📚 a-level-physics-jun-18-questionpaper2 Practical Investigation | A-Level物理2018年6月试卷2实验探究
In the June 2018 A-Level Physics Paper 2, one of the assessed practical investigations focused on determining the acceleration due to gravity, g, by measuring the time of free fall of a steel sphere over known distances. This classic experiment allows students to develop skills in data collection, linearisation of equations, graphical analysis, and evaluation of uncertainties. The following article provides a complete walkthrough of the practical task, including experimental setup, data processing, uncertainty calculation, and common sources of error, all aligned with the standard A-Level assessment objectives.
在2018年6月的A-Level物理试卷2中，一道实验探究题要求学生通过测量钢球自由下落不同距离所需的时间，来测定重力加速度g。这个经典实验能够帮助学生掌握数据收集、方程线性化、图像分析以及不确定度评估等关键技能。本文将完整呈现该实验探究过程，涵盖实验装置、数据处理、不确定度计算和常见误差来源，完全紧扣A-Level考评目标。
1. Experiment Overview | 实验概述
The aim of this investigation is to determine a value for the acceleration of free fall, g, and to evaluate the quality of the experimental data through uncertainty analysis. A steel ball is released from rest using an electromagnet, and the time taken to fall through a measured vertical distance is recorded electronically via a trapdoor switch or a pair of light gates. The relationship s = ½ g t² is exploited to obtain g from the gradient of a suitable straight‑line graph.
本实验的目标是测定自由落体加速度 g，并通过不确定度分析评估实验数据的质量。实验中使用电磁铁从静止释放钢球，利用触发开关或光电门以电子方式记录小球下落已知距离所需的时间。利用关系式 s = ½ g t²，可以通过一条适当的直线图像的斜率求出 g 值。
2. Theory and Linearisation | 原理与方程线性化
For an object starting from rest and falling under gravity, the vertical displacement s after time t is given by s = ½ g t², assuming negligible air resistance. Directly plotting s against t would yield a parabola, which makes accurate gradient determination difficult. Instead, the equation is rearranged into the form s/t² = ½ g, or more conveniently, s is plotted against t². In the s – t² graph, the data should lie on a straight line through the origin, with gradient m = ½ g. Hence g = 2m.
对于从静止开始仅在重力作用下的落体，竖直位移 s 与时间 t 的关系为 s = ½ g t²，这里忽略了空气阻力的影响。如果直接绘制 s 随 t 的变化曲线会得到一条抛物线，不便于精确求取斜率。因此我们将方程改写为 s/t² = ½ g 的形式，或者更常用的做法是绘制 s – t² 图像。在 s – t² 图中，数据点应位于一条过原点的直线上，斜率 m = ½ g，从而得到 g = 2m。
3. Apparatus Setup | 实验装置
The main components include a sturdy clamp stand, an electromagnet connected to a low‑voltage DC supply, a steel sphere, a metre ruler (or a set square with a calibrated vertical scale), and an electronic timer interfaced with a trapdoor or light‑gate sensor placed on the floor. A plumb line is used to ensure the electromagnet is directly above the centre of the receiving switch, so that the ball falls vertically without striking the sides.
实验装置主要包括一个稳固的铁架台、连接低压直流电源的电磁铁、一个钢球、一把米尺（或配有校准刻度的直角尺）以及与接在地面的触发开关或光电门相连的电子计时器。用铅垂线确保电磁铁正好位于接收开关中心的上方，使小球垂直下落且不会碰触边缘。
4. Data Collection Procedure | 数据收集步骤
The electromagnet is energised and the steel sphere is attached. The distance s from the bottom of the sphere to the top of the trapdoor (or the beam of the light gate) is measured with the metre ruler. The timer is reset, and the electromagnet circuit is broken to release the sphere. The timer records the fall time t. The process is repeated at least five times for each chosen value of s, and a range of s values from about 0.200 m to 1.000 m is used. All readings are recorded to the appropriate precision (e.g., ±0.001 m for distance and ±0.001 s for time).
接通电磁铁电源并吸住钢球。用米尺测量球底部到触发开关顶部（或光电门光束）的距离 s。计时器归零后，断开电磁铁电路释放小球，计时器记录下落时间 t。对每个选定的 s 值至少重复测量五次，s 的取值范围大约为 0.200 m 至 1.000 m。所有读数都应记录到合适的精度（例如距离为 ±0.001 m，时间为 ±0.001 s）。
5. Raw Data and Processing | 原始数据与处理
Below is a table of typical measurements. For each distance s, the average time tavg is calculated from the repeated trials. The square of time t² is then computed. The data are tabulated with their absolute uncertainties: the uncertainty in s is taken as half the smallest division of the metre ruler (±0.001 m), while the uncertainty in t is estimated from the spread of repeat readings, often expressed as ±0.002 s after averaging.
下表为一组典型测量数据。对于每个距离 s，根据重复试验计算平均时间 tavg，再计算时间的平方 t²。数据连同其绝对不确定度一并列出：s 的不确定度取米尺最小刻度的一半（±0.001 m），而时间的不确定度由重复读数的离散程度估算，平均后通常可表示为 ±0.002 s。
| s / m | t₁ / s | t₂ / s | t₃ / s | tavg / s | t² / s² |
|---|---|---|---|---|---|
| 0.200 | 0.202 | 0.204 | 0.200 | 0.202 | 0.0408 |
| 0.400 | 0.286 | 0.288 | 0.284 | 0.286 | 0.0818 |
| 0.600 | 0.350 | 0.352 | 0.348 | 0.350 | 0.1225 |
| 0.800 | 0.404 | 0.406 | 0.402 | 0.404 | 0.1632 |
| 1.000 | 0.450 | 0.454 | 0.452 | 0.452 | 0.2043 |
The uncertainty in t² is calculated using Δ(t²) = 2 t Δt, which gives values roughly between 0.0008 and 0.0018 s² for this data set.
t² 的不确定度采用 Δ(t²) = 2 t Δt 计算，对于本组数据该值约在 0.0008 至 0.0018 s² 之间。
6. Graphical Analysis and Gradient Determination | 图像分析与斜率求取
A graph of s (vertical axis) against t² (horizontal axis) is plotted on millimetre paper or using software. Error bars are added for both axes: horizontal bars represent Δ(t²) and vertical bars represent the uncertainty in s (±0.001 m). The best‑fit straight line is drawn through the points, and a worst‑acceptable line (either steepest or shallowest) is added to estimate the uncertainty in the gradient. The gradient of the best‑fit line, m = Δs / Δ(t²), is determined using a large triangle read from the graph.
在毫米方格纸或软件上绘制 s（纵轴）随 t²（横轴）变化的图像。横轴的误差棒代表 Δ(t²)，纵轴的误差棒则代表 s 的不确定度（±0.001 m）。通过数据点画出最佳拟合直线，再添加一条最差可接受线（最陡或最缓），用以估算斜率的不确定度。最佳拟合线的斜率 m = Δs / Δ(t²) 通过读取图线上一个大三角形的坐标计算。
From the sample data, the best‑fit line gives: m = (1.000 – 0.200) / (0.2043 – 0.0408) = 0.800 / 0.1635 ≈ 4.893 m/s². The worst‑acceptable line yields a gradient of about 5.125 m/s², giving Δm ≈ 0.232 m/s².
根据示例数据，最佳拟合线得出：m = (1.000 – 0.200) / (0.2043 – 0.0408) = 0.800 / 0.1635 ≈ 4.893 m/s²。最差可接受线的斜率约为 5.125 m/s²，因此 Δm ≈ 0.232 m/s²。
7. Calculating g and Its Uncertainty | 计算 g 值及其不确定度
Since g = 2m, the experimental value of g is obtained as g = 2 × 4.893 = 9.79 m/s². The absolute uncertainty is Δg = 2 × Δm = 2 × 0.232 = 0.46 m/s². Thus the result is expressed as g = 9.8 ± 0.5 m/s². The percentage uncertainty is (0.5 / 9.8) × 100% ≈ 5.1%.
因为 g = 2m，实验所得重力加速度值为 g = 2 × 4.893 = 9.79 m/s²。绝对不确定度为 Δg = 2 × Δm = 2 × 0.232 = 0.46 m/s²。因此结果可以表示为 g = 9.8 ± 0.5 m/s²。百分不确定度约为 (0.5 / 9.8) × 100% ≈ 5.1%。
This result is consistent with the accepted value of 9.81 m/s² within the experimental uncertainty, indicating that the systematic errors were reasonably controlled and that the random uncertainties have been realistically assessed.
该结果在实验不确定度范围内与公认值 9.81 m/s² 相符，表明系统误差得到了较好的控制，随机不确定度也得到了贴合实际的评估。
8. Sources of Error and Improvements | 误差来源与改进措施
Several sources of error can affect the accuracy of this experiment. The reaction time of the electromagnet after switching off may cause a slight delay in release, introducing a systematic error that makes the measured time slightly larger than the true free fall time. Using a dual light‑gate setup reduces this error by timing the ball only after it has started moving. Air resistance is negligible for the small steel sphere at low speeds but can be further minimised by using a heavier sphere and keeping the drop height moderate. Parallax error in measuring s can be reduced by using a set square and reading at eye level. Repeating measurements and averaging reduces random errors from timing fluctuations.
几项误差来源会影响实验的准确度。电磁铁断电后的剩磁可能导致释放延迟，这是系统误差，会使测得的时间略大于真实自由落体时间。采用双光电门装置可以避免这一问题，因为计时只在球开始运动后才启动。对于低速下落的小钢球，空气阻力的影响可以忽略，但使用更重的球体并控制下落高度能进一步减小其影响。测量 s 时的视差可通过使用直角尺并在视线水平位置读数来降低。多次测量并取平均值能够减小计时波动带来的随机误差。
9. Evaluation of the Linear Relationship | 线性关系的评价
The s – t² graph should pass through the origin if the ball is truly released from rest. The intercept of the best‑fit line can be checked: a non‑zero intercept may indicate a systematic error, such as the timer starting before the ball is actually released or an offset in the distance measurement. In the provided data, the intercept is very close to zero (approximately −0.002 m), which supports the validity of the assumed relationship. The small scatter of points about the best‑fit line confirms that random uncertainties are within acceptable limits for this level of practical work.
若小球真正从静止开始下落，s – t² 图像应过原点。最佳拟合线的截距可以作为检验：非零截距可能暗示存在系统误差，例如计时器在球尚未真正释放时便开始计时，或是距离测量存在零点偏移。在所给数据中，截距非常接近于零（约 −0.002 m），这支持了所假设关系的有效性。数据点围绕最佳拟合线的微小离散说明随机不确定度在本阶段实验的可接受范围之内。
10. Conclusion | 结论
The practical investigation successfully determined the value of g as 9.8 ± 0.5 m/s², which agrees with the accepted standard. The experiment highlights the importance of linearising an equation, careful choice of measurement instruments, and thorough uncertainty treatment. These skills form the foundation of experimental physics at A-Level and are regularly examined in practical‑based questions.
本实验探究成功测定了重力加速度 g = 9.8 ± 0.5 m/s²，与公认标准值一致。实验突出了方程线性化、谨慎选择测量工具以及深入处理不确定度的重要性。这些技能构成了A-Level实验物理的基础，也是以实验为背景的考题中经常考查的内容。
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