Key Formula Derivations in Physics | 物理核心公式推导

📚 Key Formula Derivations in Physics | 物理核心公式推导

In A-Level Physics, understanding how key formulas are derived is as important as memorising them. Derivations provide insight into the underlying principles and make problem-solving more intuitive. This article walks through the derivations of several essential equations from kinematics, dynamics, energy, circular motion, simple harmonic motion, and electricity.

在 A-Level 物理中，理解核心公式的推导过程与记忆公式同等重要。推导能帮助我们深入理解背后的原理，使解题更加得心应手。本文将详细介绍运动学、动力学、能量、圆周运动、简谐运动和电学中几个重要方程的推导过程。

1. Equations of Motion for Constant Acceleration | 匀加速运动学方程

When an object moves with constant acceleration a, the relationship between initial velocity u, final velocity v, and time t comes directly from the definition of acceleration: a = (v – u) / t. Rearranging gives the first equation.

当物体以恒定加速度 a 运动时，初速度 u、末速度 v 和时间 t 的关系直接从加速度的定义得出：a = (v – u) / t。移项可得第一个方程。

v = u + a t

The displacement s in time t is the area under the velocity-time graph. For constant acceleration, the graph is a straight line, and the area is a trapezium: s = (u + v) t / 2. Substituting v = u + a t into this expression yields the second equation.

在时间 t 内的位移 s 是速度-时间图下的面积。对于匀加速运动，图像是一条直线，该面积为一个梯形：s = (u + v) t / 2。将 v = u + a t 代入这个式子，可得到第二个方程。

s = u t + ½ a t²

To eliminate time t, solve the first equation for t = (v – u) / a and substitute into s = (u + v) t / 2. This gives the third equation, which links final velocity, initial velocity, acceleration, and displacement.

为了消去时间 t，可由第一式解出 t = (v – u) / a 并代入 s = (u + v) t / 2。这样便能得到联系末速度、初速度、加速度和位移的第三个方程。

v² = u² + 2 a s

2. Newton’s Second Law and Momentum | 牛顿第二定律与动量

Newton’s second law states that the resultant force F acting on a body is equal to the rate of change of its momentum p: F = dp / dt. For a constant mass m, momentum is p = m v, so the derivative becomes F = m (dv/dt) = m a. This is the widely used form of the second law.

牛顿第二定律指出，作用在物体上的合力 F 等于其动量 p 的变化率：F = dp / dt。当质量 m 恒定时，动量 p = m v，因此导数可写为 F = m (dv/dt) = m a，这就是我们常用的第二定律形式。

F = m a

In situations where mass changes (e.g. rocket propulsion), it is essential to use the full expression F = d(m v) / dt rather than treating mass as constant. The law also implies that impulse F Δt equals the change in momentum Δp.

在质量会发生变化的场景中（如火箭推进），必须使用完整的表达式 F = d(m v) / dt，而不能简单地将质量视为恒量。该定律还表明冲量 F Δt 等于动量的变化量 Δp。

3. Work and Kinetic Energy | 功与动能定理

Work done by a constant force F over a displacement s is W = F s (when force and displacement are parallel). Using F = m a and the equation v² = u² + 2 a s, we can express work in terms of velocity change: F s = m a s = m (v² – u²) / 2.

恒力 F 在位移 s 上所做的功为 W = F s（力与位移同向时）。利用 F = m a 和运动学方程 v² = u² + 2 a s，可以将功用速度变化表示：F s = m a s = m (v² – u²) / 2。

W = ½ m v² – ½ m u²

This result defines kinetic energy Eₖ = ½ m v². The net work done on an object equals its change in kinetic energy, a statement known as the work-energy principle. It is a powerful tool for solving problems without dealing with time directly.

由此定义了动能 Eₖ = ½ m v²。对一个物体所做的净功等于其动能的变化量，这就是功-能定理。它是解决不涉及时间问题的一个有力工具。

4. Gravitational Potential Energy and Conservation | 重力势能与机械能守恒

Lifting an object of mass m by a height h against gravity requires work W = m g h (where g is gravitational field strength). This work is stored as gravitational potential energy Eₚ. Therefore, ΔEₚ = m g Δh.

将质量为 m 的物体提升高度 h 以克服重力，需要做功 W = m g h（g 为重力场强度）。这些功以重力势能 Eₚ 的形式储存起来，因此 ΔEₚ = m g Δh。

Eₚ = m g h

In a closed system with only conservative forces (e.g. gravity), the total mechanical energy is conserved: Eₖ₁ + Eₚ₁ = Eₖ₂ + Eₚ₂. This principle allows us to relate speed and height without calculating the work of individual forces.

在只有保守力（如重力）的封闭系统中，总机械能守恒：Eₖ₁ + Eₚ₁ = Eₖ₂ + Eₚ₂。利用这一原理，无需逐项计算力做功就可将速度与高度联系起来。

5. Centripetal Acceleration | 向心加速度

An object moving at constant speed v in a circle of radius r experiences a continuous change in direction. Over a small time Δt, the change in velocity Δv points toward the centre. From similar triangles, Δv / v = Δs / r, where Δs is the arc length.

以恒定速率 v 在半径为 r 的圆周上运动的物体，速度方向持续变化。在很短的 Δt 内，速度变化量 Δv 指向圆心。由相似三角形可得 Δv / v = Δs / r，其中 Δs 为弧长。

a = v² / r

Dividing by Δt and using v = Δs / Δt gives the magnitude of centripetal acceleration: a = v² / r. Using the angular speed ω = v / r, this can also be written as a = ω² r. The direction is always towards the centre of the circle.

两边除以 Δt 并利用 v = Δs / Δt，得到向心加速度的大小：a = v² / r。利用角速度 ω = v / r，还可以写为 a = ω² r。加速度方向始终指向圆心。

a = ω² r

6. Simple Harmonic Motion (SHM) | 简谐运动

Simple harmonic motion occurs when the restoring force F is proportional to the displacement x from equilibrium and acts in the opposite direction: F = -k x, where k is a constant. Applying Newton’s second law F = m a gives m a = -k x, or a = -(k/m) x.

当回复力 F 与偏离平衡位置的位移 x 成正比且方向相反时，物体做简谐运动：F = -k x（k 为常数）。应用牛顿第二定律 F = m a 得 m a = -k x，即 a = -(k/m) x。

a = -ω² x

Comparing this with the standard SHM equation a = -ω² x reveals that ω² = k / m. Since the period T is related to angular frequency by T = 2π / ω, we obtain the period of a mass-spring system.

与该标准形式 a = -ω² x 对比，可知 ω² = k / m。由于周期 T 与角频率满足 T = 2π / ω，便得到弹簧振子的周期公式。

T = 2π √(m / k)

For a simple pendulum of length L, the restoring component is m g sinθ ≈ m g θ for small angles, leading to a = -(g / L) x. Hence ω² = g / L and T = 2π √(L / g).

对于长度为 L 的单摆，小角度时回复力分量为 m g sinθ ≈ m g θ，可得 a = -(g / L) x。因此 ω² = g / L，周期为 T = 2π √(L / g)。

7. Resistors in Series and Parallel | 串联与并联电阻

When resistors are connected in series, the same current I flows through each. The total potential difference V is the sum of individual p.d.s: V = V₁ + V₂ + …. Using Ohm’s law V = I R yields I R_total = I R₁ + I R₂ + …, so the equivalent resistance is simply the sum.

当电阻串联时，流过每个电阻的电流 I 相同。总电压 V 等于各电阻电压之和：V = V₁ + V₂ + …。由欧姆定律 V = I R 可得 I R_total = I R₁ + I R₂ + …，因此等效电阻就是各电阻之和。

R_total = R₁ + R₂ + R₃ + …

In a parallel arrangement, the voltage across each branch is the same, but the total current splits: I = I₁ + I₂ + …. Again applying Ohm’s law, V / R_total = V / R₁ + V / R₂ + …, which gives the reciprocal formula for parallel resistance.

在并联电路中，各支路两端电压相同，但总电流分叉：I = I₁ + I₂ + …。再一次利用欧姆定律，有 V / R_total = V / R₁ + V / R₂ + …，从而得出并联电阻的倒数公式。

1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃ + …

8. Capacitor Discharge | 电容器放电

A capacitor of capacitance C holds charge q with voltage V = q / C. During discharge through a resistor R, the current is I = dq / dt and by Kirchhoff’s voltage rule, V = -I R. Combining these gives a differential equation: dq / dt = -q / (R C).

电容为 C 的电容器储存电荷 q，其电压为 V = q / C。当通过电阻 R 放电时，电流 I = dq / dt，根据基尔霍夫电压定律 V = -I R。联立这些关系得到微分方程：dq / dt = -q / (R C)。

q = Q₀ exp(− t / (R C))

Solving this equation (by separation of variables) leads to an exponential decay: q = Q₀ e^(−t / (R C)), where Q₀ is the initial charge. The product R C is the time constant τ; after a time τ the charge falls to about 37% of its original value. The same form applies to voltage V = V₀ exp(− t / (R C)) and current.

通过分离变量法求解该方程，得到指数衰减规律：q = Q₀ e^(−t / (R C))，其中 Q₀ 是初始电荷。R C 的乘积称为时间常数 τ；经过时间 τ 后，电荷降至初始值的约 37%。电压和电流也遵循相同的形式 V = V₀ exp(− t / (R C))。

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Key Formula Derivations in Physics | 物理核心公式推导