📚 Would an Increase in Mass of a Sphere Shaped Plastic Affect Its Terminal Velocity? | 增加球形塑料的质量是否会影响其终端速度？
When a sphere is released from rest in a viscous fluid, it accelerates until the upward drag force and buoyant force together balance the downward weight. From that moment onward, the sphere falls at a constant speed known as the terminal velocity. This article explores whether altering the mass of a plastic sphere — while carefully controlling its cross-sectional area — changes its terminal velocity, and if so, what mathematical relationship links mass to terminal velocity. We will derive the physics from first principles, design a controlled experiment suitable for an IB Physics internal assessment, and analyze the data through linearization, uncertainty propagation, and critical evaluation.
当一个球体从静止状态被释放到黏性流体中时，它会加速下落，直到向上的拖曳力和浮力共同平衡向下的重力。从那一刻起，球体以恒定速度下落，称为终端速度。本文探讨改变塑料球的质量（同时仔细控制其横截面积）是否会改变其终端速度，以及如果会，质量和终端速度之间存在怎样的数学关系。我们将从第一性原理推导物理公式，设计一个适合IB物理内部评估的受控实验，并通过线性化、不确定度传播和批判性评估来分析数据。
1. Introduction to Terminal Velocity | 终端速度简介
Terminal velocity is a classic topic in mechanics that brings together Newton’s second law, fluid dynamics, and the concept of balanced forces. For a sphere falling through a fluid such as water, air, or glycerol, the interplay between weight, buoyancy, and drag determines how fast it ultimately travels. In many textbook problems, students are told that a heavier object falls faster, but the precise relationship depends on whether the object’s size remains constant. This investigation dissects that nuance by asking: if we keep the diameter of a plastic sphere unchanged and only vary its mass (by filling it with different amounts of water or sand), how does the terminal velocity change?
终端速度是力学中的一个经典课题，它汇集了牛顿第二定律、流体动力学以及平衡力的概念。对于在流体（如水、空气或甘油）中下落的球体，重力、浮力和拖曳力之间的相互作用决定了它最终的运动速度。在许多教科书问题中，学生被告知较重的物体下落得更快，但精确的关系取决于物体的大小是否保持不变。本次探究剖析了这一细微之处，提出如下问题：如果我们保持塑料球的直径不变，仅改变其质量（通过注入不同量的水或沙子），终端速度会如何变化？
2. Theoretical Background: Drag Forces in Fluids | 理论背景：流体中的拖曳力
When an object moves through a fluid, it experiences a resistive force called drag. The nature of this drag depends on the Reynolds number Re = ρvd/η, where ρ is the fluid density, v is the velocity, d is the sphere diameter, and η is the dynamic viscosity. At very low Reynolds numbers (Re < 1), the flow is laminar and the drag force obeys Stokes’ law: Fd = 6πηrv, where r is the radius of the sphere. At higher Reynolds numbers (typically Re > 10³), the flow becomes turbulent and the drag force is proportional to the square of the velocity: Fd = ½ρCdAv², where Cd is the drag coefficient (approximately 0.5 for a smooth sphere in turbulent flow) and A is the cross-sectional area. For a medium-sized plastic sphere falling in water, Re often falls in the transition zone (10² to 10³), making the v² dependence a more appropriate model than the linear Stokes regime.
当物体在流体中运动时，会受到称为拖曳力的阻力。拖曳力的性质取决于雷诺数 Re = ρvd/η，其中 ρ 是流体密度，v 是速度，d 是球体直径，η 是动力黏度。在极低雷诺数（Re < 1）下，流动为层流，拖曳力遵循斯托克斯定律：Fd = 6πηrv，其中 r 是球体半径。在较高雷诺数（通常 Re > 10³）下，流动变为湍流，拖曳力与速度的平方成正比：Fd = ½ρCdAv²，其中 Cd 是拖曳系数（对于光滑球体在湍流中约为0.5），A 是横截面积。对于中等大小的塑料球在水中下落，Re 通常落在过渡区（10² 至 10³），这使得 v² 依赖关系比线性斯托克斯模型更为合适。
3. Forces Acting on a Falling Sphere | 作用在下落球体上的力
Consider a plastic sphere of mass m and radius r falling vertically through a fluid of density ρf. Three forces act on it: the downward weight Fg = mg; the upward buoyant force Fb = ρfVg, where V = (4/3)πr³ is the volume of the sphere; and the upward drag force Fd. The net force determines the acceleration according to Newton’s second law. As the sphere accelerates, the drag force increases until the vector sum of all forces becomes zero. At this equilibrium point, the sphere has reached its terminal velocity vt.
考虑一个质量为 m、半径为 r 的塑料球在密度为 ρf 的流体中垂直下落。有三个力作用在其上：向下的重力 Fg = mg；向上的浮力 Fb = ρfVg，其中 V = (4/3)πr³ 是球体的体积；以及向上的拖曳力 Fd。根据牛顿第二定律，合力决定加速度。随着球体加速，拖曳力不断增加，直到所有力的矢量和为零。在这个平衡点，球体达到了其终端速度 vt。
4. Deriving the Terminal Velocity Equation | 推导终端速度方程
At terminal velocity, the net force is zero: Fg – Fb – Fd = 0. Assuming turbulent flow where drag is proportional to v², we substitute Fd = ½ρfCdAv². The equilibrium condition becomes mg – ρfVg – ½ρfCdAvt² = 0. Rearranging for terminal velocity yields the equation shown below. For spheres with the same radius r, the cross-sectional area A = πr² remains constant, and the volume V = (4/3)πr³ is also fixed. Therefore, if we keep r constant and only vary the mass m, the terminal velocity vt is directly proportional to the square root of the net downward force (weight minus buoyancy). Neglecting buoyancy as a small correction or treating it as a constant offset, the model predicts vt² ∝ m.
在终端速度下，合力为零：Fg – Fb – Fd = 0。假设为湍流，拖曳力与 v² 成正比，代入 Fd = ½ρfCdAv²。平衡条件变为 mg – ρfVg – ½ρfCdAvt² = 0。重新整理后得到终端速度的方程如下。对于相同半径 r 的球体，横截面积 A = πr² 保持不变，体积 V = (4/3)πr³ 也是固定的。因此，如果我们保持 r 不变而仅改变质量 m，终端速度 vt 与净向下力（重力减浮力）的平方根成正比。将浮力视为小修正量或常数偏移，模型预测 vt² ∝ m。
vt = √(2(m – ρfV)g / (ρfCdA))
5. Mass Dependence: Two Distinct Regimes | 质量依赖性：两种不同的情况
It is crucial to distinguish two scenarios when discussing how mass affects terminal velocity. Scenario A — constant volume, varying mass: the cross-sectional area A is fixed, so vt ∝ √m. This is the case we investigate here, achievable by filling identical hollow plastic spheres with different quantities of water or sand. Scenario B — constant density, varying size: if we instead use solid plastic balls of different diameters, mass increases with r³, while area increases with r². In that case, vt ∝ √(r³/r²) = √r ∝ m1/6. The predicted exponent changes dramatically from 0.5 to approximately 0.17, highlighting why controlling the cross-sectional area is the most critical variable in this experiment.
在讨论质量如何影响终端速度时，区分两种情况至关重要。情况A——体积不变，质量变化：横截面积 A 固定，因此 vt ∝ √m。这是我们在此探究的情况，可以通过向相同的空心塑料球中注入不同量的水或沙子来实现。情况B——密度不变，尺寸变化：如果我们改用不同直径的实心塑料球，质量随 r³ 增加，而面积随 r² 增加。在这种情况下，vt ∝ √(r³/r²) = √r ∝ m1/6。预测的指数从0.5急剧变化到约0.17，这凸显了为什么控制横截面积是本实验中最为关键的变量。
6. Research Question and Hypothesis | 研究问题与假设
Based on the theoretical framework established above, we formulate the research question: How does the terminal velocity vt of a plastic sphere of fixed diameter depend on its mass m when falling through water? The independent variable is the mass m of the sphere, varied by injecting known volumes of water into a hollow plastic ball of diameter 3.0 cm. The dependent variable is the terminal velocity vt, measured by tracking the sphere’s position over time using video analysis. Controlled variables include the sphere diameter (3.0 cm ± 0.05 cm), fluid temperature (20 °C ± 0.5 °C), fluid type (tap water), and the release mechanism. The hypothesis states that vt² will be directly proportional to the effective mass (m – mbuoyancy), yielding a linear relationship when vt² is plotted against m.
基于上述建立的理论框架，我们提出研究问题：固定直径的塑料球在水中下落时，其终端速度 vt 如何取决于其质量 m？自变量是球体的质量 m，通过向直径为3.0 cm的空心塑料球中注入已知体积的水来改变。因变量是终端速度 vt，通过视频分析追踪球体随时间的位移来测量。控制变量包括球体直径（3.0 cm ± 0.05 cm）、流体温度（20 °C ± 0.5 °C）、流体类型（自来水）以及释放方式。假设为：vt² 与有效质量（m – mbuoyancy）成正比，即当 vt² 对 m 作图时，将呈现线性关系。
7. Experimental Design and Variables | 实验设计与变量
A transparent acrylic tube of height 1.5 m and internal diameter 15 cm is filled with tap water and positioned vertically. A high-speed camera (120 frames per second) is placed at a distance of 1.0 m from the tube to capture the entire falling path. The sphere is released just below the water surface using a pair of tweezers to avoid imparting initial velocity. A ruler with millimeter markings is attached to the back of the tube for calibration. The experiment is repeated five times for each of six mass values (ranging from approximately 15 g to 65 g), giving a total of 30 trials. The sphere’s position is extracted from the video using Tracker software, and the terminal velocity is determined from the slope of the linear portion of the displacement-time graph.
一根高1.5 m、内径15 cm的透明亚克力管装满自来水并垂直放置。一台高速相机（每秒120帧）放置在距离管子1.0 m处，以捕捉整个下落路径。球体使用镊子在水面下方释放，以避免施加初始速度。标有毫米刻度的直尺固定在管子背面用于校准。对六个质量值（范围约15 g至65 g）各重复实验五次，共进行30次试验。使用Tracker软件从视频中提取球体的位置，终端速度由位移-时间图线性部分的斜率确定。
8. Apparatus and Materials | 仪器和材料
| Item | Specification / Quantity |
| Hollow plastic sphere | Diameter 3.00 cm ± 0.05 cm, with sealable injection port |
| Syringe with needle | 10 mL, graduation 0.2 mL |
| Electronic balance | Resolution ± 0.01 g |
| Transparent acrylic tube | Height 1.50 m, inner diameter 15 cm |
| High-speed camera | 120 fps minimum |
| Meter rule | ± 1 mm, attached to tube for calibration |
| Thermometer | ± 0.1 °C, to monitor water temperature |
| Tracker software | Free video analysis tool |
The hollow plastic sphere is modified by drilling a small hole (sealed with a rubber stopper) to allow water injection without changing its external diameter. Before each trial, the sphere is weighed on the electronic balance to confirm the total mass. The water temperature is monitored because viscosity — and hence drag — is temperature-dependent; a change of 1 °C alters water viscosity by roughly 2.5%, which could introduce systematic error.
空心塑料球通过钻一个小孔（用橡胶塞密封）进行改造，以便在不改变外径的情况下注入水。每次试验前，球体在电子天平上称重以确认总质量。水温需监测，因为黏度——进而拖曳力——与温度相关；水温每变化1 °C，水的黏度约改变2.5%，这可能引入系统误差。
9. Experimental Procedure | 实验步骤
Step 1: Fill the acrylic tube with tap water and allow it to settle for 30 minutes so that convection currents dissipate. Measure and record the water temperature at the top, middle, and bottom of the tube; verify that the temperature is uniform within ± 0.5 °C. Step 2: Prepare six mass configurations for the plastic sphere by injecting 0 mL, 10 mL, 20 mL, 30 mL, 40 mL, and 50 mL of water using the syringe. Weigh each configuration three times on the electronic balance and record the average mass. Step 3: Position the camera on a tripod so that its optical axis is perpendicular to the tube, with the entire 1.5 m falling distance visible in the frame. Place the meter rule inside the tube, aligned vertically, to serve as a calibration reference. Step 4: For each mass, use tweezers to hold the sphere just below the water surface. Release gently without imparting any initial velocity. Step 5: Record the fall using the high-speed camera. Repeat each mass configuration five times to obtain a reliable data set. Step 6: Import the video clips into Tracker software. Calibrate the scale using the meter rule, then track the sphere’s vertical position frame by frame. Step 7: For each trial, plot a displacement-time graph and identify the linear portion. The slope of this region gives the terminal velocity vt. Discard any trial where the sphere visibly touched the tube wall during descent.
步骤1：向亚克力管中注满自来水，静置30分钟以使对流消散。测量并记录管子顶部、中部和底部的水温；确认温度均匀性在 ± 0.5 °C 内。步骤2：使用注射器向塑料球中注入0 mL、10 mL、20 mL、30 mL、40 mL和50 mL水，制备六种质量配置。每种配置在电子天平上称重三次，记录平均质量。步骤3：将相机安装在三脚架上，使其光轴垂直于管子，整个1.5 m下落距离均在画面中可见。将米尺垂直放置在管子内部作为校准参考。步骤4：对于每种质量，使用镊子将球体夹持在水面正下方，轻轻释放，不施加任何初速度。步骤5：使用高速相机记录下落过程。每种质量配置重复五次以获得可靠数据集。步骤6：将视频片段导入Tracker软件。使用米尺校准比例，然后逐帧追踪球体的垂直位置。步骤7：对于每次试验，绘制位移-时间图并识别线性部分。此区域的斜率给出终端速度 vt。剔除球体在下落过程中明显触碰管壁的任何试验。
10. Data Collection and Hypothetical Results | 数据收集与假设性结果
The raw data table below shows average terminal velocities for each mass, obtained from five repeated trials. The uncertainty in mass u(m) is taken as the resolution of the electronic balance (± 0.01 g). The uncertainty in terminal velocity u(vt) is calculated as the standard deviation of the five repeated measurements divided by √5, yielding the standard error of the mean. Additional systematic uncertainties from the calibration scale (0.5%) and frame timing (0.2%) are combined in quadrature to give the total uncertainty presented.
下方的原始数据表展示了每种质量的平均终端速度，来自五次重复试验。质量的不确定度 u(m) 取为电子天平的分辨率（± 0.01 g）。终端速度的不确定度 u(vt) 计算为五次重复测量值的标准差除以 √5，即平均值的标准误差。来自校准标尺（0.5%）和帧间计时（0.2%）的额外系统不确定度以平方和根方式合成，得出所列的总不确定度。
| Mass m / g | u(m) / g | Average vt / cm s⁻¹ | u(vt) / cm s⁻¹ | vt² / cm² s⁻² |
| 15.2 | 0.01 | 18.5 | 0.4 | 342 |
| 25.4 | 0.01 | 23
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