📚 AS Physics Paper 5: Mastering Formula Derivation in Exams | AS 物理 Paper 5：掌握考试中的公式推导
AS Physics Paper 5 is a practical skills examination that tests your ability to plan experiments, analyse data, and draw conclusions. One of the most critical skills assessed is the ability to rearrange a physical equation into a linear form (y = mx + c) so that you can obtain a straight-line graph and extract meaningful constants. This article will walk you through the logic of formula derivation, common linearisation strategies, key examples, and how to use your graph to determine physical quantities along with their uncertainties.
AS 物理 Paper 5 是实践技能考试，考查你设计实验、分析数据并得出结论的能力。其中一个最关键的评估技能是将物理方程重新排列为线性形式 (y = mx + c)，从而画出直线图并提取有意义的常数。本文将带你了解公式推导的逻辑、常见的线性化策略、关键示例以及如何利用图像确定物理量及其不确定性。
1. The Importance of the Straight Line Graph | 直线图的重要性
In Paper 5, you rarely plot a raw curve to find a constant. A straight line is preferred because its gradient and intercept can be determined precisely and objectively. Any deviation from linearity is immediately visible, making it easier to spot systematic errors or invalid data points. The core task is to manipulate the physics formula so that two measurable variables become y and x, while the gradient and/or intercept become expressions containing the desired physical constants.
在 Paper 5 中，你很少直接绘制原始曲线来求常数。直线图是首选，因为其斜率和截距可以精确、客观地确定。任何偏离线性的情况都会立即显现，便于发现系统误差或无效数据点。核心任务是将物理公式变形，使两个可测变量成为 y 和 x，而斜率和/或截距成为包含待求物理常数的表达式。
2. Recognising the Relationship Type | 识别关系类型
Before you derive a linear equation, you must classify the underlying relationship. The most common types you will encounter are direct proportion (y ∝ x), inverse proportion (y ∝ 1/x), power laws (y = k xⁿ), and exponential decay or growth (y = A eᵏˣ). Each category demands a different linearisation technique. For example, a power law is best handled by taking logarithms on both sides; an exponential relationship requires natural logs.
在推导线性方程之前，你必须先判断基本关系类型。最常见的类型有正比 (y ∝ x)、反比 (y ∝ 1/x)、幂律 (y = k xⁿ) 以及指数衰减或增长 (y = A eᵏˣ)。每种类型需要不同的线性化方法。例如，幂律关系最好两边取对数；指数关系则需要取自然对数。
If you are unsure, look at the units and the way the variables are combined. Does doubling one variable double the other? Does the equation contain a square root or a square? Identifying the relationship early prevents you from applying the wrong transformation and losing marks.
如果不确定，可以观察单位和变量的组合方式。一个变量翻倍，另一个是否也翻倍？公式中是否包含平方根或平方项？尽早识别关系可以避免施加错误的变换而导致失分。
3. The Linearisation Toolkit: Substitution and Logarithms | 线性化工具包：代换与对数
There are two primary tools for converting a non-linear formula into a straight-line form: variable substitution and logarithmic transformation.
将非线性公式转化为直线形式主要有两个工具：变量代换和对数变换。
Variable substitution works when the non-linearity is a simple square, square root, or inverse. For instance, if you have s = ½ a t², you can treat t² as the x-variable, making the graph s against t² a straight line with gradient ½ a. This method keeps the physical interpretation clear and avoids the complexities of log scales.
变量代换适用于非线性仅为简单的平方、平方根或倒数关系。例如，若有 s = ½ a t²，可将 t² 当作 x 变量，则画 s–t² 图便得到斜率为 ½ a 的直线。这种方法物理意义明确，也避免了对数坐标的复杂性。
Logarithmic transformation is essential for power laws and exponentials. For y = k xⁿ, take logs: log y = n log x + log k. Plotting log y against log x yields a straight line of gradient n and intercept log k. For V = V₀ e^(-t/RC), take natural logs: ln V = ln V₀ – t/RC. A graph of ln V against t gives gradient = -1/RC and intercept = ln V₀. Always specify which log base you use; in physics, natural log (ln) is standard for exponential processes.
对数变换对于幂律和指数关系至关重要。对于 y = k xⁿ，取对数得 log y = n log x + log k。画出 log y–log x 图得到斜率为 n、截距为 log k 的直线。对于 V = V₀ e^(-t/RC)，取自然对数：ln V = ln V₀ – t/RC。画出 ln V–t 图可得斜率 = -1/RC，截距 = ln V₀。务必注明使用何种对数底数；物理中指数过程通常用自然对数 (ln)。
4. Worked Example 1: The Simple Pendulum | 案例 1：单摆
The period T of a simple pendulum is given by T = 2π √(L/g), where L is the length and g the acceleration of free fall. You cannot plot T against L directly because they are not linearly related. Square both sides to obtain T² = (4π²/g) L. Now identify y = T², x = L, gradient m = 4π²/g, and intercept c = 0. Thus, g = 4π² / m.
单摆的周期 T 由 T = 2π √(L/g) 给出，其中 L 为摆长，g 为自由落体加速度。不能直接画 T–L 图，因为它们不是线性关系。两边平方得到 T² = (4π²/g) L。此时确定 y = T²，x = L，斜率 m = 4π²/g，截距 c = 0。因此，g = 4π² / m。
In your exam planning, you would describe measuring L with a metre rule and T with a stopwatch for several lengths, computing T², and plotting T² vs L. The best-fit line should pass through the origin. If it does not, a systematic error such as an incorrectly measured length may be present. The percentage uncertainty in g equals the percentage uncertainty in the gradient, which you estimate using the steepest and shallowest plausible lines.
在考试中设计实验时，你会描述用米尺测量 L，用秒表测量多次摆动求平均周期 T，计算 T²，并绘制 T²–L 图。最佳拟合线应通过原点。如果没有通过，可能存在长度测量不准确等系统误差。g 的不确定度百分比等于斜率的不确定度百分比，可通过最陡和最浅可接受直线进行估算。
5. Worked Example 2: Resistance of a Wire | 案例 2：导线电阻
The resistance R of a uniform wire is R = ρL / A, where ρ is resistivity, L length, and A cross-sectional area. For a wire of fixed diameter, A is constant, so R = (ρ/A) L. Plotting R on the y-axis against L on the x-axis gives a gradient m = ρ/A. Hence ρ = m A. If you measure the diameter d with a micrometer, A = πd²/4, and you can calculate ρ.
均匀导线的电阻 R 为 R = ρL / A，其中 ρ 为电阻率，L 为长度，A 为横截面积。对于直径固定的导线，A 为常量，故 R = (ρ/A) L。将 R 作为 y 轴，L 作为 x 轴绘图，得到斜率 m = ρ/A。于是 ρ = m A。若用千分尺测量直径 d，则 A = πd²/4，即可计算 ρ。
This example illustrates a direct proportional relationship requiring only a simple substitution. Be careful to specify that the wire must have a constant cross-section and that temperature should remain constant to keep ρ stable. You might also include an intercept check: the line should pass through the origin, confirming zero resistance at zero length. A non-zero intercept could indicate contact resistance in the circuit.
此例展示了仅需简单代换的正比关系。注意说明导线截面积必须恒定，且温度应不变以保持 ρ 稳定。还可以检查截距：直线应通过原点，确认零长度时电阻为零。截距不为零可能表明电路中存在接触电阻。
6. Worked Example 3: Capacitor Discharge | 案例 3：电容放电
During the discharge of a capacitor through a fixed resistor, the voltage V across the capacitor follows V = V₀ e^(-t/RC). To linearise, take the natural logarithm: ln V = ln V₀ – (1/RC) t. Comparing with y = mx + c, we set y = ln V, x = t, gradient m = -1/RC, and intercept c = ln V₀. Therefore, RC = -1/m, and the initial voltage V₀ can be found from e^c.
电容通过固定电阻放电时，其两端电压 V 遵循 V = V₀ e^(-t/RC)。为线性化，取自然对数：ln V = ln V₀ – (1/RC) t。与 y = mx + c 对比，令 y = ln V，x = t，斜率 m = -1/RC，截距 c = ln V₀。因此，RC = -1/m，并且可以从 e^c 求出初始电压 V₀。
This transformation is a favourite in Paper 5 because it tests your understanding of exponential behaviour. Remember to measure V at regular time intervals and to plot ln V against t, not V against t. The gradient will be negative, giving a positive time constant RC. Estimate uncertainty in RC from the range of possible gradients, and then link it to the uncertainty of capacitor or resistor values.
这种变换是 Paper 5 的常客，因为它考查你对指数行为的理解。记得按等时间间隔测量 V，并绘制 ln V–t 图，而不是 V–t 图。斜率将为负，从而得到正的时间常数 RC。通过可行斜率范围估算 RC 的不确定度，进而关联电容或电阻值的不确定度。
7. Worked Example 4: Free Fall under Gravity | 案例 4：自由落体
Starting from the SUVAT equation s = ut + ½ a t², if an object is dropped from rest (u = 0), then s = ½ g t². Treating t² as the independent variable, we get y = s and x = t², with gradient m = ½ g. Hence g = 2m. An alternative approach uses v² = u² + 2 a s; with u = 0, v² = 2 g s. Plotting v² against s gives gradient = 2 g, so g = m/2.
根据运动学公式 s = ut + ½ a t²，若物体从静止下落 (u = 0)，则 s = ½ g t²。将 t² 作为自变量，则有 y = s，x = t²，斜率 m = ½ g。因此 g = 2m。另一种方法利用 v² = u² + 2 a s；由 u = 0 得 v² = 2 g s。画 v²–s 图可得斜率 = 2 g，故 g = m/2。
Both methods are valid, but which one you choose depends on the apparatus. If using a stopwatch and ruler, the s–t² method is simple. If using light gates to measure instantaneous speeds, v²–s may give better precision. In either case, check that the intercept passes through the origin; any offset suggests timing or distance zero errors. Always consider air resistance as a source of systematic error at higher speeds.
两种方法均有效，选择哪一种取决于仪器。如果使用秒表和尺子，s–t² 方法简单。如果使用光电门测量瞬时速度，v²–s 可能精度更高。无论哪种方法，都要检查截距是否通过原点；任何偏移都暗示计时或距离零点误差。高速时需始终考虑空气阻力作为系统误差来源。
8. Worked Example 5: Verifying Newton’s Second Law | 案例 5：验证牛顿第二定律
A standard dynamics experiment accelerates a trolley by a hanging mass m providing weight m g. The total accelerating mass M_total = mass of trolley + mass on trolley + m. The acceleration a is given by a = (m g) / M_total. By keeping M_total constant and varying m, a becomes directly proportional to the force m g. Equation: a = (1/M_total) F. Plot a against F, gradient = 1/M_total, intercept should be zero if friction is compensated.
标准动力学实验通过悬挂质量 m 所产生的重力 m g 加速小车。总加速质量 M_total = 小车质量 + 小车上的附加质量 + m。加速度 a 由 a = (m g) / M_total 给出。保持 M_total 不变并改变 m，a 与力 m g 成正比。方程：a = (1/M_total) F。画出 a–F 图，斜率 = 1/M_total，若摩擦力已补偿则截距应为零。
This derivation is subtle because the hanging mass contributes to both the driving force and the inertia. By transferring masses from the trolley to the hanger, you keep the total mass unchanged. The linear fit gradient tells you the total mass, and you can compare this with the directly measured value. A non-zero intercept may suggest the track is not level, providing a useful evaluation point in the exam.
这一推导很微妙，因为悬挂质量既贡献驱动力又贡献惯性。通过将质量从小车移至悬挂端，可保持总质量不变。直线拟合的斜率可给出总质量，并能将其与直接测量值进行比较。截距不为零可能意味着轨道未调平，这在考试中是一个有用的评价点。
9. Determining Constants and Their Uncertainties from Graphs | 从图像确定常数与不确定性
Once you have plotted your straight-line graph, draw a best-fit line and two worst-fit lines (steepest and shallowest that still fit the error bars) to find the range of possible gradients. The gradient m is calculated using a large triangle: m = Δy / Δx. The uncertainty Δm is taken as half the difference between the steepest and shallowest gradients. The intercept uncertainty can be read directly from the y-intercept spread.
画出直线图后，绘制一条最佳拟合线和两条最差拟合线（可拟合误差棒的最斜和最浅线）以找到可能的斜率范围。利用大三角形计算斜率 m：m = Δy / Δx。不确定性 Δm 取最陡和最浅斜率差的一半。截距不确定性可直接从 y 截距的散布范围读取。
For a derived quantity like g = 4π²/m, the percentage uncertainty in g is the same as the percentage uncertainty in m. So %u_g = %u_m. If g is obtained from 2m, %u_g = %u_m as well. Always quote your final answer in standard form with the appropriate number of significant figures, matching the precision of your measurements.
对于像 g = 4π²/m 这样的导出量，g 的百分比不确定度等于 m 的百分比不确定度。因此 %u_g = %u_m。如果 g 由 2m 得到，%u_g 也等于 %u_m。始终用标准形式并附上适当有效数字给出最终答案，使其与测量精度相符。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Many candidates lose marks not because they do not understand the physics, but because of avoidable graphical mistakes. Here are frequent pitfalls:
许多考生失分并非因为不理解物理，而是因为可避免的作图错误。以下为常见陷阱：
| Mistake | Consequence | How to avoid |
|---|---|---|
| Plotting raw nonlinear data without transformation | Curve cannot give a reliable gradient | Always derive y = mx + c form first |
| Forgetting to label axes with quantity and unit | Marks deducted; graph meaningless | Label as ‘T² / s²’ or ‘ln(V/V)’ etc. |
| Using too small a triangle for gradient | Large percentage uncertainty | Use at least half the line length |
| Assuming intercept must be zero | Misses systematic error clues | 更多咨询请联系16621398022（同微信）
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