Experimental Investigation of Free Fall: Measuring g | 自由落体实验探究：测量重力加速度

📚 Experimental Investigation of Free Fall: Measuring g | 自由落体实验探究：测量重力加速度

In many AS Physics Unit 2 papers, including the January 2020 session, students are required to design or analyse an experiment to determine a physical quantity. One classic investigation is measuring the acceleration of free fall, g, using a method that minimises systematic errors. This article explores a typical experiment: dropping an object past two light gates connected to a timer, recording the time interval, and calculating g from the equations of uniformly accelerated motion. We will cover the theory, apparatus, procedure, common pitfalls, and a worked example.

在许多 AS 物理单元 2 试卷中（包括 2020 年 1 月场次），学生需要设计或分析一个测量物理量的实验。一个经典的探究就是利用减小系统误差的方法测量自由落体加速度 g。本文探讨一个典型实验：让物体下落经过两个连接计时器的光门，记录时间间隔，并利用匀加速运动方程计算 g。我们将涵盖理论、仪器、步骤、常见问题及一个计算实例。

1. Introduction to the Experiment | 实验介绍

This investigation aims to measure the acceleration due to gravity, g, near the Earth’s surface. A freely falling object in a vacuum would accelerate uniformly at approximately 9.81 m s⁻². However, air resistance and measurement uncertainties require careful experimental design. The method uses two light gates positioned a known vertical distance apart. A small opaque card of known length passes through the first light gate, triggering a timer, and then through the second, stopping the timer. From the measured times and the card’s length, we can calculate the initial velocity at the first gate, the final velocity at the second, and hence g.

本探究旨在测量地球表面附近的重力加速度 g。在真空中，自由下落的物体会以约 9.81 m s⁻² 均匀加速。然而，空气阻力和测量不确定度要求精心的实验设计。该方法使用两个相隔已知垂直距离的光门。一张已知长度的不透光卡片通过第一个光门时触发计时器，通过第二个时停止计时。根据测得的时间和卡片长度，我们可以计算第一个门处的初速度、第二个门处的末速度，进而求出 g。

2. Apparatus and Setup | 仪器与装置

The essential apparatus includes: a retort stand and clamp to hold the electromagnet; an electromagnet to release the object smoothly; a small steel ball or a card attached to a falling mass; a pair of light gates with built‑in timers (or connected to a data‑logger); a metre ruler or measuring tape to measure the vertical separation between the gates; and an opaque card of known width, typically 10.0 cm. The electromagnet is placed high on the stand. The two light gates are aligned vertically below, separated by a distance of about 1.0 m. A plumb line ensures vertical alignment so the object passes through the centre of each gate.

主要仪器包括：铁架台和夹具用于固定电磁铁；一个电磁铁用于平稳释放物体；一个小钢球或附着在落体上的卡片；一对带有内置计时器的光门（或连接至数据记录器）；米尺或卷尺用于测量两个光门间的垂直距离；一张已知宽度（通常 10.0 cm）的不透光卡片。电磁铁安装在铁架台高处。两个光门垂直对齐置于下方，相距约 1.0 m。使用铅垂线确保垂直对齐，使物体能通过每个光门的中心。

3. Theory and Equations | 理论和方程

When an object falls freely under gravity, its motion obeys the kinematic equations for constant acceleration. If the object starts from rest at the electromagnet, its initial velocity at the first light gate is u, and after falling a distance s to the second gate its velocity becomes v. Using the principle that the card interrupts the light beam for a very short time, we can measure the velocity at each gate. The time for which the card cuts the beam is inversely proportional to the speed. If the card length is L and the interruption time at the top gate is t₁, then the velocity at that point is:

当物体在重力作用下自由下落时，其运动遵循匀加速运动的运动学方程。若物体从电磁铁处静止释放，它在第一个光门处的初速度为 u，下落距离 s 到达第二个光门时速度为 v。由于卡片遮挡光束的时间极短，我们可以测量每个门处的速度。卡片遮挡光束的时间与速度成反比。若卡片长度为 L，上光门处遮挡时间为 t₁，则该点的速度为：

u = L / t₁

Similarly, at the second gate, the velocity is:

类似地，在第二个光门处，速度为：

v = L / t₂

Using the equation of motion linking initial and final velocities with displacement and acceleration:

利用联系初、末速度与位移和加速度的运动方程：

v² = u² + 2 a s

Hence, the acceleration a (which is g) can be found from:

因此，加速度 a（即 g）可由下式求得：

g = (v² − u²) / (2 s)

4. Procedure | 实验步骤

First, measure the width L of the opaque card several times with a vernier calliper to obtain an average and reduce the uncertainty. Record the uncertainty in the calliper reading, usually ±0.01 cm. Set up the electromagnet at the top of the stand and ensure it is switched on so it holds the card assembly. Clamp the first light gate about 10 cm below the electromagnet; exactly measure and note this starting distance if needed for an alternative method. Then clamp the second light gate a measured distance s below the first one. The distance s should be measured between the centres of the two gates using a metre ruler, and its uncertainty recorded (±0.5 cm typical). Using a plumb line, check that the gates are vertically aligned so the card falls through their centres without touching the edges.

首先，用游标卡尺多次测量不透光卡片的宽度 L，取平均值以减小不确定度。记录卡尺读数的不确定度，通常为 ±0.01 cm。在铁架台顶端安装电磁铁，确保通电以吸住卡片组件。将第一个光门固定在电磁铁下方约 10 cm 处；若采用替代方法，需精确测量并记录此起始距离。然后将第二个光门固定在第一个光门下方测得的距离 s 处。距离 s 应使用米尺测量两门中心间的距离，并记录其不确定度（通常为 ±0.5 cm）。用铅垂线检查光门是否垂直对齐，使卡片通过各门中心且不触碰边缘。

Switch off the electromagnet to release the object. The timer starts when the card enters the top gate and stops when it enters the bottom gate. However, the timer may record the time interval Δt between the two gates. For the velocity method, you need a timer that measures the time t₁ for the card to pass the first gate and t₂ for the second gate separately. Many data‑loggers can do this. If only one gate time is available, a different formula must be used: s = u Δt + ½ g (Δt)², requiring a simultaneous equation or a graph. In this article we assume separate gate timings are available. Repeat the drop at least five times, recording t₁ and t₂ each time.

断开电磁铁释放物体。当卡片进入上光门时计时器启动，进入下光门时停止。然而，计时器可能记录两门间的时间间隔 Δt。对于速度法，需要分别测量卡片通过第一个光门的时间 t₁ 和通过第二个光门的时间 t₂ 的计时器。许多数据记录器可以做到。如果只有一个门的时间可用，则需采用不同公式：s = u Δt + ½ g (Δt)²，这需要联立方程或作图。本文假设可单独获得每个门的计时。至少重复下落五次，每次记录 t₁ 和 t₂。

5. Data Collection | 数据收集

Record all measurements in a table. For each trial, note the top gate interruption time t₁, bottom gate interruption time t₂, and the pre‑measured distance s. An example data table is shown below:

将所有测量数据记录在表格中。每次试验记录上光门遮挡时间 t₁、下光门遮挡时间 t₂，以及预先测量的距离 s。示例数据表如下：

Trial / 试验	t₁ (s) / 上光门时间	t₂ (s) / 下光门时间	s (m) / 距离
1	0.0523	0.0312	0.950
2	0.0519	0.0310	0.950
3	0.0530	0.0315	0.950
4	0.0525	0.0313	0.950
5	0.0521	0.0311	0.950

Also record the card length L and its absolute uncertainty. For example, L = 0.100 m ± 0.0005 m. The uncertainty in distance s is typically ±0.005 m.

还需记录卡片长度 L 及其绝对不确定度。例如，L = 0.100 m ± 0.0005 m。距离 s 的不确定度通常为 ±0.005 m。

6. Data Analysis and Graphing | 数据分析与作图

For each trial, compute u = L / t₁ and v = L / t₂. Calculate the mean values of u and v over five trials. Then use the mean values to determine g from the equation:

对每次试验，计算 u = L / t₁ 和 v = L / t₂。计算五次试验中 u 和 v 的平均值。然后利用平均值由下式求 g：

g = (v_mean² − u_mean²) / (2 s)

Alternatively, a graphical method can be used to minimise the impact of outliers and to estimate uncertainty. If you vary s and record the time of flight Δt between gates, the equation s = u Δt + ½ g (Δt)² can be rearranged to:

或者，可采用图解方法以减小异常值的影响并估计不确定度。如果改变 s 并记录两门间的飞行时间 Δt，方程 s = u Δt + ½ g (Δt)² 可重新整理为：

s / Δt = u + ½ g Δt

Plotting s/Δt on the y‑axis against Δt on the x‑axis yields a straight line with gradient ½ g and intercept u. This avoids precise velocity measurements from short gate times and is often more reliable. The uncertainty in g can be estimated from the line of best fit and the worst acceptable line.

将 s/Δt 作为纵轴，Δt 作为横轴作图，可得到一条直线，其斜率为 ½ g，截距为 u。这避免了对短时间门信号的精确速度测量，通常更可靠。g 的不确定度可通过最佳拟合线和最大可接受线估计。

7. Sources of Uncertainty and Error | 不确定度和误差来源

The main source of uncertainty in the direct velocity method comes from the measurement of the very short times t₁ and t₂. Even a 0.0005 s timing error can significantly affect calculated velocities because the card length is only 0.1 m. Additionally, the assumption that the average velocity while the card interrupts the beam equals the instantaneous velocity at the gate centre introduces a small systematic error. The distance s is difficult to measure precisely because it is the distance between the beam centres, not the visible edges of the gates. Air resistance slightly reduces acceleration, especially for light or low‑density objects.

在直接速度法中，不确定度的主要来源是对极短时间 t₁ 和 t₂ 的测量。即便 0.0005 s 的计时误差也会显著影响计算出的速度，因为卡片长度仅为 0.1 m。此外，假设卡片遮挡光束期间的平均速度等于光门中心处的瞬时速度会引入微小的系统误差。距离 s 难以精确测量，因为它是光束中心间的距离，而非光门外侧可见边缘。空气阻力会轻微降低加速度，尤其对于轻质或低密度物体。

Parallax error when measuring s with a metre ruler can be minimised by reading at eye level. Reaction time is irrelevant here because timers are electronically triggered. However, electromagnetic release might cause a slight delay between switching off and actual release; this affects the initial velocity if the object falls slightly before the first gate, but with two gates this is accounted for in u.

用米尺测量 s 时的视差可通过平视读数降至最低。在此实验中反应时间无关紧要，因为计时器是电子触发。但电磁释放可能会导致断电与实际释放之间的微小延迟；如果物体在到达第一个光门前已下落一小段距离，这会影响初速度，但使用双光门时，这一点已在 u 中考虑。

8. Improvements and Precautions | 改进与注意事项

To improve accuracy, use a card with a sharper edge so the beam makes and breaks cleanly. Increase the distance s to about 1.5 m to obtain larger velocity differences, but ensure the object does not hit the ground. For timing, use a digital storage oscilloscope or a fast data‑logger with microsecond resolution. Measure L with a micrometer screw gauge rather than vernier callipers for higher precision. Repeat the experiment for various values of s and use the graphical method to average out timing errors.

为提高准确度，使用边缘更锋利的卡片以使光束通断干脆。增加距离 s 至约 1.5 m 以获得更大的速度差，但确保物体不落地。计时方面，使用数字存储示波器或高分辨率（微秒级）的数据记录器。用千分尺而非游标卡尺测量 L 以获更高精度。改变不同 s 值重复实验，并采用图解方法以平均计时误差。

Use a dense metallic ball instead of a card to reduce air resistance, but then you must measure its diameter and use the interruption time of the ball. Make sure the object is not magnetised so it does not stick to the electromagnet. Perform the experiment in still air (close windows, no fans). Finally, always repeat measurements and calculate percentage differences to quantify random errors.

使用致密金属球代替卡片以减小空气阻力，但此时需测量其直径并记录球的遮挡时间。确保物体不带磁性，以免粘在电磁铁上。在静止空气中进行实验（关闭窗户，无风扇）。最后，务必重复测量并计算百分差以量化随机误差。

9. Example Calculation | 示例计算

Using the mean data from the table, suppose t₁_mean = 0.05236 s and t₂_mean = 0.03122 s. With L = 0.100 m:

利用表中的平均数据，假设 t₁_mean = 0.05236 s，t₂_mean = 0.03122 s。已知 L = 0.100 m：

u_mean = 0.100 / 0.05236 ≈ 1.910 m s⁻¹

v_mean = 0.100 / 0.03122 ≈ 3.203 m s⁻¹

Given s = 0.950 m, calculate:

已知 s = 0.950 m，计算：

g = (3.203² − 1.910²) / (2 × 0.950) = (10.259 − 3.648) / 1.90 = 6.611 / 1.90 ≈ 3.48 m s⁻²

The result is far from 9.81 m s⁻², indicating that the simple velocity method is highly sensitive to timing errors. In a real lab, such discrepancies prompt a careful re‑evaluation of raw data. Using the graphical method often yields a value closer to the accepted one. For this reason, exam questions may ask students to identify and explain such anomalously low results: systematic timing offsets, zero errors in the timer, or misalignment of gates.

结果远小于 9.81 m s⁻²，表明简单的速度法对计时误差极为敏感。在实际实验室中，这种偏差会促使我们仔细重新评估原始数据。采用图解方法通常能得到更接近公认值的数值。因此，考试题目可能会要求学生找出并解释这种异常偏低的结果：系统计时偏移、计时器零点误差或光门未对准。

Let’s assume a corrected set of data gives u = 2.80 m s⁻¹, v = 4.95 m s⁻¹, and s = 0.950 m. Then:

假设一组修正后的数据给出 u = 2.80 m s⁻¹，v = 4.95 m s⁻¹，s = 0.950 m。则：

g = (4.95² − 2.80²) / (2 × 0.95) = (24.50 − 7.84) / 1.90 = 16.66 / 1.90 ≈ 8.77 m s⁻²

This is closer to 9.81 m s⁻², with an error of about 10%. The uncertainty can be propagated from the uncertainties in L, t, and s. For the velocity method, the fractional uncertainty in g is roughly the sum of fractional uncertainties: 2ΔL/L + 2Δt/t + Δs/s. Thus, reducing timing uncertainty is critical.

这更接近 9.81 m s⁻²，误差约 10%。不确定度可由 L、t 和 s 的不确定度传递得到。对于速度法，g 的相对不确定度大致是各项相对不确定度之和：2ΔL/L + 2Δt/t + Δs/s。因此，减小计时不确定度至关重要。

10. Conclusion | 结论

The free‑fall method using two light gates is a standard AS Physics experiment that reinforces understanding of kinematic equations, measurement uncertainties, and graphical analysis. While conceptually straightforward, it presents significant practical challenges in timing and alignment. Through careful technique and error analysis, students appreciate the importance of experimental design in obtaining reliable values for fundamental constants. This investigation exemplifies the type of problem‑solving and evaluation skills assessed in Unit 2 papers.

使用两个光门的自由落体法是 AS 物理中一个标准实验，它能加深对运动学方程、测量不确定度及图解分析的理解。尽管概念上简单，但在计时和校直方面存在显著的实际挑战。通过仔细的操作和误差分析，学生能体会到在获取可靠基本常数值时实验设计的重要性。本探究体现了单元 2 试卷中所考查的解决问题与评估能力的典型题型。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）