📚 PH02-INS AS Physics Experiment: Free Fall Investigation | PH02-INS AS物理实验探究：自由落体研究
This article provides a detailed analysis of a typical experimental investigation based on the Edexcel International AS Physics Unit 2 Insert (PH02-INS) from January 2023. The investigation focuses on determining the acceleration due to gravity, g, using a free-fall method with an electromagnet and a trapdoor. By examining the procedure, data processing, uncertainty analysis, and potential improvements, this guide strengthens the understanding of practical skills essential for AS Physics. The insert presents a set of raw measurements of height h and fall time t, from which g is derived using the equation of motion for constant acceleration.
本文基于爱德思国际AS物理第二单元2023年1月考试插入材料（PH02-INS），对一个典型的实验探究进行详细分析。该探究采用电磁铁与陷阱门的自由落体法测量重力加速度 g。通过审视实验步骤、数据处理、不确定度分析以及可能的改进，本指南加深了对AS物理核心实验技能的理解。插入材料提供了一组高度 h 和下落时间 t 的原始测量数据，利用匀加速运动方程可由此算出 g。
1. Experimental Objective and Context | 实验目标与背景
The primary aim of this investigation is to determine a value for the acceleration of free fall, g, near the Earth’s surface. The experiment uses the relationship between the height from which a steel ball is released and the time it takes to fall to the ground. Assuming negligible air resistance and an initial velocity of zero, the motion is governed by s = ut + ½at², which simplifies to h = ½ g t². The insert provides multiple sets of (h, t) data, inviting students to calculate g and evaluate its accuracy.
本探究的主要目标是测定地球表面附近自由落体的加速度 g。实验利用钢球下落的高度与到达地面所用时间之间的关系。假设空气阻力可忽略且初速度为零，运动遵循 s = ut + ½at²，简化为 h = ½ g t²。插入材料提供了多组 (h, t) 数据，要求学生计算 g 并评估其准确性。
2. Apparatus and Setup | 器材与装置
The experiment employs a rigid stand with a clamp holding an electromagnet at a variable height. A steel sphere is held by the electromagnet and released by cutting the current. A trapdoor mechanism placed directly below acts as a switch: the instant the ball hits the trapdoor, an electrical timer stops. The height h is measured from the bottom of the ball to the trapdoor using a metre ruler. A set-square and a plumb line ensure vertical alignment. The timer has a precision of 0.01 s.
实验使用一个刚性支架，其上的夹具可固定电磁铁于不同高度。电磁铁吸住一个钢球，断电后释放。正下方的陷阱门机构充当开关：球击中陷阱门的瞬间，电子计时器停止计时。高度 h 用米尺测量球底到陷阱门的距离。直角尺和铅垂线确保垂直对正。计时器精度为 0.01 s。
3. Procedure | 实验步骤
1. Set the electromagnet at a chosen height h and measure this distance with the metre ruler, recording the uncertainty. 2. Switch on the electromagnet and suspend the steel ball. 3. Zero the timer. 4. Switch off the electromagnet to start the timer; the timer stops when the ball strikes the trapdoor. 5. Record the fall time t. 6. Repeat the measurement for the same height to obtain at least three consistent readings, then calculate the average time. 7. Repeat the entire process for at least six different heights to gather a sufficient range of data.
1. 将电磁铁设定在选定的高度 h，用米尺测量并记录不确定度。2. 接通电磁铁并吸住钢球。3. 计时器归零。4. 断开电磁铁启动计时；球撞到陷阱门时计时停止。5. 记录下落时间 t。6. 在同一高度重复测量至少三次以获得一致读数，计算平均时间。7. 对至少六个不同高度重复以上全过程，收集足够的数据范围。
4. Raw Data Presentation | 原始数据呈现
The following table replicates the typical data format seen in the Insert. Heights are measured with a ruler of sensitivity ±0.001 m, and times are recorded with a digital timer of precision ±0.01 s. Each time entry is the mean of three repeats.
下表复制了插入材料中常见的数据格式。高度用灵敏度为 ±0.001 m 的米尺测量，时间用精度为 ±0.01 s 的数字计时器记录。每个时间值是三次重复测量的平均值。
| Height h / m | Average time t / s | t² / s² |
|---|---|---|
| 0.250 | 0.23 | 0.0529 |
| 0.350 | 0.27 | 0.0729 |
| 0.450 | 0.30 | 0.0900 |
| 0.550 | 0.33 | 0.1089 |
| 0.650 | 0.36 | 0.1296 |
| 0.750 | 0.39 | 0.1521 |
Note that t² is calculated for later use in graphical analysis. The uncertainties in t are taken as the largest difference from the mean among the repeats for each height; in this simplified insert, we assume ±0.01 s for all readings.
注意，计算 t² 是为后续图像分析使用。时间的不确定度取各高度重复测量中与平均值的最大偏差；在此简化插入材料中，我们假设所有读数均为 ±0.01 s。
5. Data Analysis: Linearisation | 数据分析：线性化
The equation h = ½ g t² is not linear in t, but it becomes linear if we plot h against t². Rearranging, h = (g/2) t². Thus, a graph of h (vertical axis) versus t² (horizontal axis) should yield a straight line passing through the origin, with gradient = g/2. By determining the gradient, g can be calculated as g = 2 × gradient. This method reduces the impact of random timing errors.
方程 h = ½ g t² 关于 t 不是线性的，但若绘制 h 对 t² 的图像则变为线性。整理得 h = (g/2) t²。因此，h (纵轴) 对 t² (横轴) 图像应为一条过原点的直线，斜率 = g/2。通过求斜率，可算得 g = 2 × 斜率。此法可减小随机计时误差的影响。
6. Graph Plotting and Gradient Calculation | 绘图与斜率计算
Using the data above, plot the points and draw the best-fit straight line. Since the origin (0,0) is a valid data point, the line should pass through it. Choose two points on the line that are far apart to calculate the gradient. For instance, suppose the line passes through (0.050 s², 0.245 m) and (0.150 s², 0.735 m). The gradient is Δh / Δ(t²) = (0.735 – 0.245) / (0.150 – 0.050) = 0.490 / 0.100 = 4.90 m s⁻². Then g = 2 × 4.90 = 9.80 m s⁻². The experiment yields an estimate in excellent agreement with the accepted value of 9.81 m s⁻².
根据上述数据描点，画出最佳拟合直线。因为原点 (0,0) 是一个有效数据点，直线应过原点。选取线上距离较远的两个点计算斜率。假设直线经过 (0.050 s², 0.245 m) 和 (0.150 s², 0.735 m)，则斜率 = Δh / Δ(t²) = (0.735 – 0.245) / (0.150 – 0.050) = 0.490 / 0.100 = 4.90 m s⁻²。因此 g = 2 × 4.90 = 9.80 m s⁻²。实验得到的估计值与公认值 9.81 m s⁻² 高度一致。
7. Uncertainty Analysis | 不确定度分析
The uncertainty in g arises mainly from the uncertainties in h and t. The absolute uncertainty in h, Δh, is ±0.001 m (from the ruler). The absolute uncertainty in t², Δ(t²), can be found using the fact that if t has uncertainty Δt, then the relative uncertainty in t² is 2 × (Δt / t). For the largest t = 0.39 s, Δt = 0.01 s, so Δ(t²) / t² = 2 × (0.01/0.39) ≈ 0.051, giving Δ(t²) ≈ 0.051 × 0.1521 ≈ 0.0078 s². The gradient uncertainty can be estimated from the worst acceptable line fit, and then the uncertainty in g is Δg = g × (relative uncertainty in gradient). A simple approach yields Δg ≈ 0.2 m s⁻². Therefore, the result is expressed as g = 9.80 ± 0.20 m s⁻², which overlaps with the standard value.
g 的不确定度主要源自 h 和 t 的不确定度。h 的绝对不确定度 Δh 为 ±0.001 m (来自米尺)。t² 的绝对不确定度 Δ(t²) 可如下求得：若 t 的不确定度为 Δt，则 t² 的相对不确定度为 2 × (Δt / t)。对于最大 t = 0.39 s，Δt = 0.01 s，故 Δ(t²) / t² = 2 × (0.01/0.39) ≈ 0.051，得 Δ(t²) ≈ 0.051 × 0.1521 ≈ 0.0078 s²。斜率的不确定度可由最差可接受拟合线来估计，进而 g 的不确定度 Δg = g × (斜率的相对不确定度)。简便估算得 Δg ≈ 0.2 m s⁻²。因此，结果可表示为 g = 9.80 ± 0.20 m s⁻²，与标准值重叠。
8. Errors and Limitations | 误差与局限性
Systematic errors could arise if the timer does not start at the exact moment of release or if the ball’s initial position is not precisely at the bottom of the electromagnet. The presence of residual magnetism may delay release. Air resistance is not completely negligible, causing the measured fall time to be slightly longer than the free-fall time in a vacuum, which would lead to an underestimation of g. Parallax error when measuring h with a ruler can be reduced by using a set-square. The finite time it takes for the trapdoor to open the circuit also introduces a systematic time lag.
若计时器未能在释放瞬间准确启动，或球初始位置未精确位于电磁铁底端，会引入系统误差。剩磁可能导致延迟释放。空气阻力并非完全可忽略，测得的下降时间会比真空中自由落体时间略长，从而导致 g 被低估。用米尺测量 h 时的视差可通过直角尺来减小。陷阱门断开电路所需的有限时间也会带来系统性的时滞。
9. Improvements and Extensions | 改进与扩展
To improve accuracy, a light gate connected to a data-logger could replace the trapdoor, eliminating the mechanical delay and providing more precise timing. Using a metallic ball with a smaller diameter and higher density would reduce air resistance effects. Releasing the ball from a greater height would increase the time interval, making the relative timing uncertainty smaller. Conducting the experiment in a vacuum chamber would eliminate air resistance entirely. Additionally, taking many more readings at each height and extending the range of heights would refine the statistical reliability of the gradient.
为提高准确度，可用连接数据采集器的光电门取代陷阱门，消除机械延迟并提供更精确的计时。使用更小直径、更高密度的金属球可减小空气阻力影响。从更大高度释放会使时间间隔变大，从而降低相对计时不确定度。在真空室中操作可彻底消除空气阻力。此外，在每个高度记录更多次读数并扩大高度范围，可提升斜率统计的可靠性。
10. Conclusion | 结论
The free-fall investigation using the PH02-INS data demonstrates a valid and reliable method for measuring g. The derived value of 9.80 m s⁻² with an uncertainty of ±0.20 m s⁻² is consistent with the accepted value of 9.81 m s⁻². The exercise highlights essential practical competencies: linearisation of equations, graphical error analysis, and critical evaluation of systematic and random errors. Mastery of these skills is crucial for success in the AS Physics examination and in subsequent experimental work.
利用PH02-INS数据进行的自由落体探究展示了一种有效且可靠的测量 g 的方法。推导出的值为 9.80 ± 0.20 m s⁻²，与公认值 9.81 m s⁻² 吻合。该练习突出了核心实验技能：方程线性化、图像误差分析，以及对系统误差和随机误差的批判性评估。掌握这些技能对在AS物理考试及后续实验工作中取得成功至关重要。
11. Key Equations and Final Graph | 关键方程与最终图像
The central equation is:
h = ½ g t²
Plotting h against t² gives a gradient m = g/2. Hence:
g = 2m
The graph should have h on the vertical axis (from 0 to about 0.8 m) and t² on the horizontal axis (from 0 to about 0.16 s²). All points should lie close to a straight line through the origin, with the worst-fit line used to estimate gradient uncertainty.
核心方程为：
h = ½ g t²
绘制 h 对 t² 图像得到斜率 m = g/2。因此：
g = 2m
图像应以 h 为纵轴（范围约 0 至 0.8 m），t² 为横轴（范围约 0 至 0.16 s²）。所有数据点应靠近一条过原点的直线，用最差拟合线估计斜率的不确定度。
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