships are essential for carbon dating and nuclear medicine applications.
这些关系在碳定年和核医学应用中至关重要。
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