📚 Determining velocity and acceleration in the laboratory | 在实验室中测定速度与加速度
In A-Level physics, you are expected not only to define velocity and acceleration but also to obtain reliable values for them in the laboratory. This article reviews the main experimental techniques, including ticker timers, photogates, air tracks and free-fall apparatus, and shows how to process data into velocity-time graphs.
在 A-Level 物理中，你不仅要定义速度和加速度，还要能够在实验室中获得可靠的测量值。本文回顾主要的实验技术，包括打点计时器、光电门、气垫导轨和自由落体装置，并说明如何将数据处理成速度-时间图像。
1. Core definitions and equations | 核心定义与方程
Velocity is the rate of change of displacement, so it is a vector quantity, while speed is the rate of change of distance. Acceleration is the rate of change of velocity with respect to time.
速度是位移的变化率，因此是矢量；而速率是路程的变化率。加速度是速度对时间的变化率。
For motion in a straight line, the key relationships are v = Δs / Δt and a = Δv / Δt. For uniformly accelerated motion starting from rest, s = ½at² and v² = 2as are often used.
对于直线运动，关键关系式为 v = Δs / Δt 和 a = Δv / Δt。对于从静止开始的匀加速运动，常用 s = ½at² 和 v² = 2as。
In the laboratory, ‘velocity’ is usually calculated from a small displacement divided by the corresponding short time interval, which gives an average velocity over that interval.
在实验室中，“速度”通常用一段小位移除以相应短时间间隔来计算，从而得到该间隔内的平均速度。
2. Apparatus overview: ticker timer, photogates and air track | 仪器概述：打点计时器、光电门和气垫导轨
A ticker timer prints dots at a known frequency, usually 50 Hz, onto a paper tape pulled by the moving object. The spacing of the dots is a record of displacement at equal time intervals.
打点计时器以已知频率（通常为 50 Hz）在运动物体拉动的纸带上打点。点间距记录了相等时间间隔内的位移。
A photogate contains a light source and a detector. When a card of known length interrupts the beam, the gate measures the interruption time, allowing the average speed through the gate to be found.
光电门包含光源和探测器。当已知长度的挡光片遮断光束时，光电门测量遮光时间，从而求出通过光电门的平均速度。
An air track supports a glider on a cushion of air, greatly reducing friction. It is ideal for studying nearly constant velocity and constant acceleration produced by a small slope or a pulley system.
气垫导轨利用气垫支撑滑块，大大减少摩擦。它非常适合研究近似的匀速运动和由微小倾斜或滑轮系统产生的匀加速运动。
3. Using a ticker timer | 使用打点计时器
Set up the ticker timer at one end of the bench and attach the tape to the moving trolley. Switch on the timer, then release the trolley so the tape is pulled through the timer without jerking.
将打点计时器固定在工作台一端，把纸带连接到运动小车上。打开计时器，然后释放小车，使纸带平稳地通过计时器。
Because the timer operates at 50 Hz, the time between two consecutive dots is 0.02 s. A common trick is to count every fifth dot, giving a time interval of 0.10 s between marked points.
由于计时器以 50 Hz 工作，相邻两个点之间的时间为 0.02 s。常用技巧是每五个点取一个标记点，这样标记点之间的时间间隔为 0.10 s。
Label the tape with the direction of motion and mark the chosen intervals clearly. Measurements should be taken with a metre rule placed along the tape to reduce parallax error.
在纸带上标出运动方向，并清晰标记所选间隔。测量时应将米尺沿纸带放置，以减少视差误差。
4. Measuring velocity from ticker tape | 从纸带测量速度
For a chosen interval, measure the distance between the start and end marks. Divide this distance by the time for that interval to obtain the average velocity.
对于所选间隔，测量起点和终点标记之间的距离。将该距离除以该间隔的时间，即可得到平均速度。
If the dots get farther apart, the object is speeding up; if they get closer together, it is slowing down. A constant spacing indicates constant velocity within experimental uncertainty.
如果点间距逐渐变大，说明物体在加速；如果点间距逐渐变小，说明物体在减速。间距保持不变则表明在实验不确定度内速度恒定。
For more accurate values, measure each interval separately and calculate a series of average velocities. Each velocity is assigned to the midpoint time of its interval when plotting a graph.
为了获得更准确的结果，应分别测量每个间隔并计算一系列平均速度。在绘图时，每个速度应对应其间隔的中点时刻。
5. Measuring acceleration from ticker tape | 从纸带测量加速度
Acceleration can be found by calculating two or more velocities at different times and using a = Δv / Δt. However, using many intervals and drawing a velocity-time graph gives a more reliable value.
可以通过计算不同时刻的两个或多个速度并用 a = Δv / Δt 来求加速度。但使用多个间隔并绘制速度-时间图像会得到更可靠的结果。
For uniformly accelerated motion, the displacement in successive equal time intervals increases by a constant amount. The acceleration is given by a = (sₙ₊₁ – sₙ) / T², where T is the time per interval.
对于匀加速运动，相邻相等时间间隔内的位移增量恒定。加速度由 a = (sₙ₊₁ – sₙ) / T² 给出，其中 T 是每段间隔的时间。
When using the velocity-time graph, the gradient is the acceleration. Calculate the gradient from a large triangle on the best-fit line, not from individual data points.
使用速度-时间图像时，斜率为加速度。应从最佳拟合线上取较大三角形计算斜率，而不是从单个数据点计算。
6. Using photogates and a data logger | 使用光电门与数据记录仪
Photogates are often used with a card of width d fixed on the moving object. The timer records the duration t for the card to pass through the beam, so the average speed is v = d / t.
光电门通常与固定在运动物体上的宽度为 d 的挡光片配合使用。计时器记录挡光片通过光束的持续时间 t，因此平均速度为 v = d / t。
This method is useful because it removes the reaction-time error associated with manually starting and stopping a stopwatch. The measurement is triggered electrically.
该方法很有用，因为它消除了人工启动和停止秒表所带来的反应时间误差。测量由电信号自动触发。
To determine acceleration, place two photogates a known distance apart. The data logger records the speed at each gate and the time taken to travel between them, allowing a = (v₂ – v₁) / t.
为了测定加速度，将两个光电门相隔已知距离放置。数据记录仪记录每个光电门处的速度和通过两门之间的时间，从而求得 a = (v₂ – v₁) / t。
Some systems use a single photogate with a double-bladed card, which measures two interruption times and the gap time in one pass.
有些系统使用带双挡片的卡片配合单个光电门，在一次通过中测出两次遮光时间和间隔时间。
7. Air track experiments for uniform acceleration | 气垫导轨匀加速实验
To obtain acceleration, the air track can be tilted slightly so that the component of gravity along the track accelerates the glider. Alternatively, a small mass hanging over a pulley can provide a constant net force.
为了获得加速度，可以将气垫导轨稍微倾斜，使重力沿导轨的分量加速滑块。或者，用悬挂在滑轮上的小质量提供恒定的合外力。
The glider carries a card of known length. Two photogates measure the speeds u and v, and a timer or data logger records the time t between them. Then a = (v – u) / t.
滑块上固定已知长度的挡光片。两个光电门测量速度 u 和 v，计时器或数据记录仪记录两门之间的时间 t。则 a = (v – u) / t。
To verify that acceleration is constant, repeat the measurement with different starting positions or different gate separations. A velocity-time graph should be a straight line of constant gradient.
为了验证加速度恒定，可改变起始位置或两门间距重复测量。速度-时间图像应为一条斜率恒定的直线。
Because air friction is very small, the glider’s motion closely matches the predictions of the equations of motion, making this a preferred method for quantitative work.
由于空气摩擦很小，滑块的运动与运动学方程的预测非常接近，因此这是定量实验的首选方法。
8. Free fall with a trapdoor and electromagnet | 自由落体：电磁铁与陷阱门
For measuring the acceleration of free fall g, a steel ball is held by an electromagnet. When the current is switched off, the ball falls and a timer starts automatically; when it hits a trapdoor, the timer stops.
为了测量自由落体加速度 g，钢球由电磁铁吸住。断电后钢球下落，计时器自动启动；当钢球撞击陷阱门时计时停止。
If the ball falls from rest through a measured height s in time t, then s = ½gt². Rearranging gives g = 2s / t².
如果钢球从静止下落已知高度 s，所用时间为 t，则 s = ½gt²。整理得 g = 2s / t²。
Repeating the experiment for several heights and plotting s against t² gives a straight line through the origin. The gradient is ½g, so g = 2 × gradient.
对不同高度重复实验，并绘制 s 对 t² 的图像，可得到一条过原点的直线。斜率为 ½g，因此 g = 2 × 斜率。
The main advantage is that timing starts and stops electronically, reducing random errors from human reaction time.
主要优点是计时由电子方式开始和停止，从而减少了
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