📚 A-Level Physics June 2018 Paper 3: Experimental Determination of g | A-Level 物理 2018年6月卷3：重力加速度实验探究
The June 2018 A-Level Physics Paper 3 is a practical skills assessment designed to test candidates’ ability to plan experiments, collect and process data, and evaluate uncertainties. One typical investigation in this paper involves measuring the acceleration of free fall, g, using a steel ball falling through two light gates connected to a timer. The experiment cleverly eliminates the need to know the initial velocity by applying the kinematic equation s = ut + ½gt² and analysing a linear graph of s/t against t.
2018年6月A-Level物理试卷3是一份实验技能试卷，旨在考查考生设计实验、采集处理数据以及评估不确定度的能力。该试卷中一个典型的探究实验是利用钢球下落穿过两个连接计时器的光电门来测量重力加速度g。该实验巧妙地通过运动学方程s = ut + ½gt²，并分析s/t对t的线性图像，消除了必须已知初速度的困难。
1. Overview of the Investigation | 实验探究概述
In this investigation, a small steel ball is released from an electromagnet positioned a few centimetres above a first light gate. The ball interrupts the first light gate, starting a digital timer, and later interrupts a second light gate, stopping the timer. The distance s between the two light gates is varied, and the corresponding fall time t is recorded. Because the ball already has an unknown velocity u when it reaches the first light gate, the data are not directly analysed using s = ½gt². Instead, the equation is rearranged into a linear form: s/t = u + ½ g · t.
在该探究中，一个小钢球从一个位于第一光电门上方几厘米处的电磁铁释放。钢球先后遮断第一和第二个光电门，从而启动和停止数字计时器。改变两光电门之间的距离 s，记录相应的下落时间 t。由于钢球到达第一光电门时已具有未知的速度u，数据不能直接用s = ½gt²处理。因此，将方程变形为线性形式：s/t = u + ½ g · t。
By plotting a graph of s/t (on the y-axis) against t (on the x-axis), the gradient equals ½ g, and the y-intercept equals u. This allows g to be determined without needing to measure u or to ensure the ball is released from exactly zero initial speed at the first light gate. The method also provides a clear route for estimating the uncertainty in g from the best-fit and worst-fit lines.
通过绘制s/t（y轴）对t（x轴）的图像，斜率等于½ g，y轴截距等于u。这样既不需要测量u，也不要求钢球在第一光电门处的初速严格为零，就能够求得g。该方法还提供了通过最佳拟合线和最差拟合线估算g的不确定度的清晰途径。
2. Apparatus and Setup | 实验装置与搭建
The apparatus includes an electromagnet connected to a low-voltage DC supply, a steel ball bearing (approximately 1–2 cm diameter), two light gates interfaced with a digital timer (resolution 0.001 s or better), a metre rule and a vernier calliper for measuring s, a micrometer screw gauge for determining the ball’s diameter, a plumb line to ensure vertical alignment, and a retort stand with clamps to hold the light gates. The second light gate is fixed at a known height, while the first light gate can be moved to vary s.
实验装置包括：连接低压直流电源的电磁铁，一个直径约1–2 cm的钢球，两个与数字计时器（分辨率为0.001 s或更优）连接的光电门，测量s的米尺和游标卡尺，测量钢球直径的螺旋测微器，确保垂直对齐的铅垂线，以及用于固定光电门的铁架台和夹具。第二光电门固定在一个已知高度，第一光电门可以移动以改变s。
The electromagnet must hold the ball securely and release it cleanly without imparting any sideways motion. To minimise the effect of residual magnetism, the current is switched off rapidly using a toggle switch, and the ball is allowed to drop under gravity. The light gates are aligned so that the ball passes through the centre of the infrared beams, and the timer is set to start on the first interruption and stop on the second.
电磁铁必须牢固地吸附钢球，并在切断电流时干净地释放，不能对钢球施加任何侧向运动。为了减小剩磁的影响，使用拨动开关快速断开电流，让钢球在重力作用下下落。两个光电门需要对齐，使钢球通过红外光束的中心位置，计时器设定为第一次遮断时启动、第二次遮断时停止。
3. Experimental Procedure | 实验步骤
1. Use the micrometer to measure the diameter of the steel ball at several different orientations. Record at least five readings, calculate the mean diameter, and determine the absolute uncertainty (e.g. ± ½ range) for the diameter.
1. 使用螺旋测微器在多个不同方向测量钢球直径。至少记录五组读数，计算平均直径，并确定直径的绝对不确定度（例如，± ½ 极差）。
2. Set up the two light gates vertically, one below the other, with the electromagnet positioned just above the top light gate. Use the plumb line to check that the line through the centres of both light gates is vertical.
2. 将两个光电门垂直放置，一个在另一个的下方，电磁铁则紧贴在上方光电门的上方。使用铅垂线检查通过两个光电门中心点的连线是否竖直。
3. Place the first light gate such that the distance s between the two gates is at its maximum (e.g. 0.800 m). Measure s using a metre rule, ensuring your eye is level with the scale to avoid parallax error. Record the value of s with its uncertainty (e.g. ±0.001 m if using a precise scale).
3. 放置第一个光电门，使两光电门之间的距离s为最大预设值（例如0.800 m）。使用米尺测量s，确保视线与刻度齐平以避免视差。记录s值及其不确定度（如使用精密刻度可为±0.001 m）。
4. Switch on the electromagnet, attach the steel ball, and then switch off the electromagnet to release the ball. Record the time t displayed on the timer for the fall between the two light gates. Repeat this measurement three times under the same conditions to obtain t1, t2, t3, and calculate the mean time.
4. 接通电磁铁，吸附钢球，然后切断电磁铁电流释放钢球。记录计时器上显示的两个光电门之间的下落时间t。在相同条件下重复该测量三次，得到t1、t2、t3，并计算平均时间。
5. Decrease the distance s by moving the first light gate downward in about 0.100 m decrements (or as directed in the question). After each adjustment, remeasure s, and repeat the timing experiment as before. Record all readings in a suitable table.
5. 通过向下移动第一个光电门来减小距离s，每次大约减小0.100 m（或按照题目要求）。每次调整后重新测量s，并重复上述计时实验。将所有读数记录在合适的表格中。
6. For each s, calculate the mean time t and the quantity s/t. The raw data and processed columns are now ready for graphing.
6. 对每个s，计算平均时间t和量s/t。原始数据和处理后的列数据现已准备就绪，可以绘图。
4. Data Collection and Table Design | 数据收集与表格设计
A typical results table for this investigation is shown below. The columns must be clearly labelled with the quantity and unit, and uncertainties should be stated in the header. The precision of the recorded values should reflect the resolution of the measuring instruments used.
该探究中典型的实验表格如下所示。各列必须明确标出量和单位，表头中应说明不确定度。记录值的精确度需反映所用测量仪器的分辨率。
| s / m (±0.001 m) | t₁ / s | t₂ / s | t₃ / s | Mean t / s | s/t / m s⁻¹ |
|---|---|---|---|---|---|
| 0.800 | 0.247 | 0.248 | 0.248 | 0.2477 | 3.23 |
| 0.700 | 0.224 | 0.225 | 0.225 | 0.2247 | 3.11 |
| 0.600 | 0.199 | 0.200 | 0.200 | 0.1997 | 3.00 |
| 0.500 | 0.172 | 0.173 | 0.173 | 0.1727 | 2.90 |
For each raw time measurement, the repeat readings typically agree to within ±0.001 s for a digital timer with millisecond resolution. The absolute uncertainty in the mean time is taken as the larger of the half-range or the instrument resolution, while the uncertainty in s/t is propagated from the uncertainties in s and t using the percentage method.
对于每一个时间原始测量值，对于具有毫秒分辨率的数字计时器，重复读数通常在±0.001 s内一致。平均时间的绝对不确定度取半极差与仪器分辨率中的较大者，而s/t的不确定度则由s和t的不确定度通过百分差法传递求得。
5. Graph Plotting and Analysis | 作图与图像分析
A graph of s/t (vertical axis) against t (horizontal axis) should be plotted on standard graph paper. The axes must be labelled with the quantity and unit, and the scales should be chosen so that the data points occupy at least half the grid area in both directions. Each data point is marked with a small cross, and error bars for both s/t and t are added if required by the question.
在标准坐标纸上绘制s/t（纵轴）对t（横轴）的图像。坐标轴必须标出量和单位，所选比例应使数据点在两个方向上都至少占据网格区域的一半。每个数据点用小十字标出，如果题目要求，还需添加s/t和t的误差棒。
As the relationship is linear (s/t = u + ½g · t), a straight best-fit line should be drawn through the points, ensuring an even spread of points above and below the line. Two additional worst-acceptable lines should be drawn: one with the steepest gradient and one with the shallowest gradient that still reasonably fit the data, taking the error bars into account.
由于该关系是线性的（s/t = u + ½g · t），应通过数据点绘制一条最佳拟合直线，并确保点在直线上下均匀分布。另需绘制两条最差可接受线：一条具有最大斜率，另一条具有最小斜率，并且需考虑误差棒，使它们仍能合理地拟合数据。
The gradient of the best-fit line, m = ½g, is calculated by taking a large triangle on the graph. For the worst lines, the gradients m_max and m_min are similarly found. The absolute uncertainty in the gradient, Δm, is given by (m_max – m_min)/2. The value of g is then determined as g = 2m, with an uncertainty Δg = 2Δm.
最佳拟合线的斜率 m = ½g，通过在图线上选取一个大三角形计算得到。对最差线同样求出斜率 m_max 和 m_min。斜率的绝对不确定度 Δm 由 (m_max – m_min)/2 给出。然后由 g = 2m 求出 g 值，不确定度 Δg = 2Δm。
6. Calculating the Acceleration g | 计算重力加速度
From the graph, a typical best-fit gradient might be m = 4.90 m s⁻². This gives an experimental value for g:
由图像得到的典型最佳拟合斜率可能为 m = 4.90 m s⁻²。这给出了实验测得的g值：
g = 2 × 4.90 = 9.80 m s⁻²
If the worst-acceptable slopes are m_max = 5.05 m s⁻² and m_min = 4.75 m s⁻², the uncertainty in the gradient is Δm = (5.05 – 4.75)/2 = 0.15 m s⁻². Hence, Δg = 2 × 0.15 = 0.30 m s⁻². The final result is expressed as g = (9.80 ± 0.30) m s⁻².
如果最差可接受斜率分别为 m_max = 5.05 m s⁻² 和 m_min = 4.75 m s⁻²，则斜率的不确定度为 Δm = (5.05 – 4.75)/2 = 0.15 m s⁻²。因此，Δg = 2 × 0.15 = 0.30 m s⁻²。最终结果表示为 g = (9.80 ± 0.30) m s⁻²。
This value should be compared with the accepted local value of 9.81 m s⁻². The percentage difference is calculated and comments are made on the agreement. The y-intercept (u) from the graph can be checked for consistency with an independent estimate of the velocity at the first light gate, perhaps from v = s/t at the smallest t, or by using the equation v² = u² + 2as.
应将此值与当地公认值9.81 m s⁻²进行比较。计算百分差，并就结果的一致性进行评述。从图像获得的y轴截距u也可以通过与第一光电门处速度的独立估算值（例如用最小t值时的s/t，或利用v² = u² + 2as）进行对比，以检验一致性。
7. Uncertainty Estimation and Propagation | 不确定度估算与传递
Uncertainties arise from several sources in this experiment. The absolute uncertainty in s (Δs) is determined by the metre rule or vernier calliper; a typical value is ±0.001 m. The uncertainty in the mean time t (Δt) combines the reaction time (negligible with light gates) and the spread of repeat readings. For the data above, half-range for t might be 0.0005 s, which is less than the timer resolution, so Δt is taken as ±0.001 s.
该实验的不确定度来自多个来源。s的绝对不确定度（Δs）由米尺或游标卡尺决定，典型值为±0.001 m。平均时间t的不确定度（Δt）则结合了反应时间（使用光电门可忽略）和重复读数的离散度。对于上述数据，t的半极差可能为0.0005 s，小于计时器分辨率，因此Δt取±0.001 s。
The percentage uncertainties in s/t are found by adding the percentage uncertainty in s to the percentage uncertainty in t (%U(s/t) = %U(s) + %U(t)), as division is involved. For example, if s = 0.800 ± 0.001 m, then %U(s) = 0.125%. If t = 0.2477 ± 0.001 s, then %U(t) = 0.404%. Therefore, %U(s/t) ≈ 0.529%. The absolute uncertainty in s/t is then 0.529% of 3.23 m s⁻¹ ≈ 0.02 m s⁻¹. These values are used for error bars on the graph.
s/t的百分不确定度由s的百分不确定度与t的百分不确定度相加得到（%U(s/t) = %U(s) + %U(t)），因为涉及除法运算。例如，若 s = 0.800 ± 0.001 m，则%U(s) = 0.125%。若 t = 0.2477 ± 0.001 s，则%U(t) = 0.404%。因此，%U(s/t) ≈ 0.529%。进而算出s/t的绝对不确定度为3.23 m s⁻¹的0.529%，约等于0.02 m s⁻¹。这些值用于在图像上绘制误差棒。
Propagating these uncertainties through the graph analysis, as described in Section 5, is more reliable than analytical propagation because it includes systematic effects that might shift the best-fit line. The method of extreme lines is the standard approach in A-Level practical papers.
如第5节所述，通过图像分析传递这些不确定度比解析传递更为可靠，因为它涵盖了可能使最佳拟合线偏移的系统效应。极端线法是A-Level实验试卷中的标准方法。
8. Sources of Error and Possible Improvements | 误差来源与改进措施
One significant source of systematic error is the residual magnetism of the electromagnet. After the current is switched off, the ball may still be held momentarily, causing a delay in release and an apparently larger g. Tapping the electromagnet or using a non-magnetic release mechanism can minimise this effect.
一个重要的系统误差来源是电磁铁的剩磁。电流切断后，钢球可能仍被短暂吸住，导致释放延迟，使得测得的g偏大。轻敲电磁铁或使用非磁性释放机构可将此影响降至最低。
Air resistance acts on the ball, especially at higher speeds near the second light gate. This slightly reduces the acceleration, causing a systematic underestimate of g. Using a dense, smooth steel ball and keeping the fall distance moderate helps to reduce the effect.
空气阻力作用于钢球，特别是在接近第二光电门的高速段，这会略微减小加速度，导致g的系统性低估。使用密度大、表面光滑的钢球并保持下落距离适中有助于减小此效应。
Another error arises if the light gates are not perfectly horizontal, so that the ball does not cut the beam at the same point each time, or if the ball’s diameter is comparable to the beam width. Ensuring careful alignment and using a ball diameter large relative to the beam width improves consistency.
如果光电门未能完全保持水平，钢球每次遮断光束的位置将不同，或者钢球直径与光束宽度相当，都会引入误差。仔细对准并使用直径相对于光束宽度较大的钢球可提高重复性。
Parallax error when measuring s can be reduced by using a vernier calliper for smaller separations and by clamping the metre rule firmly in place. Timing errors from the digital timer are usually small, but the timer should be checked for zero error before starting the experiment.
测量s时的视差可通过使用游标卡尺测量较小间距以及将米尺夹紧固定来减小。数字计时器的计时误差通常很小，但在实验开始前应检查计时器的零点误差。
Finally, repeating the experiment for a wider range of s and taking more repeat readings would allow a more reliable estimate of the mean time and a better-defined best-fit line, improving the overall accuracy of g.
最后，在更大的s范围内重复实验，并进行更多次重复读数，可以得到更可靠的平均时间估计和更明确的最佳拟合线，从而提高g的整体准确度。
9. Safety Precautions | 安全注意事项
Although the steel ball is small, it can cause injury if it falls onto a foot or onto sensitive equipment. Ensure that a foam pad or a box of sand is placed beneath the second light gate to catch the ball safely. Keep the area below the electromagnet clear during the release.
尽管钢球很小，但如果掉落到脚上或精密设备上仍可能造成伤害。应在第二光电门下方放置泡沫垫或沙盒，安全接住钢球。释放期间，保持电磁铁下方区域畅通无阻。
When moving the light gates or adjusting the apparatus, take care not to drop the retort stand or to trap fingers between clamps. Electrical connections to the electromagnet should be secure, and the low-voltage DC supply must be treated with respect, avoiding short circuits.
移动光电门或调整装置时，注意不要使铁架台掉落，也不要将手指夹在夹具间。电磁铁的电气连接应牢固，对待低压直流电源也应谨慎，避免短路。
10. Common Mistakes in Paper 3 | 卷3常见错误
Candidates often forget to include units in table headers or to state the absolute uncertainty next to the measured value. In the graph, a frequent error is plotting s against t instead of s/t against t, which would not yield a straight line. Examiners expect the linearised relationship to be clearly identified and used.
考生经常忘记在表格表头中注明单位，或在测量值旁注明绝对不确定度。在图像绘制中，一个常见错误是直接绘制s对t的图像，而非s/t对t，这并不能得到一条直线。考官期望明确识别并使用线性化后的关系式。
When drawing the lines of worst fit, some candidates either ignore the error bars or draw lines that are too steep or too shallow to be plausible. The worst lines must still be reasonably representative of the data spread. Also, the gradient triangle should be large — at least half the length of the drawn line — to minimise read-off error.
在绘制最差拟合线时，一些考生或者忽略误差棒，或者画出斜度过大或过小以至于不合理的最差线。最差线必须仍能合理地代表数据分布。此外，用于计算斜率的三角形应足够大——至少是所画直线长度的一半——以尽可能减小读取误差。
Another mistake is failing to check the y-intercept calculation or misreading the timer display. It is advisable
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