📚 AS-Level Physics Paper 1: Report on Exams – Formula Derivations | AS 物理 Paper 1 考试报告：公式推导
Welcome to this detailed exploration of the most important formula derivations that frequently appear in AS-Level Physics Paper 1 exams. In this article, we break down the reasoning behind kinematics, dynamics, energy, circular motion, gravitation, electricity and waves equations. Mastering these derivations deepens your understanding of physical principles and prepares you for the conceptual questions that examiners love to include in multiple-choice and short-answer sections.
欢迎详细探讨 AS 物理 Paper 1 考试中频繁出现的最重要公式推导。本文将分解运动学、动力学、能量、圆周运动、引力、电学和波动方程背后的推理过程。掌握这些推导能加深你对物理原理的理解，并为选择题和简答题中考生喜欢考查的概念题做好准备。
1. Deriving the Equations of Motion from Definitions | 从定义推导运动方程
For uniformly accelerated motion, acceleration a is constant. By definition, a = (v – u) / t, where u is initial velocity, v is final velocity and t is time taken. Rearranging gives the first equation: v = u + a t.
对于匀加速直线运动，加速度 a 恒定。依据定义，a = (v – u) / t，其中 u 为初速度，v 为末速度，t 为时间。整理可得第一个方程：v = u + a t。
v = u + a t
Displacement s is the area under a velocity–time graph, or equivalently the product of average velocity and time. Average velocity for constant acceleration is (u + v)/2. Therefore s = ((u + v)/2) × t. Substituting v = u + a t gives s = u t + ½ a t².
位移 s 是速度–时间图像下的面积，或者等于平均速度与时间的乘积。匀加速运动的平均速度为 (u + v)/2。因此 s = ((u + v)/2) × t。代入 v = u + a t 得到 s = u t + ½ a t²。
s = u t + ½ a t²
To obtain the third equation, eliminate t from the first two. From v = u + a t we have t = (v – u)/a. Substituting into s = (u + v)/2 × t yields s = (u + v)(v – u)/(2a) = (v² – u²)/(2a). Hence v² = u² + 2 a s.
为得到第三个方程，从前两个式子中消去 t。由 v = u + a t 得 t = (v – u)/a。代入 s = (u + v)/2 × t 得 s = (u + v)(v – u)/(2a) = (v² – u²)/(2a)。因此 v² = u² + 2 a s。
v² = u² + 2 a s
2. Deriving the Projectile Range Formula | 推导抛体水平射程公式
Consider a projectile launched from ground level with speed u at angle θ to the horizontal. Resolving the velocity: horizontal component uₓ = u cos θ, vertical component u_y = u sin θ. Ignoring air resistance, horizontal motion is uniform, vertical motion is uniformly accelerated with a = -g.
考虑从地面以速率 u、与水平方向成 θ 角发射的抛体。速度分解：水平分量 uₓ = u cos θ，竖直分量 u_y = u sin θ。忽略空气阻力，水平方向为匀速运动，竖直方向为加速度 a = -g 的匀加速运动。
Time of flight T is found from vertical motion. The projectile returns to the ground when vertical displacement = 0: s_y = u_y T – ½ g T² = 0. Solving gives T = 2 u_y / g = 2 u sin θ / g.
飞行时间 T 由竖直运动求得。抛体返回地面时竖直位移为 0：s_y = u_y T – ½ g T² = 0。解得 T = 2 u_y / g = 2 u sin θ / g。
Horizontal range R is the horizontal distance travelled in time T: R = uₓ T = (u cos θ) × (2 u sin θ / g) = (u² × 2 sin θ cos θ) / g. Using the identity 2 sin θ cos θ = sin 2θ gives the well-known range equation.
水平射程 R 是在时间 T 内的水平位移：R = uₓ T = (u cos θ) × (2 u sin θ / g) = (u² × 2 sin θ cos θ) / g。利用三角恒等式 2 sin θ cos θ = sin 2θ，就得到著名的射程公式。
R = (u² sin 2θ) / g
3. Deriving F = ma from Momentum Change | 从动量变化推导 F = ma
Newton’s second law in its most general form states that the resultant force equals the rate of change of momentum: F = Δp / Δt. For a particle of constant mass m, momentum p = m v. If the velocity changes from u to v in time Δt, the change in momentum is Δp = m v – m u = m (v – u).
牛顿第二定律最普遍的形式为合外力等于动量的变化率：F = Δp / Δt。对于质量 m 恒定的质点，动量 p = m v。若在 Δt 时间内速度由 u 变为 v，动量变化量为 Δp = m v – m u = m (v – u)。
Divide by Δt: F = m (v – u) / Δt. But (v – u)/Δt is the definition of acceleration a, provided the force and acceleration are in the same direction. Hence F = m a. This shows that F = ma is a special case for constant mass systems.
除以 Δt：F = m (v – u) / Δt。而 (v – u)/Δt 正是加速度 a 的定义，前提是力与加速度同向。因此 F = m a。这表明 F = ma 是质量不变系统的特殊情况。
F = m a
4. Deriving Kinetic Energy = ½ m v² | 推导动能 = ½ m v²
The work done by a resultant force F acting over a displacement s is W = F s. Using F = m a, work becomes W = m a s. Now relate a s to velocities using v² = u² + 2 a s. For a particle starting from rest (u = 0), the equation simplifies to v² = 2 a s, so a s = v²/2.
合外力 F 在位移 s 上做的功为 W = F s。利用 F = m a，功变为 W = m a s。现在就 a s 与速度通过 v² = u² + 2 a s 关联起来。对于从静止开始的质点（u = 0），方程简化为 v² = 2 a s，即 a s = v²/2。
Substituting gives W = m × (v²/2) = ½ m v². This work done by the net force is entirely converted into kinetic energy, so the kinetic energy KE of a body moving at speed v is ½ m v². If the object had an initial speed u, the work done equals the change in kinetic energy: W = ½ m v² – ½ m u².
代入得 W = m × (v²/2) = ½ m v²。合外力做的功全部转化为动能，因此以速率 v 运动的物体的动能 KE 为 ½ m v²。若物体具有初速率 u，则功等于动能变化量：W = ½ m v² – ½ m u²。
KE = ½ m v²
5. Deriving Centripetal Acceleration a = v² / r | 推导向心加速度 a = v² / r
Consider an object moving in a circle of radius r at constant speed v. Although the speed is constant, the direction changes continuously, so the velocity vector changes. In a short time Δt, the object moves through an arc of length v Δt, subtending an angle Δθ = v Δt / r.
考虑物体以恒定速率 v 在半径为 r 的圆周上运动。尽管速率恒定，方向不断变化，因此速度矢量发生变化。在很短的时间 Δt 内，物体经过弧长 v Δt，对应的圆心角为 Δθ = v Δt / r。
The velocity vectors at the start and end of this interval have equal magnitude v but differ in direction by Δθ. The change in velocity Δv forms the base of an isosceles triangle with sides v. For small Δθ, the magnitude Δv ≈ v Δθ. Thus Δv ≈ v × (v Δt / r) = v² Δt / r.
该间隔起点与终点的速度矢量大小均为 v，方向相差 Δθ。速度变化量 Δv 构成以 v 为腰的等腰三角形的底边。对于很小的 Δθ，Δv 的大小近似为 v Δθ。因此 Δv ≈ v × (v Δt / r) = v² Δt / r。
Centripetal acceleration a is the magnitude of Δv / Δt: a = Δv / Δt ≈ v² / r. In the limit Δt → 0, the approximation becomes exact. The direction of Δv, and hence acceleration, is towards the centre of the circle. Therefore a = v² / r directed radially inward.
向心加速度 a 的大小为 Δv / Δt：a = Δv / Δt ≈ v² / r。在 Δt → 0 的极限下，该近似变为精确。Δv 的方向指向圆心，因此加速度也指向圆心。所以 a = v² / r，方向沿径向向内。
a = v² / r
6. Deriving Gravitational Field Strength g = G M / r² | 推导引力场强度 g = G M / r²
Newton’s law of universal gravitation gives the force between two point masses M and m separated by distance r: F = G M m / r². For a planet of mass M and a small object of mass m on its surface, this force is the weight of the object, mg.
牛顿万有引力定律给出两个相距 r 的质点 M 与 m 之间的引力：F = G M m / r²。对于质量为 M 的行星和其表面质量为 m 的小物体，此力即物体的重量 mg。
Equate the two expressions for force: mg = G M m / r². Cancel the small mass m (assuming m is much smaller than M) to obtain g = G M / r². This shows that gravitational field strength at a point is independent of the test mass and depends only on the source mass M and distance r from its centre.
将两个力的表达式等同起来：mg = G M m / r²。消去小质量 m（假设 m 远小于 M）得到 g = G M / r²。这表明某点的引力场强度与检验质量无关，仅取决于场源质量 M 及到场源中心的距离 r。
g = G M / r²
This relationship is frequently used to calculate the mass of a planet from surface g and radius, or to show that g decreases with altitude.
该关系常被用于通过表面 g 和半径计算行星质量，或证明 g 随高度增加而减小。
7. Deriving the Resistivity Formula R = ρ L / A | 推导电阻率公式 R = ρ L / A
For a uniform conductor of length L and cross-sectional area A, experiment shows that resistance R is directly proportional to L and inversely proportional to A. Hence R ∝ L / A. Introducing the constant of proportionality, the resistivity ρ, we write R = ρ L / A.
对于长为 L、横截面积为 A 的均匀导体，实验表明电阻 R 与 L 成正比，与 A 成反比。因此 R ∝ L / A。引入比例常数——电阻率 ρ，我们可以写出 R = ρ L / A。
Microscopically, ρ is related to the number density of charge carriers n, their charge e, and the mean drift velocity vₔ via the current I = n A vₔ e. Combined with Ohm’s law V = I R and the expression for electric field E = V / L, one can show that resistivity is an intrinsic material property. However, for Paper 1, the macroscopic proportionality is the essential derivation.
从微观上看，ρ 与载流子数密度 n、载流子电荷 e 以及平均漂移速度 vₔ 通过电流 I = n A vₔ e 相关联。结合欧姆定律 V = I R 和电场 E = V / L，可以证明电阻率为材料的本征属性。不过对 Paper 1 而言，宏观比例关系才是核心推导。
R = ρ L / A
8. Deriving Series and Parallel Resistance Formulas | 推导串联和并联电阻公式
Series: When resistors are connected end-to-end, the same current I flows through each. The total potential difference V across the combination is the sum of individual p.d.s: V = V₁ + V₂ + V₃. Applying Ohm’s law to each, V₁ = I R₁, V₂ = I R₂, V₃ = I R₃, and the total V = I R_total. Hence I R_total = I R₁ + I R₂ + I R₃, so R_total = R₁ + R₂ + R₃.
串联：当电阻首尾相接时，相同电流 I 流过每个电阻。组合两端总电势差 V 等于各电势差之和：V = V₁ + V₂ + V₃。对每个电阻应用欧姆定律，V₁ = I R₁，V₂ = I R₂，V₃ = I R₃，而总电压 V = I R_total。因此 I R_total = I R₁ + I R₂ + I R₃，故 R_total = R₁ + R₂ + R₃。
R_total = R₁ + R₂ + R₃
Parallel: Resistors in parallel share the same p.d. V across them, but the total current I splits: I = I₁ + I₂ + I₃. Using I = V / R for each branch, I_total = V / R_total = V / R₁ + V / R₂ + V / R₃. Cancelling V gives 1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃.
并联：并联电阻两端电势差 V 相同，总电流 I 则分流：I = I₁ + I₂ + I₃。利用每个支路 I = V / R，得 I_total = V / R_total = V / R₁ + V / R₂ + V / R₃。消去 V 得到 1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃。
1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃
9. Deriving Power in Electrical Circuits: P = I V, P = I² R | 推导电功率：P = I V，P = I² R
Electrical power is the rate at which energy is transferred. The energy transferred when a charge Q moves through a potential difference V is E = Q V. Therefore power P = E / t = (Q / t) V. Since current I = Q / t, we immediately obtain P = I V.
电功率是能量转化的速率。当电荷 Q 经过电势差 V 时，转移的能量为 E = Q V。因此功率 P = E / t = (Q / t) V。由于电流 I = Q / t，我们立即得到 P = I V。
P = I V
For a resistor obeying Ohm’s law, V = I R. Substituting into P = I V yields P = I (I R) = I² R. Alternatively, expressing I = V / R gives P = V² / R. These forms are essential for analysing heating in components.
对于遵循欧姆定律的电阻器，V = I R。代入 P = I V 得到 P = I (I R) = I² R。同理，代入 I = V / R 可得 P = V² / R。这些形式对分析元件发热至关重要。
P = I² R = V² / R
10. Deriving Wave Speed v = f λ | 推导波速 v = f λ
A wave travels one full wavelength λ in one time period T. Speed is defined as distance divided by time, so the wave speed is v = λ / T. Frequency f is the reciprocal of period: f = 1 / T. Combining these gives the fundamental wave equation v = f λ.
波在一个周期 T 内传播一个完整的波长 λ。速度定义为距离除以时间，因此波速为 v = λ / T。频率 f 是周期的倒数：f = 1 / T。将两者结合就得到基本波动方程 v = f λ。
v = f λ
This relationship holds for all types of waves – sound, light, water waves – because it stems purely from the definitions of wavelength, frequency and speed. When waves move from one medium to another, frequency remains constant, so wavelength and speed change proportionally.
该关系适用于所有类型的波——声波、光波、水波——因为它完全来自波长、频率和速度的定义。当波从一种介质进入另一种介质时，频率保持不变，因此波长和速度按比例变化。
11. Deriving the Elastic Potential Energy Stored in a Spring: E = ½ F x = ½ k x² | 推导弹簧弹性势能：E = ½ F x = ½ k x²
When stretching a spring within its elastic limit, Hooke’s law states that the force F is proportional to extension x: F = k x, where k is the spring constant. The work done in stretching the spring by a small additional extension Δx is ΔW = F Δx. Because the force varies from 0 to F_max, the total work is the area under the force–extension graph, which is a triangle. Thus W = ½ × maximum force × total extension = ½ F x.
在弹性限度内拉伸弹簧时，胡克定律指出力 F 与伸长量 x 成正比：F = k x，其中 k 为劲度系数。拉伸弹簧微小附加长度 Δx 所做的功为 ΔW = F Δx。由于力从 0 变至最大值 F_max，总功等于力-伸长图下的面积，即一个三角形。因此 W = ½ × 最大力 × 总伸长量 = ½ F x。
E = ½ F x
Substituting F = k x into this expression gives the elastic potential energy stored: E = ½ (k x) x = ½ k x². This energy is recoverable when the spring returns to its original length, assuming no plastic deformation.
将 F = k x 代入上述表达式，得到储存的弹性势能：E = ½ (k x) x = ½ k x²。假设没有塑性形变，当弹簧恢复原长时，该能量可以释放。
E = ½ k x²
12. Deriving the Refractive Index Relation n = c / v | 推导折射率关系 n = c / v
When a light wave passes from vacuum into a medium, its speed decreases while frequency remains constant. The refractive index n of the medium is defined as the ratio of the speed of light in vacuum c to its speed in the medium v: n = c / v. This is a definition, but it can also be linked to wave properties using v = f λ. In vacuum: c = f λ₀; in medium: v = f λ. Since f is invariant, dividing gives c / v = λ₀ / λ, meaning n = λ₀ / λ. Therefore the wavelength in the medium is λ = λ₀ / n.
当光波从真空进入介质时，其速度减小而频率保持不变。介质的折射率 n 定义为真空中光速 c 与介质中光速 v 之比：n = c / v。这是一个定义，但也可以通过波的性质与 v = f λ 关联。在真空中：c = f λ₀；在介质中：v = f λ。由于 f 不变，相除得 c / v = λ₀ / λ，即 n = λ₀ / λ。因此介质中的波长为 λ = λ₀ / n。
n = c / v = λ₀ / λ
This simple derivation explains why light bends towards the normal when entering a denser medium – the reduced speed necessitates a smaller wavelength and, via wavefront continuity, a change in direction. The same principle underpins Snell’s law, n₁ sin θ₁ = n₂ sin θ₂.
这一简单推导解释了光进入光密介质时为何向法线偏折——速度减小导致波长变短，并通过波前连续性要求方向改变。同样的原理也是斯涅尔定律 n₁ sin θ₁ = n₂ sin θ₂ 的基础。
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