📚 A-Level Physics: Formula Derivation from Unit 3 Insert Jan 19 | A-Level 物理：Unit 3 插页公式推导 (2019年1月)
In Edexcel International A-Level Physics Unit 3 (Experimental Physics), the January 2019 examination insert provided a classic set of experimental data and required candidates to derive a physical quantity from first principles. This article unpacks the logical steps behind such formula derivation, using the free-fall determination of gravitational acceleration g as a representative scenario. You will learn how to move from raw measurements to a final derived value, mastering linearisation, graphical analysis, and uncertainty propagation along the way.
在 Edexcel 国际 A-Level 物理 Unit 3（实验物理）的 2019 年 1 月考试插页中，提供了一组经典实验数据，要求考生从基本原理出发推导物理量。本文以自由落体测定重力加速度 g 为代表场景，逐步拆解此类公式推导背后的逻辑步骤。你将学会如何从原始测量数据走向最终的推导值，并在过程中掌握线性化、图解分析和不确定度传递等核心技能。
1. Understanding the Unit 3 Insert | 理解 Unit 3 插页
The January 2019 insert was no mere formula sheet; it contained a real set of experimental readings obtained by releasing a steel sphere from various heights and measuring the fall time. In the exam, this insert acts as a starting point – the raw material you must process, plot, and interpret in order to uncover a hidden physical constant. Every number in those columns is a clue that, when rearranged correctly, tells a clear story.
2019 年 1 月的插页并非普通的公式表；它包含了一组真实的实验读数，通过从不同高度释放钢球并测量下落时间来获得。在考试中，这张插页是一个起点——你必须对其加工、绘图并解读，才能揭示隐藏的物理常数。每一栏中的数据都是一条线索，只要正确重组，就能讲述一个清晰的故事。
2. The Role of Data in Experimental Physics | 数据在实验物理中的作用
Raw data by itself is meaningless without a theoretical framework. In experimental physics, we do not simply collect numbers; we seek mathematical patterns that link variables according to known laws. For free fall under gravity, the kinematic equation s = ut + ½at² reduces to h = ½gt² when the initial velocity u is zero. The aim is to test whether the data supports this relationship and to extract a value for g with a credible uncertainty.
如果没有理论框架，原始数据本身毫无意义。在实验物理中，我们不只是收集数字；我们要寻找那些按照已知定律将变量联系起来的数学模式。对于自由落体运动，当初始速度 u 为零时，运动学方程 s = ut + ½at² 可简化为 h = ½gt²。我们的目标就是检验数据是否支持这一关系，并以可靠的不确定度提取出 g 的值。
3. Linearising Equations: The Key to Derivation | 线性化方程：推导的关键
The relationship h = ½gt² is non-linear in t, but it becomes linear when we cleverly choose what to plot. By treating t² as the independent variable, we can rewrite the equation as h = (g/2) t². This is now in the form y = mx + c, where y = h, x = t², the slope m = g/2, and the intercept c = 0. Linearisation not only makes hand-drawn graphs possible but also allows us to use simple gradient calculations to find g.
h = ½gt² 这个关系对 t 而言是非线性的，但如果我们巧妙地选择绘图对象，它就能变成线性关系。将 t² 视为自变量，我们可以把方程重写为 h = (g/2) t²。这现在就是 y = mx + c 的形式，其中 y = h，x = t²，斜率 m = g/2，截距 c = 0。线性化不仅使手绘图形成为可能，也让我们能用简单的梯度计算来求出 g。
h = ½gt² → h = (g/2) t²
In practice, you would calculate t² for every measured time, enter these transformed values into a table, and then plot h on the vertical axis against t² on the horizontal axis. A straight line passing through the origin confirms the theoretical model.
实际操作时，你需要为每一个测量的时间计算 t²，将这些转换后的数值填入表格，然后把 h 画在纵轴上，t² 画在横轴上。一条通过原点的直线能够证实理论模型。
4. Example from Jan 19: Free Fall Experiment | Jan 19 实例：自由落体实验
The insert from January 2019 likely displayed a dataset similar to the one below, obtained using an electromagnet, a trap-door switch and a digital timer. A typical set of readings might look like this:
2019 年 1 月的插页很可能展示了类似下面的数据集，这些数据是用电磁铁、陷阱门开关和数字计时器获取的。一组典型读数可能如下所示：
| h / m | t / s | t² / s² |
|---|---|---|
| 0.200 | 0.203 | 0.0412 |
| 0.400 | 0.286 | 0.0818 |
| 0.600 | 0.350 | 0.1225 |
| 0.800 | 0.404 | 0.1632 |
| 1.000 | 0.452 | 0.2043 |
Notice how the raw times alone do not immediately reveal a linear pattern. The addition of a t² column is the first step in the derivation process – and a classic exam expectation.
注意，仅仅看原始时间数据并不能立即显现线性规律。添加 t² 这一列是推导过程的第一步，也是经典的考试要求。
5. Deriving Gravitational Acceleration g | 推导重力加速度 g
Plotting these points on a graph of h vs t² yields a set of coordinates that should lie very close to a straight line through the origin. By drawing the best-fit line, you can select two well-separated points (x₁, y₁) and (x₂, y₂) on the line to calculate the gradient m = (y₂ – y₁) / (x₂ – x₁). Suppose the gradient of the best-fit line is 4.90 m s⁻². According to our linearised equation, m = g/2, so g = 2 × m = 2 × 4.90 = 9.80 m s⁻². This derived value is an experimental determination of the acceleration due to gravity.
把这些点画在 h–t² 图上，会得到一组应当非常接近一条穿过原点的直线的坐标。画出最佳拟合线后，你可以在线上选取两个相隔较远的点 (x₁, y₁) 和 (x₂, y₂)，计算梯度 m = (y₂ – y₁) / (x₂ – x₁)。假设最佳拟合线的梯度为 4.90 m s⁻²。根据线性化后的方程，m = g/2，所以 g = 2 × m = 2 × 4.90 = 9.80 m s⁻²。这个推导值就是对重力加速度的实验测定。
g = 2 × gradient
It is important to state your derived answer with the correct units and a suitable number of significant figures, reflecting the precision of the original measurements. If the raw times were recorded to three decimal places, g should be quoted to three significant figures, e.g. 9.80 m s⁻².
重要的是，你在陈述推导答案时要使用正确的单位和适当的小数位数，以反映原始测量的精度。如果原始时间记录到小数点后三位，那么 g 应当给出三位有效数字，例如 9.80 m s⁻²。
6. Error Analysis and Uncertainty Propagation | 误差分析与不确定度传递
Every experimental measurement carries an uncertainty. In the Jan 19 insert, typical uncertainties might be ±0.001 m in height and ±0.01 s in time. To determine the uncertainty in g, you can use the worst-line method: draw the steepest and shallowest possible fit lines that still pass through all error bars, compute their gradients mᵘᵖ and mˡᵒʷ, and then calculate g_{\text{min}} and g_{\text{max}}. The absolute uncertainty Δg is half the range, giving g = 9.80 ± Δg m s⁻².
每一个实验测量都带有不确定度。在 2019 年 1 月的插页中，典型的不确定度可能是高度 ±0.001 m 和时间 ±0.01 s。要确定 g 的不确定度，你可以使用最差线法：画出仍然穿过所有误差棒的最陡和最浅的可能拟合线，计算它们的梯度 mᵘᵖ 和 mˡᵒʷ，然后算出 g_{\text{min}} 和 g_{\text{max}}。绝对不确定度 Δg 是范围的一半，结果表示为 g = 9.80 ± Δg m s⁻²。
Alternatively, you can propagate percentage uncertainties. The percentage uncertainty in t² is twice that of t, since t² = t × t. If time has an uncertainty of 0.01 s and a typical reading is 0.40 s, the percentage uncertainty is (0.01/0.40)×100% = 2.5%. Then the percentage uncertainty in t² becomes 2 × 2.5% = 5.0%. Adding the percentage uncertainty in height (e.g., 0.5% for 0.001 m in 0.200 m) gives a total percentage uncertainty in g of about 5.5%. This technique is particularly useful when the graph does not pass perfectly through the origin.
另一种方法是传递百分不确定度。t² 的百分不确定度是 t 的两倍，因为 t² = t × t。如果时间的不确定度为 0.01 s，典型读数为 0.40 s，则百分不确定度为 (0.01/0.40)×100% = 2.5%。那么 t² 的百分不确定度变为 2 × 2.5% = 5.0%。加上高度的百分不确定度（例如 0.200 m 中 0.001 m 对应 0.5%），得出 g 的总百分不确定度约为 5.5%。当图形并不完美地通过原点时，这种技巧特别有用。
7. Graphical Analysis of Gradient and Intercept | 图解法分析斜率和截距
A good graph is the foundation of reliable derivation. Always label axes with quantity and unit, choose scales that occupy more than half the graph paper, and plot data points with small, clearly visible crosses. The best-fit line should have an even distribution of points on either side. If the line does not go through the origin, you need to comment on any systematic error – perhaps the timer started slightly before or after release, or the height measurement included the sphere’s radius.
一幅好的图形是可靠推导的基础。始终用物理量和单位标注坐标轴，选择能占据画图纸一半以上的标度，并用小而清晰可见的十字叉标出数据点。最佳拟合线应当使两侧的数据点均匀分布。如果直线没有通过原点，你就需要讨论可能的系统误差——也许是计时器在释放前或释放后稍微启动了，或者高度测量包含了球的半径。
The gradient calculated from a large triangle drawn on the graph reduces the impact of random reading errors. Always show your working: mark the triangle, write down the coordinates used, and state the gradient value clearly. This is exactly what examiners look for in Unit 3 scripts.
利用在图上画出的大三角形来计算梯度，可以减小随机读数误差的影响。一定要展示你的推导过程：标出三角形，写下所用坐标，并清晰地写出梯度值。这正是考官在 Unit 3 答卷中寻找的。
8. Practical Tips for the Exam | 考试实用技巧
When you open the insert, first scan the column headings and identify which variable is being changed (independent) and which is being measured (dependent). For free fall, height is the independent variable and time the dependent. Next, look for any cues in the question that ask you to plot a derived quantity, such as t² or h/t. Remember that linearisation is a common requirement. Jot down the target formula and the corresponding y = mx form before you start plotting.
当你打开插页时，首先浏览各栏标题，确定哪个是被改变的变量（自变量），哪个是被测量的变量（因变量）。对于自由落体，高度是自变量，时间是因变量。接着，寻找题目中任何要求你绘制导出量的提示，例如 t² 或 h/t。记住线性化是一个常见要求。在开始绘图前，草草写下目标公式及其对应的 y = mx 形式。
Use a pencil for all graphs and lines, keep your calculations neat on the facing page, and double-check that your derived g is sensible (around 9.8 m s⁻²). If it comes out as 5 m s⁻² or 15 m s⁻², you have likely made an algebraic slip in the derivation – revisit your rearranged equation immediately.
所有图形和线条都使用铅笔，在对开页上保持计算整洁，并再次核查你推导出的 g 是否合理（约 9.8 m s⁻²）。如果算出 5 m s⁻² 或 15 m s⁻²，你很可能在推导中犯了代数错误——立即回头检查你重新排列的方程。
9. Common Pitfalls in Formula Derivation | 公式推导常见误区
One frequent mistake is to plot h against t instead of t², which produces a parabola and makes gradient analysis impossible. Another is forgetting to square the time uncertainty when propagating errors. Some students also treat the intercept as irrelevant when it is clearly non-zero, missing an opportunity to discuss systematic error. Additionally, never use raw data points to calculate a gradient directly with a two-point formula; always use the best-fit line.
一个常见错误是把 h 对着 t 而不是 t² 画图，这样得到一个抛物线，使得梯度分析无法进行。另一个错误是在传递误差时忘记了对时间的不确定度取平方。有些学生还把明显不为零的截距视为无关紧要，从而错失了讨论系统误差的机会。此外，永远不要直接用原始数据点通过两点公式计算梯度；一定要使用最佳拟合线。
Finally, be wary of unit mismatches. If height is recorded in cm but the formula expects metres, you must convert before plotting. A graph of h (cm) versus t² (s²) will still be linear, but the gradient will be in cm s⁻² and g will be derived wrongly if you forget to convert back to SI units.
最后，要警惕单位不匹配的问题。如果高度以 cm 记录但公式需要的是米，你就必须在绘图前进行换算。h (cm) 与 t² (s²) 的图形仍然是线性的，但梯度单位会是 cm s⁻²，如果你忘记转换回国际单位制，推导出的 g 就会出错。
10. Bringing It All Together | 综合运用
To excel in the Unit 3 formula derivation task, you must combine three skills: theoretical understanding of the underlying physics, confident handling of linearisation and graphs, and rigorous uncertainty analysis. The Jan 19 insert tested exactly these abilities, and future papers will continue to do so. Practice with past datasets, draw clean graphs, and always interpret your computed constant in the context of the original experiment.
要在 Unit 3 公式推导任务中脱颖而出，你必须结合三种能力：对基础物理的理论理解、自信地处理线性化和图形，以及严谨的不确定度分析。2019 年 1 月的插页恰恰考查了这些能力，未来的试卷也将延续这一做法。用过往数据集进行练习，画出干净的图形，并且始终将你计算出的常数放在原始实验的背景下进行解读。
Whether you are deriving g, the Young modulus, or the resistivity of a wire, the logical flow remains the same – identify variables, linearise, plot, calculate gradient, derive target quantity, and evaluate uncertainties. This structured approach transforms a seemingly complex exam question into a straightforward analytical routine.
无论你是在推导 g、杨氏模量还是导线电阻率，其逻辑流程都是相同的——确定变量、线性化、绘图、计算梯度、推导目标量、评估不确定度。这种结构化的方法将看似复杂的试题转变为直截了当的分析流程。
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