A-Level Physics: Experimental Investigation – Free Fall Method (WPH11/01 Jan 2021) | A-Level物理：实验探究——自由落体法（WPH11/01 2021年1月卷）

📚 A-Level Physics: Experimental Investigation – Free Fall Method (WPH11/01 Jan 2021) | A-Level物理：实验探究——自由落体法（WPH11/01 2021年1月卷）

This article explores the core experimental investigation from the January 2021 Edexcel International A-Level Physics Unit 1 paper (WPH11/01). The focus is on determining the acceleration due to gravity, g, using the free fall method. We will examine the underlying theory, apparatus, procedure, data analysis, graphical treatment, uncertainty evaluation, and common sources of error, providing a complete revision resource for students.

本文深入探讨2021年1月Edexcel国际A-Level物理单元1试卷（WPH11/01）中的核心实验探究题——利用自由落体法测量重力加速度g。我们将全面梳理实验原理、仪器、步骤、数据处理、图像分析、不确定度评价以及常见误差来源，为同学们提供一份完整的复习指南。

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1. Introduction to the Experiment | 实验简介

In Unit 1 of the Edexcel IAL Physics specification, experimental skills are tested through a data analysis and evaluation question. The January 2021 paper presented a classic experiment in which a steel ball is released from an electromagnet and falls through a light gate connected to a digital timer. The aim is to obtain a value for g by measuring the time taken for the ball to fall through a known height.

在Edexcel国际A-Level物理单元1的考试中，实验技能通过一道数据分析与评估题来考查。2021年1月的试卷给出了一个经典实验：钢球从电磁铁上释放，下落后通过一个连接数字计时器的光电门。实验目的是通过测量小球下落已知高度所需的时间来测得重力加速度g的值。

This experiment relies on the laws of motion under uniform acceleration. Because the ball starts from rest, the relationship between height h and time t is particularly simple, allowing students to plot a linear graph and extract g from the slope. Understanding every aspect of this investigation is crucial for answering examination questions that involve planning, data analysis, and evaluation.

该实验依赖匀加速直线运动的规律。由于小球从静止开始下落，高度h与时间t的关系非常简单，学生可以绘制线性图像并从斜率中提取g值。透彻理解这一探究的各个方面，对于解答涉及实验设计、数据分析和评估的考试题目至关重要。

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2. Theoretical Background | 理论背景

When an object is released from rest and falls freely under gravity, it experiences a uniform downward acceleration g. Assuming negligible air resistance, the equation of motion that links distance fallen h, initial velocity u = 0, and time t is: h = u t + ½ g t² = ½ g t².

当物体从静止开始仅在重力作用下自由下落时，它获得恒定的向下加速度g。假设空气阻力可忽略，关联下落距离h、初速度u=0和时间t的运动方程为：h = u t + ½ g t² = ½ g t²。

Rearranging gives t² = (2/g) h. If we plot h on the y‑axis and t² on the x‑axis, the graph should be a straight line passing through the origin. The gradient of this line is equal to ½ g, so g can be determined from g = 2 × gradient. If the line does not pass through the origin, it suggests a systematic error such as a time delay or an incorrect zero position.

将公式变形得到 t² = (2/g) h。若以h为纵轴、t²为横轴作图，图像应为一条过原点的直线。该直线的斜率等于½ g，因此g可由 g = 2 × 斜率 求得。如果直线不过原点，则暗示存在系统误差，例如时间延迟或零点位置不正确。

h = ½ g t²  →  g = 2 × (slope of h vs t² graph)

It is essential to use the mean time from several repeats for each height to minimize random uncertainty in t, and to square this mean value before plotting.

关键是要使用每个高度多次重复测量的平均时间，以减小时间t的随机不确定度，并在绘图前对该平均值进行平方。

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3. Apparatus and Setup | 仪器与装置

The typical apparatus used in the Jan 2021 free fall experiment consists of: a steel ball, an electromagnet, a switch, a light gate, a digital timer or interface, a metre rule, and a plumb line. The electromagnet holds the ball at a measured height above the light gate. When the switch is opened, the ball starts to fall and simultaneously triggers the timer. The timer stops when the ball passes through the light gate.

2021年1月自由落体实验常用的仪器包括：一个钢球、一个电磁铁、一个开关、一个光电门、一个数字计时器或数据采集接口、一把米尺和一个铅垂线。电磁铁将钢球固定在距光电门某一已测量的高度处。断开开关时，钢球开始下落，同时启动计时器。当钢球穿过光电门时，计时器停止计时。

Using a plumb line ensures that the ball’s path is directly vertical and that it passes cleanly through the centre of the light gate. The height h is measured from the bottom of the ball when it is attached to the electromagnet to the centre of the light gate beam. A metre rule with millimetre markings is sufficient, but a vernier calliper can be used to check the diameter of the ball if a correction is needed.

使用铅垂线可确保小球的下落路径严格竖直，且能准确通过光电门中央。高度h是从电磁铁吸附小球时小球的底部量至光电门光束中心。带有毫米刻度的米尺足以满足要求，但如果需要修正，可用游标卡尺测量小球的直径。

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4. Experimental Procedure | 实验步骤

The procedure must be systematic and designed to reduce both random and systematic uncertainties.

实验步骤必须系统化，并旨在减小随机和系统不确定度。

First, set up the electromagnet and light gate so that the ball falls vertically through the gate. Adjust the height using the metre rule and record the initial height h₁. Close the switch to energise the electromagnet and attach the ball gently. Open the switch to release the ball and record the time t displayed on the timer. Repeat the measurement at least three times for this height and calculate the mean t. Increase the height to h₂ and repeat the process for six to eight different heights, covering as wide a range as possible.

首先，安装电磁铁和光电门，使小球能竖直穿过光电门。用米尺调整高度并记录初始高度 h₁。闭合开关为电磁铁通电，轻轻吸附小球。断开开关释放小球，记录计时器显示的时间t。在当前高度重复测量至少三次，计算平均时间t。将高度增加至 h₂，重复上述过程，总共测量六到八个不同的高度，尽可能覆盖较宽的范围。

To minimise the effect of reaction time, the ball release and timer start are synchronised automatically. The switch should be operated cleanly, and the timer must be reset between trials. A typical range of heights might span from 0.200 m to 1.000 m. For each height, record all individual times and then calculate the mean before squaring it for the graph.

为减少反应时间的影响，小球释放与计时启动是同步自动完成的。开关应果断操作，每次试验间需重置计时器。典型的测量高度范围可从0.200 m到1.000 m。针对每个高度，记录所有单次时间，然后计算平均值并平方，用于绘图。

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5. Data Collection | 数据记录

Data should be tabulated clearly, with columns for height h, individual time readings, mean time, and t². An example of a properly designed table is shown below.

数据应清晰地用表格呈现，包含高度h、各次时间读数、平均时间及 t²。下表展示了一个设计恰当的表格范例。

h / m	t₁ / s	t₂ / s	t₃ / s	Mean t / s	t² / s²
0.200	0.202	0.199	0.204	0.202	0.0408
0.400	0.286	0.284	0.288	0.286	0.0818
0.600	0.350	0.348	0.352	0.350	0.1225
0.800	0.404	0.402	0.406	0.404	0.1632
1.000	0.452	0.450	0.453	0.452	0.2043

Notice that the mean time is calculated to three significant figures, consistent with the precision of the digital timer. The t² values are computed from the mean times before rounding. In the exam, candidates may be asked to complete such a table, calculate mean values, or process data to the appropriate number of significant figures.

请注意，平均时间计算至三位有效数字，与数字计时器的精度一致。t² 值是根据平均时间计算后再行舍入。考试中，考生可能会被要求补全此类表格、计算平均值或以适当的有效数字处理数据。

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6. Graphical Analysis and Calculation of g | 图像分析与g的计算

After recording the data, plot a graph of h (vertical axis) against t² (horizontal axis). Choose scales that use more than half of the graph paper in both directions. Label the axes clearly with quantities and units. Because h = ½ g t², the expected graph is a straight line through the origin with gradient m = ½ g.

记录数据后，绘制以 h（纵轴）对 t²（横轴）的图像。选择能使图像在两个方向上都占据超过一半图纸的坐标尺度。清晰标注坐标轴名称和单位。由于 h = ½ g t²，预期的图像是一条过原点的直线，斜率 m = ½ g。

Draw a best‑fit line that passes as close as possible to all plotted points. If a point is anomalous, circle it but do not include it when drawing the line. To determine the gradient, select two points well separated on the line (not data points) and use: gradient = (h₂ − h₁) / (t²₂ − t²₁). Then g = 2 × gradient.

画出一条尽可能靠近所有数据点的最佳拟合线。若某个点异常，将其圈出，但在连线时不应包含它。为求斜率，在直线上选取两个相距较远的点（非原始数据点），用公式：斜率 = (h₂ − h₁) / (t²₂ − t²₁)。然后 g = 2 × 斜率。

gradient = Δh / Δ(t²) → g = 2 × gradient

For example, using the line, a gradient of 4.85 m s⁻² would give g = 9.70 m s⁻². This value can then be compared with the accepted value of 9.81 m s⁻² to find the percentage error.

例如，从直线上求得斜率为 4.85 m s⁻²，则 g = 9.70 m s⁻²。可将此值与公认值 9.81 m s⁻² 作比较，求出百分比误差。

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7. Uncertainty and Percentage Error | 不确定度与百分比误差

The percentage error is calculated by: percentage error = |experimental g − accepted g| / accepted g × 100%. In the above example, percentage error ≈ |9.70 − 9.81|/9.81 × 100% ≈ 1.12%. This gives a measure of the accuracy of the experiment.

百分比误差的计算公式为：百分比误差 = |实验值 g − 公认值 g| / 公认值 g × 100%。在上述示例中，百分比误差 ≈ |9.70 − 9.81|/9.81 × 100% ≈ 1.12%。这可以衡量实验的准确度。

Uncertainties in the measurements of h and t are key. A metre rule typically gives an uncertainty of ±1 mm, but when measuring from the bottom of the ball held by an electromagnet, the uncertainty may be as large as ±2 mm due to parallax and the ball’s slight wobble. For time, the digital timer may have a resolution of 0.001 s, but the random uncertainty is better estimated from the spread of repeated readings, often about ±0.002 s to ±0.005 s.

h 和 t 的测量不确定度是关键。米尺通常给出的不确定度为 ±1 mm，但在测量电磁铁吸附小球底部的高度时，由于视差和小球的轻微晃动，不确定度可能大至 ±2 mm。对于时间，数字计时器的分辨率可达 0.001 s，但随机不确定度通过重复读数的离散程度更能反映，通常在 ±0.002 s 到 ±0.005 s 之间。

To combine uncertainties into the final result, one can estimate the percentage uncertainty in t² as twice the percentage uncertainty in t (since t is squared). The percentage uncertainty in the gradient can be visualised by drawing ‘worst acceptable’ lines—the steepest and the shallowest lines that still fit the error bars—and calculating their gradients. Half the difference gives the absolute uncertainty in the gradient.

要将不确定度合成到最终结果中，可估计 t² 的百分不确定度约为t的百分不确定度的两倍（因为平方关系）。斜率的百分不确定度可通过绘制“最坏可接受”线来直观展示，即仍然穿过误差棒的最陡和最缓的两条线，算出它们的斜率，其半差即为斜率的绝对不确定度。

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8. Systematic Errors and Their Reduction | 系统误差及其减小方法

One major systematic error in this setup arises from the electromagnet. After the switch is opened, the electromagnet may retain some residual magnetism and hold the ball for a few milliseconds. This causes a time delay Δt that makes every recorded time slightly larger than the true free‑fall time. As a result, the graph of h against t² does not pass through the origin; it has a positive intercept on the t² axis.

此装置的一个主要系统误差来源于电磁铁。断开开关后，电磁铁可能保留一些剩磁，将小球吸持几毫秒。这导致一个时间延迟 Δt，使每个记录的时间略大于真实的自由下落时间。因此，h 对 t² 的图像不通过原点，而是在 t² 轴上出现正截距。

This error can be identified and corrected by using two light gates placed a known distance apart, which eliminates the need for knowing the exact moment of release. An alternative is to plot h against t² and use the intercept to deduce the delay time. Another systematic error is air resistance, which becomes more significant at high speeds and for low‑density balls, reducing the acceleration and giving a smaller value of g.

这个误差可以通过使用两个光电门（相隔已知距离）来识别和修正，这样就不需要知道精确的释放时刻。另一种方法是通过绘制 h 对 t² 图像，利用截距推算出延迟时间。另一个系统误差是空气阻力，它在速度较高或小球密度较低时变得更为显著，会减小加速度，从而得出偏小的 g 值。

Parallax error when measuring height with a metre rule can be reduced by using a set square or by positioning the eye at the same level as the bottom of the ball. Ensuring the ball passes cleanly through the centre of the light gate avoids the ball interrupting the beam partially, which would cause erratic time readings.

用米尺测量高度时的视差可通过使用直角尺或使视线与小球底部齐平来减小。确保小球从光电门正中央穿过，可避免小球部分遮挡光束引起的时间读数波动。

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9. Modifications for More Accurate Results | 提高准确度的改进方案

To improve accuracy, use a longer drop height to increase t and reduce the percentage uncertainty in timing. However, at very large heights air resistance becomes more problematic, so a compromise is needed. Replacing the steel ball with a smaller, denser one (e.g. lead) can reduce air drag effects. Using a vacuum pump housing can eliminate air resistance entirely, as in the classic ‘guinea and feather’ experiment.

为提高准确度，可使用更大的下落高度以增加 t，从而降低计时的百分不确定度。然而，高度很大时空气阻力问题更突出，需要折中。改用密度更大、体积更小的球体（如铅球）可减弱空气阻力效应。使用真空罩完全消除空气阻力，正如经典的“钱币与羽毛”实验。

A more sophisticated method involves two light gates so that the time to fall between the gates is measured, removing the uncertainty of release time. Using a data logger with a high sampling rate gives more precise time values than a manual timer. The distance between light gates can be measured with a vernier calliper or a travelling microscope if precision is critical.

更精密的方案是使用两个光电门，测量经过两门之间距离的时间，从而消除释放时刻的不确定性。采用高采样率的数据记录器可获得比手动计时器更精确的时间值。若精度要求极高，光电门之间的距离可以用游标卡尺或移测显微镜测量。

When plotting the graph, including error bars for both h and t² provides a visual indication of reliability and allows the examiner to assess your judgement. Always state the modifications in clear, practical terms and explain how each addresses a specific source of error.

绘图时添加 h 和 t² 的误差棒，可提供可靠性的直观指示，并便于考官评估你的判断。永远要用清晰、切实的语言陈述改进方案，解释每项改进如何针对某特定误差源。

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10. Exam Tips for Experimental Investigation Questions | 实验探究题考试技巧

The Jan 2021 Unit 1 paper expects students to demonstrate the following skills: describe a suitable method with controlled variables, tabulate results correctly, select appropriate graphs, calculate gradients and intercepts, estimate percentage error, and evaluate systematic and random uncertainties.

2021年1月的单元1试卷希望考生展示以下技能：描述包含控制变量的合适方法，正确列表呈现结果，选择合适的图像，计算斜率和截距，估计百分比误差，以及评估系统误差和随机不确定度。

When asked to ‘comment on the reliability of your value’, always compare percentage error or the scatter of points with a benchmark. Mention whether the graph supports the theoretical relationship. Use phrases such as ‘the straight line passing close to the origin validates h = ½ g t²’ or ‘the best‑fit line has a small positive intercept, suggesting a systematic delay’.

当被要求“评价你所测 g 值的可靠性”时，务必用百分比误差或点的分散程度与基准作比较。要提及图像是否支撑理论关系。使用如“靠近原点的直线验证了 h = ½ g t²”或“最佳拟合线有小正截距，说明存在系统延迟”之类的表述。

Always write significant figures correctly: calculated g should be given to the same or one more significant figure than the least precise measurement. If h is known to 3 s.f., g should be quoted to 3 s.f. (e.g. 9.70 m s⁻²). Show working clearly for gradient calculations, including the coordinates of the two points used.

始终正确书写有效数字：计算出的 g 的有效数字位数应与最不精确的测量值相同或略多一位。若 h 为三位有效数字，g 应给至三位有效数字（例如 9.70 m s⁻²）。清晰展示斜率计算过程，包括所用两点的坐标。

Finally, when proposing modifications, avoid generic statements like ‘be more careful’. Specify instruments (‘use a micrometer to measure the ball diameter to 0.01 mm’) and procedural changes (‘use a data logger to record time at 1000 Hz to reduce random error’).

最后，在提出改进建议时，避免“再细心些”之类的笼统说法。要指明具体仪器（“使用测微计以0.01 mm精度测量小球直径”）和步骤变更（“使用数据记录器以1000 Hz记录时间以减少随机误差”）。

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