Measuring Velocity and Acceleration | 测量速度与加速度

📚 Measuring Velocity and Acceleration | 测量速度与加速度

Measuring velocity and acceleration is central to mechanics. In CIE A-Level Physics, you are expected not only to recall definitions and equations but also to describe practical methods, identify limiting factors, and evaluate uncertainties. This article reviews standard laboratory techniques including stopwatch-ruler measurements, light gates, ticker timers, motion sensors, and video analysis, and shows how to extract reliable values of velocity and acceleration.

测量速度和加速度是力学的核心内容。在 CIE A-Level 物理中，你不仅要记住定义和公式，还要能描述实验方法、找出限制因素并评估不确定度。本文复习秒表-刻度尺、光门、打点计时器、运动传感器和视频分析等常用实验技术，并展示如何获得可靠的速度和加速度数值。

1. Definitions and Core Equations | 定义与核心公式

Velocity is the rate of change of displacement, so it is a vector quantity. Speed is the scalar magnitude of velocity. For an object moving in a straight line, average velocity v is given by v = Δs ÷ Δt, where Δs is the displacement and Δt is the time interval. Acceleration a is the rate of change of velocity, a = Δv ÷ Δt. The SI units are m s⁻¹ for velocity and m s⁻² for acceleration.

速度是位移的变化率，因此是矢量。速率是速度的标量大小。对于沿直线运动的物体，平均速度 v 由 v = Δs ÷ Δt 给出，其中 Δs 是位移，Δt 是时间间隔。加速度 a 是速度的变化率，a = Δv ÷ Δt。速度的 SI 单位是 m s⁻¹，加速度的单位是 m s⁻²。

For constant acceleration, the equations of motion are:

v = u + at    s = ut + ½at²    v² = u² + 2as

where u is the initial velocity, v the final velocity, s the displacement, and t the time. These equations are valid only when acceleration is uniform.

对于匀加速运动，运动学公式为：v = u + at，s = ut + ½at²，v² = u² + 2as。其中 u 是初速度，v 是末速度，s 是位移，t 是时间。这些公式只在加速度恒定时成立。

2. Stopwatch and Ruler Measurement | 用秒表和刻度尺测量速度

The simplest method uses a metre ruler or tape measure to mark a known distance, and a stopwatch to record the time taken. For example, a trolley travels 2.00 m along a runway. If the average time is 1.60 s, the average velocity is v = 2.00 m ÷ 1.60 s = 1.25 m s⁻¹. This gives only the average velocity over the interval, not the instantaneous velocity.

最简单的方法是用米尺或卷尺标出已知距离，并用秒表记录所需时间。例如，一辆小车沿轨道运动 2.00 m。如果平均时间为 1.60 s，则平均速度为 v = 2.00 m ÷ 1.60 s = 1.25 m s⁻¹。这种方法只能给出该区间内的平均速度，不能给出瞬时速度。

The main uncertainties come from reaction time in starting and stopping the stopwatch, parallax error in aligning the eye with the marker, and the difficulty of releasing the object cleanly. Repeating the measurement and using two observers can reduce random scatter. To improve accuracy, use the longest convenient distance and a digital stopwatch with a resolution of at least 0.01 s.

主要不确定度来自启动和停止秒表时的反应时间、眼睛与标记未正对产生的视差，以及难以干净地释放物体。重复测量并使用两名观察者可以减少随机离散。为了提高精确度，应在方便条件下使用尽量长的距离，并使用分辨率至少为 0.01 s 的数字秒表。

3. Light Gates and Photogate Timing | 光门与光电计时

A light gate is an electronic sensor containing a beam of light and a detector. When an object passes through, the beam is interrupted. If a card of width L interrupts the beam for a time t, the average speed through the gate is v = L ÷ t. If the card is very narrow, this approximates the instantaneous velocity. Two light gates placed a known distance apart can record two velocities.

光门是一种电子传感器，含有一束光和探测器。当物体经过时，光束被遮挡。如果宽度为 L 的挡光片遮光时间为 t，则通过光门的平均速度为 v = L ÷ t。若挡光片很窄，该值近似为瞬时速度。两个相距已知距离的光门可以记录两个速度。

From the two velocities u and v, acceleration can be found using a = (v – u) ÷ t, where t is the time taken between the gates, or using v² = u² + 2as if the separation s is known. Light gates avoid stopwatch reaction time and are well suited to fast dynamics experiments. The main source of error is uncertainty in the card length L and the electronic timing resolution.

由两个速度 u 和 v 可以求得加速度，可使用 a = (v – u) ÷ t，其中 t 为两光门之间的时间，或若已知间距 s，可用 v² = u² + 2as。光门避免了秒表的反应时间，非常适合快速动力学实验。主要误差来源是挡光片长度 L 的不确定度以及电子计时分辨率。

4. Ticker-Tape Timer Analysis | 打点计时器分析

A ticker-tape timer makes a dot on a paper tape at fixed intervals. A common frequency is 50 Hz, so the time between dots is 1 ÷ 50 = 0.020 s. By measuring the length of tape containing n intervals, the average velocity is v = length ÷ (n × 0.020 s). For example, if 10 intervals cover 12.0 cm, then v = 0.120 m ÷ 0.20 s = 0.60 m s⁻¹.

打点计时器在纸带上以固定间隔打点。常见频率为 50 Hz，因此相邻点之间的时间为 1 ÷ 50 = 0.020 s。测量包含 n 个间隔的纸带长度，则平均速度为 v = 长度 ÷ (n × 0.020 s)。例如，若 10 个间隔对应 12.0 cm，则 v = 0.120 m ÷ 0.20 s = 0.60 m s⁻¹。

To obtain acceleration, cut the tape into segments each containing the same number of intervals. Each segment gives the average velocity in that equal time block. If the segment velocities increase linearly, the acceleration can be found from the slope of a velocity-time graph or by evaluating a = (v₂ – v₁) ÷ t, where v₁ and v₂ are two segment velocities separated by a known time interval.

为了得到加速度，把纸带剪成每段包含相同间隔数的若干段。每段给出该相等时间块内的平均速度。如果各段速度线性增加，可以从速度-时间图的斜率求得加速度，或使用 a = (v₂ – v₁) ÷ t，其中 v₁ 和 v₂ 是相隔已知时间的两个分段速度。

5. Motion Sensors and Data Loggers | 运动传感器与数据记录仪

An ultrasonic motion sensor emits high-frequency sound pulses and receives echoes from a moving object. A rotary motion sensor measures the angle turned by a pulley attached to a string. Both devices feed data to a data logger, which records position at regular time intervals. The software can plot displacement-time, velocity-time, and acceleration-time graphs directly.

超声波运动传感器发射高频声脉冲并接收运动物体反射的回声。旋转运动传感器测量与细绳相连的滑轮转过的角度。两种装置都把数据传给数据记录仪，按等时间间隔记录位置。软件可直接绘制位移-时间、速度-时间和加速度-时间图。

The main advantage

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