📚 Determining Velocity and Acceleration in the Laboratory | 在实验室中测定速度和加速度
Kinematics is the branch of physics that describes motion in terms of displacement, velocity and acceleration without considering the forces that cause it. In the A‑Level laboratory, students measure these quantities using a variety of apparatus, from simple rulers and stopwatches to electronic sensors. This article examines the principal experimental methods, data processing and sources of uncertainty, equipping you to evaluate and improve practical determinations of velocity and acceleration.
运动学是物理学的分支，用位移、速度和加速度描述物体的运动，而不考虑引起运动的力。在A‑Level实验室中，学生们使用从简单的尺子和秒表到电子传感器的各种仪器来测量这些量。本文探讨了主要的实验方法、数据处理以及不确定度的来源，帮助你评估并改进速度和加速度的实际测定。
1. Basic Tools for Kinematic Measurements | 运动学测量的基本工具
Measuring velocity and acceleration always begins with the measurement of displacement and time. A metre rule or tape measure provides displacement readings to the nearest millimetre, while a digital stopwatch offers a resolution of 0.01 s, although human reaction time (typically 0.1–0.3 s) limits its accuracy. For more precise timing, electronic devices such as light gates or motion sensors are preferred.
测量速度和加速度总是从位移和时间的测量开始。米尺或卷尺提供的位移读数精确到毫米，数字秒表的分辨率为0.01秒，但通常人的反应时间（约0.1–0.3秒）限制了其准确度。为了更精确地计时，人们更倾向于使用光电门或运动传感器等电子设备。
ht gate, the gate records the time interval Δt for which the beam is broken. The average speed over the short distance is taken as a good approximation of the instantaneous velocity at the midpoint of the gate, given by v = Δx / Δt.
实验室中测量瞬时速度常借助遮断光束的方法。当一块已知宽度Δx的挡光片通过光电门时，光电门记录光束被遮断的时间间隔Δt。在短距离内的平均速度可作为光电门中点处瞬时速度的良好近似，即 v = Δx / Δt。
2. Measuring Average Velocity Directly | 直接测量平均速度
The simplest kinematics experiment places a trolley on a horizontal or inclined track. Students mark a start and finish line a known distance s apart, release the trolley and use a stopwatch to measure the time t taken to travel between them. The average velocity is then calculated as vav ghtforward, this method is vulnerable to large uncertainties because of the manual timing.
最简单的运动学实验是将小车放在水平或倾斜的轨道上。学生标出相距已知距离s的起点线和终点线，释放小车并用秒表测量经过这段距离所用的时间t。然后计算平均速度 vav = s / t。这个方法虽然直接，但由于手动计时，容易产生较大的不确定度。
en the start and finish positions. The time interval is then frames / frame rate, yielding a timer uncertainty of ± half a frame interval. This approach improves the reliability of average velocity data without requiring specialised equipment.
为了减小人为反应误差，你可以用智能手机以已知帧率拍摄运动过程，并计算起点和终点之间的帧数。时间间隔就是 帧数 / 帧率，计时的不确定度为 ± 半个帧间隔。这种方法无需专用设备就能提高平均速度数据的可靠性。
3. Instantaneous Velocity Using a Single Light Gate | 用单光电门测量瞬时速度
te the velocity of a trolley at the gate’s position, use the relationship v = d / t. The card should be as narrow as practical while still being measurable accurately; a typical value is d = 5.0 cm, measured with vernier calipers to ±0.1 mm.
连接数据采集器或接口的光电门可以测量不透明挡光片宽度d遮断光束的时间。若要估算小车在光电门位置的速度，可以使用关系式 v = d / t。挡光片应尽可能窄，但同时要能精确测量；典型值是 d = 5.0 cm，用游标卡尺测量的不确定度为 ±0.1 mm。
For best results, position the light gate exactly at the point of interest and ensure the card cuts the beam perpendicularly. If the card enters at an angle, the effective width intercepted by the beam changes, introducing a systematic error. Repeating the measurement several times and taking the mean minimises random timing fluctuations.
为了获得最佳结果，应将光电门精确放置在关注点，并确保挡光片垂直切断光束。如果挡光片以一定角度进入，光束截断的实际宽度就会改变，从而引入系统误差。多次重复测量并取平均值可以减小随机的计时波动。
4. Determining Acceleration with Two Light Gates | 用双光电门测定加速度
To find the acceleration of a trolley, set up two light gates a distance s apart along the track. Each gate measures the time for a single card to pass through, giving the initial velocity u at the first gate and the final velocity v at the second gate. The data logger also records the time interval T taken for the card to travel from one gate to the other.
为了测量小车的加速度，可沿轨道相隔距离s设置两个光电门。每个光电门测量同一块挡光片通过的时间，由此得到第一个光电门处的初速度u和第二个光电门处的末速度v。数据采集器还会记录挡光片从一个光电门运动到另一个光电门所用的时间间隔T。
t of manual stopwatch readings. Alternatively, if the time between the gates is not directly recorded, you can use the kinematic equation v² = u² + 2as, provided friction and air resistance are negligible, and solve for a.
然后可以根据定义 a = (v − u) / T 计算加速度。由于T是用电子方式测量的，其不确定度远小于手动秒表读数。另外，如果没有直接记录两个光电门之间的时间，只要摩擦和空气阻力可忽略，你也可以使用运动学方程 v² = u² + 2as，并求解a。
a = (v − u) / T
5. Using a Ticker Timer | 使用打点计时器
A ticker timer produces dots on a paper tape at a known frequency, usually 50 Hz, giving a time interval of 0.02 s between consecutive dots. Attach the tape to a moving trolley; as the trolley moves, the timer prints a series of dots that record the displacement at equal time steps. The tape thus becomes a permanent record of the motion.
打点计时器以已知频率（通常为50 Hz）在纸带上打点，两个连续点之间的时间间隔为0.02 s。将纸带连接在运动的小车上；随着小车移动，计时器打印出一系列点，记录了相等时间步长内的位移。因此纸带成为运动的永久记录。
To construct a velocity– ngth of each strip divided by 0.10 s gives the average velocity during that interval. Plot these velocities against the mid‑ ip. If the graph is a straight line, the acceleration is constant and equals the gradient.
为了构建速度–时间图，可将纸带切成每5个点一段（Δt = 0.10 s）。每段纸带的长度除以0.10 s就是该时间间隔内的平均速度。将这些速度值相对于每段纸带的中间时刻作图。如果图形是一条直线，则加速度恒定且等于其斜率。
vstrip = length / 0.10 s
6. Motion Sensors and Data Logging | 运动传感器与数据记录
Ultrasonic or infrared motion sensors emit pulses that reflect off the moving object, and the sensor calculates distance from the time‑of‑flight. A data logger records position at regular intervals, typically 0.05 s or 0.02 s, generating a table of (t, x) data. Software then differentiates the data numerically to produce velocity–time and acceleration–time graphs in real time.
超声波或红外运动传感器发射脉冲，脉冲经运动物体反射，传感器根据飞行时间计算出距离。数据采集器以固定间隔（通常为0.05 s 或 0.02 s）记录位置，生成 (t, x) 数据表。然后软件对数据进行数值微分，实时生成速度–时间图和加速度–时间图。
This method is extremely convenient for studying non‑uniform acceleration, such as the motion of a mass on a spring. The main source of error is the beam’s divergence, which can pick up reflections from nearby objects. Align the sensor squarely with the line of motion and remove clutter from the sensing zone to obtain clean data.
这种方法对于研究非匀加速度运动（例如弹簧振子的运动）非常方便。主要的误差来源是波束的发散，可能会接收到附近物体的反射信号。应将传感器正对运动方向，并清除感应区域内的杂物，以获得干净的数据。
7. Stroboscopic Photography | 频闪摄影法
Stroboscopic photography uses a camera with its shutter held open in a darkened room while a stroboscope illuminates the moving object with short, regular flashes. The resulting image shows multiple exposures of the object at equal time intervals. A reference scale placed in the same plane allows displacement measurements directly from the photograph.
频闪摄影法使用快门常开的相机，在暗室中用频闪仪以短促而规则的闪光照亮运动物体。得到的照片显示出物体在相等时间间隔的多次曝光影像。在同一平面内放置的参考标尺使得可以直接从照片上测量位移。
he average velocity for each interval, v = Δx / Δt. The acceleration can be found from the change in these velocities over the recorded sequence. This technique is especially useful for analysing projectile motion.
如果闪光频率为 f，则相邻影像之间的时间间隔为 Δt = 1/f。测量相邻位置间的距离 Δx 可得到每个间隔的平均速度 v = Δx / Δt。通过记录序列中速度的变化可以求出加速度。这种方法特别适合分析抛体运动。
8. Building Velocity–Time Graphs from Raw Data | 由原始数据绘制速度–时间图
Regardless of the data collection method, the most powerful tool for determining acceleration is the velocity–time graph. After calculating a set of instantaneous or average velocities at known times, plot velocity on the vertical axis and time on the horizontal axis. Draw a line of best fit; its slope represents acceleration.
无论采用哪种数据采集方法，确定加速度最有力的工具就是速度–时间图。计算出一组已知时刻的瞬时速度或平均速度后，将速度标在纵轴、时间标在横轴上。画一条最佳拟合线，其斜率就代表加速度。
luster around a straight line. The gradient is calculated as Δv / Δt by selecting two well‑separated points on the best‑ ata points). Avoid using the origin unless the motion genuinely starts from rest at t = 0.
对于在斜面上匀加速运动的小车，数据点应围绕一条直线分布。通过选择最佳拟合线上两个间隔较远的点（不一定是数据点）可计算出斜率 Δv / Δt。除非运动确实从静止开始且t = 0，否则不要强制使用原点。
9. Calculating Acceleration from the Gradient | 通过斜率计算加速度
t₁). Always include units: if velocity is in m s⁻¹ and time in s, acceleration has units m s⁻². The line’s general equation is v = u + a t, where the y‑intercept u is the initial velocity.
画出最佳拟合线后，选取线上两点的坐标 (t₁, v₁) 和 (t₂, v₂)，并用 a = (v₂ − v₁) / (t₂ − t₁) 确定加速度a。务必标明单位：若速度单位为 m s⁻¹、时间单位为 s，则加速度单位为 m s⁻²。该直线的通式为 v = u + a t，其中y轴截距u为初速度。
If the v–t graph is curved, the acceleration is not uniform. In such cases, drawing a tangent to the curve at a specific time gives the instantaneous acceleration at that moment. Digital software can perform this tangent calculation automatically and display an acceleration–time graph.
如果v–t图是曲线，则加速度并不均匀。此时，在特定时刻作曲线的切线，可得到该时刻的瞬时加速度。数字软件可以自动完成切线计算并显示加速度–时间图。
10. Checking Constant Acceleration with Kinetic Equations | 用运动学方程验证匀加速度
est (u = 0) and travelling a distance s, the equation s = ½ a t² applies. Plotting a graph of s against t² yields a straight line through the origin, and the gradient is ½ a. This method avoids measuring instantaneous velocities altogether.
另一种方法是验证匀加速假设，即检验其中一个SUVAT方程。对于从静止开始（u = 0）并移动距离 s 的小车，公式 s = ½ a t² 成立。绘制 s 与 t² 的关系图，将得到一条过原点的直线，其斜率为 ½ a。这种方法完全避免了测量瞬时速度。
Similarly, if the initial velocity is not zero but the final velocity v is measured for different displacements s, plotting v² against s gives a gradient of 2a. The linearity of the graph provides strong evidence that the acceleration is constant and friction forces are small.
类似地，如果初速度不为零，但测量了不同位移 s 对应的末速度 v，则绘制 v² 与 s 的关系图会得到斜率为 2a 的直线。图像的线性有力地表明加速度恒定且摩擦力很小。
s = ½ a t² ; v² = u² + 2 a s
11. Error Analysis and Uncertainty | 误差分析与不确定度
centage uncertainty can be estimated by combining the percentage uncertainties in d and t: %U(v) ≈ %U(d) + %U(t). If d = 5.0 cm ± 0.1 cm (2%) and t = 0.250 s ± 0.001 s (0.4%), the velocity uncertainty is about 2.4% ties, the uncertainty approximately doubles because subtraction magnifies relative errors.
每一次测量都带有不确定度。对于单光电门速度 v = d / t，可通过合成d和t的百分比不确定度来估算其百分比不确定度：%U(v) ≈ %U(d) + %U(t)。若 d = 5.0 cm ± 0.1 cm (2%)，t = 0.250 s ± 0.001 s (0.4%)，则速度不确定度约为2.4%。对于由两个这样的速度计算出的加速度，由于减法运算会放大相对误差，不确定度大约会翻倍。
Systematic errors, such as a misaligned light gate or a tilted track causing a component of gravitational acceleration, shift all readings in one direction. They cannot be reduced by averaging but can be identified by comparing your result with a theoretical value or using an independent method. Always discuss both systematic and random uncertainties in your evaluation.
系统误差，例如光电门未对准或轨道倾斜导致重力加速度分量，会使所有读数向同一方向偏移。这类误差不能通过取平均值来减小，但可通过与理论值比较或使用独立方法来识别。在实验评估中，务必同时讨论系统不确定度和随机不确定度。
12. Practical Improvements and Extension | 实际改进与拓展
To reduce random timing errors in light‑gate experiments, use a double‑card interruption technique: attach two cards of known separation to the trolley. The same gate records two interruptions, giving two velocities and the time between them, eliminating the need for a second gate and reducing misalignment errors. This setup improves the precision of acceleration measurements significantly.
为了减小光电门实验中的随机计时误差，可以采用双挡光片遮断技术：在小车上安装两块已知间距的挡光片。同一个光电门记录两次遮断，从而得到两个速度以及它们之间的时间差，这样就不再需要第二个光电门，还能减小对准误差。这种配置可以显著提高加速度测量的精度。
Friction compensation is essential when trolleys run on tracks. Slightly tilt the track so that the trolley, given a gentle push, moves with constant velocity. This means the component of weight down the slope balances the friction. When the track is properly compensated, the applied force (e.g., a hanging mass) produces the full acceleration expected from Newton’s second law, making data analysis more straightforward.
当小车在轨道上行驶时，摩擦补偿至关重要。将轨道稍微倾斜，使小车在轻推后能匀速运动。这意味着重力沿斜面的分量与摩擦力平衡。当轨道得到恰当补偿时，施加的外力（例如悬挂的重物）就能产生符合牛顿第二定律的全部加速度，从而使数据分析更加简单。
Finally, consolidating your data in a table and plotting graphs with error bars allows you to assess the reliability of your conclusions. Use the scatter of points about the line of best fit to estimate the uncertainty in the gradient, and hence in the acceleration. This rigorous treatment mirrors the skills expected in CIE Paper 3 and Paper 5 practical assessments.
最后，将数据整理成表格并绘制带误差棒的图线，可以让你评估结论的可靠性。利用数据点围绕最佳拟合线的离散程度来估计斜率的不确定度，进而得到加速度的不确定度。这种严谨的处理方式反映了CIE卷3和卷5实验评估所要求的技能。
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