AS Physics Unit 1 Insert Jan19: Formula Derivations | AS物理单元1 Jan19公式推导

📚 AS Physics Unit 1 Insert Jan19: Formula Derivations | AS物理单元1 Jan19公式推导

The January 2019 AS Physics Unit 1 insert provides a list of essential formulas for mechanics, materials and waves. Memorising them is not enough—you need to understand how they arise from fundamental definitions and principles. This article steps through the derivations of the key equations so that you can apply them with confidence and insight. We will cover kinematics, forces, momentum, energy, power and material properties, exactly as they appear on the insert.

2019年1月AS物理单元1的公式插页列出了力学、材料与波的基本方程。死记硬背远远不够——你需要理解它们如何从基本定义和原理推导而来。本文将逐步推演关键公式的由来，帮助你满怀信心、深入透彻地应用它们。我们将涵盖运动学、力、动量、能量、功率和材料性质，完全对应公式表上的内容。

1. Definitions and v = u + at | 运动学定义与 v = u + at

Acceleration a is the rate at which velocity changes. For an object moving in a straight line with initial velocity u and final velocity v after time t, the definition states a = (v – u) / t. This is precise only when acceleration is constant, which is assumed throughout the insert. Rearranging yields the first kinematic equation: v = u + at. It tells us how velocity evolves uniformly over time and is the starting point for all other motion formulas.

加速度 a 是速度变化的快慢。对于做直线运动、初速度为 u 且经时间 t 后末速度为 v 的物体，定义式为 a = (v – u) / t。该式仅在加速度恒定时精确成立，而整个公式表都基于这一假设。移项即得第一个运动学方程：v = u + at。它描述了速度如何随时间均匀变化，是所有其他运动公式的出发点。

v = u + at

2. Displacement and Average Velocity | 位移与平均速度

The area under a velocity–time graph represents displacement s. For constant acceleration, the graph is a straight line, so the area is a trapezoid. The area of a trapezoid equals the average of the parallel sides multiplied by the base. The parallel sides are the initial and final velocities, u and v, and the base is t. Hence s = ((u + v) / 2) × t. This equation appears directly on the insert and is often used together with v = u + at.

速度–时间图线下的面积代表位移 s。加速度恒定意味着图线为直线，因此面积是一个梯形。梯形面积等于两平行边的平均值乘以底边长度。两平行边分别为初速度 u 和末速度 v，底边为时间 t。因此 s = ((u + v) / 2) × t。该方程直接出现在公式表中，常与 v = u + at 联合使用。

s = (u + v) t / 2

3. Derivation of s = ut + ½at² | s = ut + ½at² 的推导

To eliminate the final velocity from the displacement equation, substitute v = u + at into s = ((u + v)/2) t. This gives s = ((u + u + at)/2) t = ((2u + at)/2) t = (u + ½at) t. Simplifying yields s = ut + ½at². This is the second equation of motion printed on the insert. It expresses displacement purely in terms of initial velocity, acceleration and time, and is especially handy when the final velocity is unknown.

为从位移方程中消去末速度，将 v = u + at 代入 s = ((u + v)/2) t。得到 s = ((u + u + at)/2) t = ((2u + at)/2) t = (u + ½at) t。化简得 s = ut + ½at²。这是公式表上的第二个运动方程，用初速度、加速度和时间直接表示位移，在末速度未知时特别方便。

s = ut + ½at²

4. Derivation of v² = u² + 2as | v² = u² + 2as 的推导

Sometimes it is useful to have a kinematic relation that does not involve time. Start from v = u + at and solve for t: t = (v – u)/a. Substitute this expression into s = ((u + v)/2) t to get s = ((u + v)/2) × ((v – u)/a) = (v² – u²) / (2a). Multiplying through by 2a gives v² = u² + 2as. This equation is perfect for questions where you are given displacement but not time.

有时我们需要一个不含时间的运动学关系。从 v = u + at 解出时间：t = (v – u)/a。将其代入 s = ((u + v)/2) t，得 s = ((u + v)/2) × ((v – u)/a) = (v² – u²) / (2a)。两边乘以 2a 即得 v² = u² + 2as。对于已知位移但不知时间的问题，这个方程堪称完美。

v² = u² + 2as

5. Newton’s Second Law and Weight | 牛顿第二定律与重力

The insert may list Newton’s second law as F = Δp/Δt, the rate of change of momentum. When mass m is constant, Δp/Δt = mΔv/Δt = ma, which reduces to F = ma. The weight of an object is simply the gravitational force acting on it: W = mg, where g is the gravitational field strength (9.81 N kg⁻¹ on Earth). Thus weight is just a specific case of F = ma with a = g.

公式表可能将牛顿第二定律列为 F = Δp/Δt，即动量的变化率。当质量 m

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