Deriving g from Light Gate Data in PH03 January 2023 | 从PH03 2023年1月光门数据推导重力加速度g

📚 Deriving g from Light Gate Data in PH03 January 2023 | 从PH03 2023年1月光门数据推导重力加速度g

The International A Level Physics Unit 3 paper (PH03) from January 2023 presented a practical scenario where students used a light gate to measure the acceleration of free fall, g. One of the core tasks was to derive a linear equation relating recorded variables and subsequently calculate g with its uncertainty. This article unpacks the necessary formula derivations and illustrates the complete analytical process.

2023年1月的国际A Level物理单元三试卷（PH03）给出了一道使用光门测量自由落体加速度g的实验题。核心任务之一是从测量量推导线性关系式，进而计算g及其不确定度。本文拆解必要的公式推导，并展示完整的分析过程。

1. Experimental Setup and Principle | 实验装置与原理

The experiment typically involves a card of known width d attached to a falling mass. A single light gate is positioned a distance s below the initial release point. When the card interrupts the beam, the gate records the time t for which the beam is broken, enabling the calculation of the average speed v = d/t as the card passes through.

实验通常使用已知宽度d的遮光片固定在落体上。一个光门放置在释放点下方距离s处。遮光片通过时打断光束，光门记录遮光时间t，从而可计算通过光门时的平均速度 v = d/t。

The falling object starts from rest (initial velocity u = 0) and accelerates uniformly under gravity with acceleration g. The acceleration can be linked to the distance fallen and the final speed.

落体由静止释放（初速度u=0），在重力作用下以加速度g匀加速。加速度可以与下落距离及末速度联系起来。

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2. Key Equations Involved | 涉及的关键方程

From the equations of uniformly accelerated motion, the speed v of an object after falling a distance s from rest is given by v² = u² + 2as. Since u = 0 and a = g, we obtain:

由匀加速运动方程，物体从静止下落距离s后的速度v满足 v² = u² + 2as。代入u=0，a=g，可得：

v² = 2gs

Substituting the light gate measurement v = d/t gives an equation linking the directly measured quantities t and s:

代入光门测量值 v = d/t，得到直测量t和s的关系式：

(d/t)² = 2gs

This can be rearranged into a linear form, which is the central focus of the derivation in the PH03 paper.

该式可变形为直线形式，这正是PH03试卷推导的核心。

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3. Derivation of the Linearised Equation | 线性化方程的推导

To plot a graph that yields a straight line, we must express the equation in the form y = mx + c, where m is the slope and c the intercept. Starting from d²/t² = 2gs, we can divide both sides by d² to isolate 1/t²:

为了绘制一条直线，我们需要将方程转化为y = mx + c的形式，其中m为斜率，c为截距。从 d²/t² = 2gs 出发，两边同除以d²以分离1/t²：

1/t² = (2g/d²) s

Here the independent variable is the distance s, and the dependent variable is 1/t². The slope m of such a graph is therefore 2g/d². Theoretically, the intercept should be zero, because when s = 0, the time t becomes infinite (1/t² → 0).

这里自变量为距离s，因变量为1/t²。该图线的斜率m即为 2g/d²。理论上截距应为零，因为s=0时，时间t无穷大，1/t²趋向于0。

This linearisation step is crucial in the PH03 paper and allows students to extract g from the gradient of a best-fit line.

这一步线性化在PH03试卷中至关重要，使学生能够通过最佳拟合线的斜率求出g。

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4. Understanding the Slope and Intercept | 理解斜率和截距

The gradient m = 2g/d² contains the unknown g and the known card width d. By measuring d with a micrometer or known value, g can be calculated. If the intercept is not zero, it suggests a systematic error, such as the release point not being exactly at the light gate’s zero position, or a delay in triggering.

斜率m = 2g/d² 包含未知量g和已知的遮光片宽度d。通过用千分尺测量d或使用已知值，即可计算g。如果截距不为零，则表明存在系统误差，例如释放点没有精确对准光门的零点，或触发延迟。

In the PH03 January 2023 paper, typical data might show a small positive or negative intercept, and students were expected to discuss possible causes.

在2023年1月的PH03试卷中，典型数据可能显示一个小的正或负截距，并要求学生讨论可能的原因。

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5. Calculating g from the Slope | 根据斜率计算g

Once the slope m is determined from a least-squares fit or from a manually drawn line, the value of g is obtained by rearrangement:

一旦通过最小二乘法或手动画线确定斜率m，可通过重新排列公式求得g：

g = m × d² / 2

For example, if d = 0.100 m and the slope from the 1/t² vs s graph is found to be 1960 m⁻¹ s⁻², then g = 1960 × (0.100)² / 2 = 9.80 m s⁻².

例如，若遮光片宽度 d = 0.100 m，从 1/t² 对 s 的图线得到斜率 m = 1960 m⁻¹ s⁻²，则 g = 1960 × (0.100)² / 2 = 9.80 m s⁻²。

It is important to check the units throughout the calculation to avoid mistakes. The derived g should be close to 9.81 m s⁻², depending on experimental accuracy.

整个计算过程必须检查单位，避免出错。得出的g值应接近9.81 m s⁻²，具体取决于实验精度。

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6. Estimating Uncertainties in the Derived Quantities | 估算导出量的不确定度

The uncertainty in g, Δg, can be estimated by propagating the uncertainties in the slope (Δm) and in the card width (Δd). Using the product–quotient rule:

g的不确定度Δg可以通过传递斜率不确定度Δm和遮光片宽度不确定度Δd来估算。利用乘除法则：

Δg / g = Δm / m + 2 (Δd / d)

This assumes the usual worst‑case combination. In the PH03 paper, students often have to calculate percentage uncertainties from given instrument resolutions and graph plotting.

这是最不利情况下的叠加。在PH03试卷中，学生常需根据给定的仪器分辨率和绘图

Published by TutorHao | Physics Revision Series | aleveler.com

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