📚 Key Formula Derivations in Cambridge Lower Secondary Complete Physics (2nd Edition) | 剑桥初中物理第二版重要公式推导
The Cambridge Lower Secondary Complete Physics Student Book (2nd Edition) introduces fundamental physical concepts through clear explanations and derivations. Understanding how key formulas are derived not only aids memorisation but also deepens comprehension of the underlying principles. This article walks through the derivations of essential equations, linking them to observable phenomena and logical reasoning.
剑桥初中物理学生用书（第二版）通过清晰的讲解与推导介绍基础物理概念。理解核心公式的推导过程不仅有助于记忆，还能加深对原理的领悟。本文将梳理关键方程的推导思路，将公式与可观察的现象和逻辑推理联系起来。
1. Speed and Average Speed | 速度与平均速度
Speed is defined as the distance travelled per unit of time. If an object moves uniformly, covering a distance s in a time interval t, the average speed v is given by the simple ratio of distance to time.
速度定义为单位时间内通过的距离。如果物体匀速运动，在时间 t 内通过距离 s，则平均速度 v 就是距离与时间的比值。
v = s ÷ t
This relationship emerges from the direct proportionality between distance and time when speed is constant. Rearranging the formula yields s = v × t, which allows us to predict how far an object travels. The derivation is based on the definition of speed and does not require calculus; it is a fundamental building block for kinematics.
这一关系源于匀速时距离与时间的正比性。将公式变形可得 s = v × t，用以预测物体运动的距离。该推导基于速度的定义，无需微积分，是运动学的基本构建块。
2. Density: Mass and Volume | 密度：质量与体积
Density measures how much mass is packed into a given volume. For a uniform material, mass m is found to increase linearly with volume V. Thus, density ρ (rho) is the constant of proportionality.
密度衡量单位体积内所含的质量。对均匀物质而言，质量 m 随体积 V 线性增加。因此，密度 ρ 就是该比例常数。
ρ = m ÷ V
Experimental data show that doubling the volume of the same substance doubles its mass, so the ratio remains constant. This derivation highlights that density is an intrinsic property of a material, independent of the sample size. The formula can be rearranged to find mass or volume when the other quantities are known.
实验数据显示，同种物质的体积加倍时质量也加倍，因此比值保持不变。该推导强调密度是物质的内禀性质，与样品大小无关。公式可变形，用于在已知其他量的情况下求质量或体积。
3. Force, Mass, and Acceleration | 力、质量与加速度
Newton’s second law of motion states that the resultant force acting on an object is directly proportional to the acceleration it produces, and inversely proportional to the object’s mass. Through experiments using trolleys and ticker timers, we observe F ∝ m a.
牛顿第二运动定律指出，作用在物体上的合力与其产生的加速度成正比，与物体质量成反比。通过使用小车和打点计时器的实验，我们观察到 F ∝ m a。
By choosing a suitable unit of force (the newton), we eliminate the proportionality constant, leading to the familiar vector equation:
通过选择合适的力的单位（牛顿），消去比例系数，得到熟悉的矢量公式：
F = m × a
This derivation links the concepts of inertia and change in motion. When mass is constant, a larger force causes a larger acceleration; when force is constant, a larger mass results in a smaller acceleration. The formula is central to all force calculations in dynamics.
这一推导将惯性与运动变化联系起来。质量不变时，力越大加速度越大；力不变时，质量越大加速度越小。该公式是动力学所有力计算的核心。
4. Weight and Gravitational Field Strength | 重力与引力场强度
Weight is the gravitational force experienced by a mass. Near the Earth’s surface, the gravitational field strength g is approximately 9.8 N/kg. Substituting g for acceleration in F = m a gives the weight formula directly.
重量是物体由于引力而受到的力。在地球表面附近，引力场强度 g 约为9.8 N/kg。将 g 代入 F = m a 中的加速度，直接得到重量公式。
W = m × g
This derivation shows that weight is a special case of Newton’s second law where the acceleration is due to gravity. The value of g can vary slightly with altitude and latitude, but on Earth’s surface it is treated as constant for most problems. Understanding this relationship helps distinguish mass from weight.
这一推导表明重量是牛顿第二定律在加速度为重力加速度时的特例。g 值随海拔和纬度略有变化，但在地表的大多数问题中都视为常数。理解该关系有助于区分质量与重量。
5. Pressure in Solids | 固体压强
Pressure is defined as the force acting perpendicularly per unit area. When a solid object rests on a surface, the force exerted equals its weight (if no other forces are present), distributed over the contact area A.
压强定义为垂直于单位面积上的力。当固体静止在平面上时，施加的力等于其重量（若无其他力），分布在接触面积 A 上。
p = F ÷ A
The derivation follows from the idea of ‘spreading’ the force. A sharp object has a small area, producing high pressure with the same force; a blunt object has a larger area, reducing the pressure. This principle explains why knives cut and why wide tyres prevent sinking into soft ground.
该推导源于“分散”力的概念。尖锐物体面积小，相同力产生高压；钝物体面积大，压强降低。该原理解释了为什么刀能切割以及宽轮胎为何能防止陷入软地。
6. Pressure in Liquids | 液体压强
Liquid pressure increases with depth because of the weight of the fluid above. To derive the pressure at depth h, consider a column of liquid with cross-sectional area A, height h, and density ρ.
液体压强随深度增加，这是由上方液体的重量引起的。为推导深度 h 处的压强，考虑一个截面积为 A、高为 h、密度为 ρ 的液柱。
The mass of the column is m = ρ × A × h. Its weight is W = m g = ρ A h g. This weight acts on the base, so the pressure is:
液柱质量为 m = ρ × A × h，重量为 W = m g = ρ A h g。该重量作用在底部，因此压强为：
p = W ÷ A = ρ g h
This elegant derivation shows that liquid pressure depends only on density, gravitational field strength, and depth, not on the total mass or area of the container. It correctly predicts that pressure is the same at all points at the same horizontal level in a connected fluid.
这一简洁的推导表明，液体压强仅取决于密度、引力场强度和深度，而与总质量或容器面积无关。它正确预测了连通流体中同一水平面上各点压强相等。
7. Work Done by a Force | 力所做的功
Work is done when a force moves an object in the direction of the force. The simplest derivation considers a constant force F acting over a displacement d parallel to the force.
当力使物体沿力的方向移动时，做了功。最简单的推导假设恒力 F 作用在平行于力的位移 d 上。
W = F × d
The work done equals the product of the force magnitude and the distance moved in the direction of the force. If the force is not parallel, only the component along the displacement is used. This definition links energy transfer to mechanical action and is measured in joules (J).
所做的功等于力的大小与沿力方向移动距离的乘积。若力不平行，只取沿位移方向的分量。该定义将能量转移与机械作用联系起来，单位为焦耳 (J)。
8. Gravitational Potential Energy | 重力势能
When an object is lifted vertically at constant speed, the lifting force must balance its weight. The work done against gravity is stored as gravitational potential energy (Ep).
当物体匀速竖直提升时，提升的力必须与其重力平衡。克服重力所做的功储存为重力势能 (Ep)。
The minimum force required is F = m g. The distance raised is h. Therefore, the work done, and hence the gain in gravitational potential energy, is:
所需最小力为 F = m g，提升高度为 h。因此，所做的功及获得的重力势能为：
Ep = m g h
This derivation uses the work formula directly. It assumes g is constant, which is a good approximation near the Earth’s surface. The equation shows that gravitational potential energy depends on mass, height, and the strength of the gravitational field.
该推导直接使用了功的公式，并假设 g 为常数，这在地表附近是良好的近似。方程表明重力势能取决于质量、高度和引力场强度。
9. Kinetic Energy | 动能
Kinetic energy is the energy an object possesses due to its motion. To derive its formula, consider a constant net force F accelerating a mass m from rest over a distance s.
动能是物体因运动而具有的能量。为推导其公式，考虑一个恒定合力 F 将质量 m 从静止加速一段距离 s。
The work done by the force is W = F s. Using F = m a, we have W = m a s. From the equation of motion v2 = u2 + 2 a s and setting initial speed u = 0, we obtain a s = v2 / 2. Substituting this back:
力做的功为 W = F s。代入 F = m a 得 W = m a s。由运动方程 v2 = u2 + 2 a s 并令初速度 u = 0，得到 a s = v2 / 2。代回得：
W = m × (v2 / 2) = ½ m v2
This work done on the object becomes its kinetic energy. Therefore,
对物体所做的功转化为它的动能。因此，
Ek = ½ m v2
The derivation elegantly connects force, work, and motion. It shows that kinetic energy depends on the square of the speed, so doubling the speed quadruples the energy. This result is fundamental to understanding collisions and energy conservation.
该推导巧妙地将力、功和运动联系起来。它表明动能与速度的平方成正比，因此速度加倍则能量变为四倍。该结果是理解碰撞和能量守恒的基础。
10. Power | 功率
Power measures the rate at which work is done or energy is transferred. By definition, if a work W is done in a time interval t, the average power P is simple division.
功率衡量做功或能量转移的快慢。根据定义，如果在时间 t 内做了功 W，则平均功率 P 就是简单的相除。
P = W ÷ t
This formula can be combined with the work definition to give P = F d / t = F v, where v is the constant speed of the object. Thus, for a constant force acting on a moving object, power is the product of force and velocity. The unit of power is the watt (W), equivalent to one joule per second.
该公式可与功的定义结合得到 P = F d / t = F v，其中 v 是物体的恒定速度。因此，对于作用在运动物体上的恒力，功率是力与速度的乘积。功率单位为瓦特 (W)，相当于每秒一焦耳。
11. Ohm’s Law | 欧姆定律
Ohm’s law describes how the current through a conductor depends on the voltage across it. Experiments with a fixed resistor at constant temperature reveal a direct proportionality between current I and voltage V.
欧姆定律描述了通过导体的电流如何取决于其两端的电压。在恒定温度下对定值电阻的实验揭示出电流 I 与电压 V 的正比关系。
V = I × R
Here, R is the resistance, which acts as the proportionality constant. The equation can be rearranged to I = V / R or R = V / I. The derivation is based on plotting a graph of V against I, which yields a straight line through the origin, confirming the proportional relationship. This law is fundamental to circuit analysis.
这里 R 是电阻，充当比例常数。方程可变形为 I = V / R 或 R = V / I。推导基于绘制 V-I 图像，得到一条过原点的直线，验证了正比关系。该定律是电路分析的基础。
12. Resistors in Series | 串联电阻
When resistors are connected in series, the same current I flows through each one. The total voltage across the combination is the sum of the individual voltages: Vtotal = V1 + V2.
电阻串联时，每个电阻流过相同的电流 I。总电压等于各电阻电压之和：Vtotal = V1 + V2。
Applying Ohm’s law to each resistor and to the equivalent resistance Rtotal, we have V1 = I R1, V2 = I R2, and Vtotal = I Rtotal. Substituting into the voltage sum gives:
对每个电阻和等效电阻 Rtotal 应用欧姆定律，有 V1 = I R1、V2 = I R2 和 Vtotal = I Rtotal。代入电压求和式得：
I Rtotal = I R1 + I R2
Cancelling the common current I yields the series resistance formula:
约去公因子 I 得到串联电阻公式：
Rtotal = R1 + R2
This derivation can be extended to any number of resistors. It highlights how adding resistors in series increases the total opposition to current, as the same current must overcome all resistors sequentially.
该推导可推广至任意数量的电阻。它说明了串联电阻如何增大对电流的总阻力，因为同一电流必须依次克服所有电阻。
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