📚 AS Physics Insert (June 2022) Formula Derivations | AS物理2022年6月公式表核心推导
Mastering the AS Physics formula sheet involves more than memorisation — it requires understanding where each equation comes from. In this article, we derive the key formulas found in the June 2022 AS Physics Insert, step by step. These derivations reinforce core principles in mechanics, waves, electricity, and materials, and will help you apply formulas confidently in exams.
掌握AS物理公式表不仅仅是记忆——还需要理解每个公式的来源。本文将逐步推导2022年6月AS物理公式表中的核心公式。这些推导过程能够强化力学、波、电学和材料学的基本原理，帮助你在考试中自信地运用各个公式。
1. Deriving v = u + at from Acceleration | 从加速度定义推导 v = u + at
Acceleration is defined as the rate of change of velocity: a = (v – u) / t. Rearranging this definition directly gives the first equation of motion: v = u + at. This relation assumes constant acceleration.
加速度定义为速度的变化率：a = (v – u) / t。直接整理这个定义式即可得到第一个运动学公式：v = u + at。此关系式假设加速度恒定。
v = u + at
This equation allows us to find final velocity when initial velocity, acceleration, and time are known. It underpins velocity–time graphs, where the gradient is acceleration and the velocity intercept is u.
该公式使我们能够在已知初速度、加速度和时间的情况下求出末速度。它为速度–时间图像奠定了基础，图中斜率为加速度，纵截距为初速度u。
2. Deriving s = ut + ½at² using Average Velocity | 用平均速度推导 s = ut + ½at²
If acceleration is constant, the average velocity during a time interval t is (u + v)/2. Displacement s is average velocity multiplied by time: s = ((u + v)/2) × t. Substituting v = u + at gives s = ((u + u + at)/2) × t = ut + ½at².
若加速度恒定，则时间t内的平均速度为(u + v)/2。位移s等于平均速度乘以时间：s = ((u + v)/2) × t。代入v = u + at得到s = ((u + u + at)/2) × t = ut + ½at²。
s = ut + ½at²
This is the second equation of motion. It is particularly useful when time is known but final velocity is not required. The term ut represents the displacement due to initial motion, while ½at² accounts for the additional displacement caused by acceleration.
这是第二个运动学公式。当已知时间但无需末速度时，此公式特别有用。ut项表示由初始运动产生的位移，而½at²则代表由加速度引起的附加位移。
3. Deriving v² = u² + 2as without Time | 消去时间推导 v² = u² + 2as
Starting with v = u + at, we solve for t to get t = (v – u)/a. Substitute this into s = ut + ½at²: s = u(v – u)/a + ½a((v – u)/a)². Simplifying yields s = (uv – u²)/a + (v² – 2uv + u²)/(2a) = (2uv – 2u² + v² – 2uv + u²)/(2a) = (v² – u²)/(2a). Multiplying both sides by 2a gives v² = u² + 2as.
从v = u + at出发，解出t = (v – u)/a。将其代入s = ut + ½at²：s = u(v – u)/a + ½a((v – u)/a)²。化简后得s = (uv – u²)/a + (v² – 2uv + u²)/(2a) = (v² – u²)/(2a)。两边同乘2a即得v² = u² + 2as。
v² = u² + 2as
This time-independent equation is powerful for linking initial and final speeds with displacement when acceleration is constant, without needing to know the time interval.
这个不含时间的公式可以在加速度恒定的情况下，将初、末速度与位移联系起来，而无需知道时间间隔，十分有用。
4. Deriving F = ma from Momentum Change | 从动量变化推导 F = ma
Newton’s second law states that resultant force equals the rate of change of momentum: F = Δp / Δt. Momentum p = mv, so Δp = mv – mu (if mass is constant). Therefore, F = (mv – mu) / t = m (v – u)/t = ma. This shows F = ma is a special case for constant mass.
牛顿第二定律指出合力等于动量的变化率：F = Δp / Δt。动量p = mv，所以Δp = mv – mu（质量不变时）。因此，F = (mv – mu) / t = m (v – u)/t = ma。这说明F = ma是质量不变时的特殊情况。
F = ma
In the AS insert, the formula often appears as F = ma, which is used extensively in dynamics. When mass changes, the more general form F = Δp/Δt should be used.
在AS公式表中，该式通常以F = ma的形式出现，广泛应用于动力学。当质量变化时，应使用更一般的形式F = Δp/Δt。
5. Deriving Elastic Potential Energy E = ½kx² | 弹性势能 E = ½kx² 的推导
According to Hooke’s law, the force needed to extend or compress a spring is F = kx, where x is the extension and k is the spring constant. The work done to stretch the spring from 0 to an extension X is the area under the force–extension graph, which is a triangle. Thus, work W = ½ × base × height = ½ × X × kX = ½kX². This work is stored as elastic potential energy: E = ½kx².
根据胡克定律，拉伸或压缩弹簧所需的力为F = kx，其中x为形变量，k为劲度系数。将弹簧从0拉伸至形变量X所做的功等于力–伸长图下的面积，即一个三角形的面积。因此，功W = ½ × 底 × 高 = ½ × X × kX = ½kX²。这部分功以弹性势能的形式储存：E = ½kx²。
E = ½kx²
This derivation highlights that elastic potential energy is not simply force times distance, because the force increases linearly with extension.
这一推导强调，弹性势能不是简单的力乘以距离，因为力随伸长量线性增加。
6. Deriving Power as P = Fv | 功率 P = Fv 的推导
Power is the rate of doing work: P = W / t. When a constant force F moves an object at constant speed v, the work done in a small time interval Δt is F × Δs (where Δs = vΔt). Thus, P = FΔs / Δt = Fv. This equation is valid for instantaneous power when force and velocity are in the same direction.
功率是做功的速率：P = W / t。当一个恒力F使物体以恒定速度v运动时，在微小时间Δt内所做的功为F × Δs（其中Δs = vΔt）。因此，P = FΔs / Δt = Fv。当力与速度方向相同时，该式对瞬时功率成立。
P = Fv
This result is especially useful in transport problems, where the maximum speed of a vehicle can be found by equating the driving power to P = Fv with resistive forces.
这一结果在交通问题中特别有用，例如通过使驱动力功率与阻力功率P = Fv相等，可求出车辆的最大速度。
7. Deriving the Wave Equation v = fλ | 波动方程 v = fλ 的推导
A wave’s frequency f is the number of complete cycles per second, and its wavelength λ is the distance between adjacent points in phase. In one period T = 1/f, the wave travels one wavelength. Hence, speed v = distance/time = λ / T = λf. This gives the fundamental relationship v = fλ for all wave types.
波的频率f是每秒完整振动的次数，波长λ是相邻同相位点之间的距离。在一个周期T = 1/f内，波前进一个波长的距离。因此，波速v = 距离/时间 = λ / T = λf。这便是适用于所有波的基本关系式 v = fλ。
v = fλ
This equation is central to wave phenomena, linking the speed of propagation to the wave’s temporal and spatial characteristics.
该公式是波动现象的核心，将传播速度与波的时间特性和空间特性联系起来。
8. Deriving Snell’s Law n₁ sin θ₁ = n₂ sin θ₂ | 斯涅尔定律 n₁ sin θ₁ = n₂ sin θ₂ 的推导
Snell’s law arises from the ratio of wave speeds in two media. Consider a wavefront incident on a boundary. The time for the wavefront to cross the interface must be the same for both sides. Using geometry, sin θ₁ = v₁t / AB and sin θ₂ = v₂t / AB, where AB is the distance along the boundary. Dividing gives sin θ₁ / sin θ₂ = v₁ / v₂. Refractive index n is defined as c/v, so n₁ sin θ₁ = (c/v₁) sin θ₁. However, a more direct derivation: since n₁ = c/v₁ and n₂ = c/v₂, then v₁/v₂ = n₂/n₁. Hence, sin θ₁ / sin θ₂ = n₂/n₁, which rearranges to n₁ sin θ₁ = n₂ sin θ₂.
斯涅尔定律源自两种介质中波速之比。考虑一个入射到界面上的波前。波前通过界面所需的时间在两介质中必须相等。通过几何关系，得到 sin θ₁ = v₁t / AB 以及 sin θ₂ = v₂t / AB，其中AB为沿界面的距离。两式相除得 sin θ₁ / sin θ₂ = v₁ / v₂。折射率定义为 n = c/v，于是 n₁ sin θ₁ = (c/v₁) sin θ₁。但更直接的推演为：因 n₁ = c/v₁、n₂ = c/v₂，故 v₁/v₂ = n₂/n₁。于是 sin θ₁ / sin θ₂ = n₂/n₁，整理得 n₁ sin θ₁ = n₂ sin θ₂。
n₁ sin θ₁ = n₂ sin θ₂
This law is fundamental in optics and forms the basis for understanding refraction, total internal reflection, and lens behaviour.
该定律是光学的基础，是理解折射、全内反射和透镜行为的关键。
9. Deriving Resistors in Series: R = R₁ + R₂ | 串联电阻推导：R = R₁ + R₂
In a series circuit, the current I through each resistor is the same. The total potential difference V across the combination is the sum of the individual p.d.s: V = V₁ + V₂. Using Ohm’s law V = IR, we have IR = IR₁ + IR₂. Cancelling I gives R = R₁ + R₂. This shows that total resistance in series is simply the algebraic sum of individual resistances.
在串联电路中，通过每个电阻的电流 I 相同。组合两端的总电势差 V 等于各个电势差之和：V = V₁ + V₂。利用欧姆定律 V = IR，得到 IR = IR₁ + IR₂。约去 I 即得 R = R₁ + R₂。这表明串联总电阻等于各电阻的代数和。
R = R₁ + R₂ (series)
For more than two resistors, the rule extends to R = R₁ + R₂ + R₃ + …
对于两个以上的电阻，该规则扩展为 R = R₁ + R₂ + R₃ + …
10. Deriving Resistors in Parallel: 1/R = 1/R₁ + 1/R₂ | 并联电阻推导：1/R = 1/R₁ + 1/R₂
In a parallel circuit, the potential difference V across each branch is the same. The total current I is the sum of the branch currents: I = I₁ + I₂. Applying Ohm’s law to each branch gives I₁ = V/R₁ and I₂ = V/R₂, and for the equivalent resistance R, I = V/R. Substituting, V/R = V/R₁ + V/R₂. Dividing through by V yields 1/R = 1/R₁ + 1/R₂. Thus, the reciprocal of the total resistance equals the sum of the reciprocals of individual resistances.
在并联电路中，每条支路两端的电势差 V 相同。总电流 I 等于各支路电流之和：I = I₁ + I₂。对每条支路应用欧姆定律得 I₁ = V/R₁，I₂ = V/R₂；对于等效电阻 R，有 I = V/R。代入后得 V/R = V/R₁ + V/R₂。两边同除以 V，得到 1/R = 1/R₁ + 1/R₂。因此，总电阻的倒数等于各电阻倒数之和。
1/R = 1/R₁ + 1/R₂ (parallel)
It is important to remember that parallel combinations always result in a total resistance lower than the smallest individual resistor.
需要记住，并联组合的总电阻总是小于其中最小的单个电阻。
11. Deriving Electrical Power Formulas P = IV, P = I²R, P = V²/R | 电功率公式 P = IV、P = I²R、P = V²/R 的推导
Electrical power is the rate at which energy is transferred. When a charge Q moves through a potential difference V, the energy transferred is E = QV. Power P = E/t = (QV)/t. Since current I = Q/t, we obtain P = IV. Using Ohm’s law V = IR, substituting gives P = I × (IR) = I²R. Alternatively, substituting I = V/R yields P = (V/R) × V = V²/R. These three forms are equivalent for a resistor.
电功率是能量转移的速率。当电荷 Q 通过电势差 V 时，转移的能量为 E = QV。功率 P = E/t = (QV)/t。由于电流 I = Q/t，我们得到 P = IV。利用欧姆定律 V = IR 代入，得到 P = I × (IR) = I²R。或者，代入 I = V/R 得到 P = (V/R) × V = V²/R。对于电阻，这三种形式是等价的。
P = IV P = I²R P = V²/R
Select the most convenient form based on the known quantities in a given circuit problem.
根据电路问题中已知的量，选择最方便的形式使用。
12. Deriving the Young Modulus E = Stress / Strain | 杨氏模量 E = 应力 / 应变 的推导
The Young modulus E quantifies the stiffness of a material. It is defined as the ratio of tensile stress to tensile strain within the limit of proportionality. Tensile stress = F/A, where F is the applied force and A is the cross-sectional area. Tensile strain = ΔL/L, where ΔL is the extension and L is the original length. Therefore, E = (F/A) / (ΔL/L), which is often rearranged for calculations. This relationship holds only for materials obeying Hooke’s law in the linear region.
杨氏模量 E 量化了材料的刚度，其定义为在比例极限内，拉伸应力与拉伸应变之比。拉伸应力 = F/A，其中 F 为施加的力，A 为截面积。拉伸应变 = ΔL/L，其中 ΔL 为伸长量，L 为原始长度。因此，E = (F/A) / (ΔL/L)，该式在计算时常进行变形使用。此关系仅适用于在线性区域内遵循胡克定律的材料。
E = (F/A) ÷ (ΔL/L) or E = FL / AΔL
The Young modulus appears on the AS insert as a fundamental material property, important for comparing different materials’ elastic behaviour.
杨氏模量作为一项基本的材料属性出现在AS公式表中，对于比较不同材料的弹性行为至关重要。
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