📚 Determining the Drag Coefficient of a Round Parachute | 圆形降落伞阻力系数的测定
When an object falls through a fluid, it experiences a drag force that opposes its motion. For a parachute, this drag force is crucial, and the drag coefficient Cd tells us how efficiently the shape converts kinetic energy into fluid resistance. In this IB Physics investigation, we determine the drag coefficient of a round parachute by measuring its terminal velocity and applying the drag force equation. The experiment combines hands-on construction with video analysis, allowing a robust exploration of air resistance and data processing.
当物体在流体中下落时，会受到与其运动方向相反的阻力。对于降落伞，这种阻力至关重要，而阻力系数 Cd 则表明其形状将动能转化为流体阻力的效率。在这个 IB 物理探究中，我们通过测量圆形降落伞的终端速度并应用阻力方程来确定其阻力系数。该实验将动手制作与视频分析相结合，从而对空气阻力和数据处理进行深入的探索。
1. Introduction and Theory | 引言与理论
The drag force acting on an object moving through a fluid is given by Fd = ½ ρ v² Cd A, where ρ is the fluid density, v is the speed, A is the cross‑sectional area, and Cd is the drag coefficient – a dimensionless constant that depends on the shape of the object. When a parachute falls, it quickly reaches a terminal velocity vt where the upward drag force balances the downward weight. Setting m g = ½ ρ vt² Cd A allows us to solve for Cd if the other quantities are known. For a round parachute that forms a hemispherical cup during descent, the relevant area is the projected area of the mouth, A = π r², where r is the radius of the flat fabric circle before gathering the lines.
作用在流体中运动物体上的阻力由公式 Fd = ½ ρ v² Cd A 给出，其中 ρ 是流体密度，v 是速率，A 是横截面积，Cd 是阻力系数——一个取决于物体形状的无量纲常数。当降落伞下落时，它会很快达到终端速度 vt，此时向上的阻力与向下的重力平衡。令 m g = ½ ρ vt² Cd A，若其他量已知，即可求出 Cd。对于在下落过程中形成半球形杯状的圆形降落伞，相关面积是其伞口的投影面积，A = π r²，其中 r 是伞布在收拢伞绳之前的平铺半径。
2. The Drag Equation and Terminal Velocity | 阻力方程与终端速度
The terminal velocity condition gives the working equation for this investigation:
Cd = (2 m g) / (ρ vt² π r²)
Here m is the total falling mass (parachute + payload), g is the acceleration due to gravity (9.81 m s⁻²), and ρ is the density of air (about 1.2 kg m⁻³ at room temperature). By measuring m, r and vt experimentally, Cd can be deduced. A proper understanding of this relationship is essential before designing the procedure and selecting instruments.
终端速度条件给出了本次探究的工作方程：Cd = (2 m g) / (ρ vt² π r²)。其中 m 是总下落质量（降落伞与负载），g 是重力加速度（9.81 m s⁻²），ρ 是空气密度（室温下约为 1.2 kg m⁻³）。通过实验测量 m、r 和 vt，便可推算出 Cd。在设计步骤和选择仪器之前，正确理解这一关系至关重要。
3. Experimental Setup | 实验装置
The round parachute was made from a lightweight plastic bag. A circle of known radius r was cut, and four equally‑spaced thin strings were attached to its edge. The strings were tied together under the parachute and connected to a metal weight that served as the payload. A metre rule was placed in the field of view of a smartphone camera to calibrate distances. The camera was set to record slow‑motion video at 120 fps to capture the fall. The video was later analysed using Tracker software to obtain the position–time data and extract the terminal velocity.
该圆形降落伞使用轻质塑料袋制作。剪出一个已知半径 r 的圆形，并在其边缘系上四根等距的细线。细线在伞下汇总并系在一个作为负载的金属砝码上。在智能手机摄像头的视野内放置一把米尺，用于标定距离。手机设置为 120 fps 慢动作录制下落过程，随后用 Tracker 软件对视频进行分析，获得位置–时间数据并提取终端速度。
The mass m of the complete system (canopy, lines and payload) was measured with an electronic balance (resolution 0.1 g). The radius r was verified with a vernier calliper while the fabric was laid flat and unstretched. Air density ρ was taken as 1.20 kg m⁻³ based on the lab temperature and pressure; a more precise value could be obtained from weather data if desired.
整套系统（伞衣、伞绳和负载）的质量 m 用电子天平（分度值 0.1 g）测量。半径 r 在伞布平铺且未拉伸的状态下用游标卡尺核实。空气密度 ρ 根据实验室温度与气压取为 1.20 kg m⁻³；若需要更精确的数值，可从气象数据中获取。
4. Procedure | 实验步骤
Step 1: Cut a plastic circle of radius r = 0.200 m. Attach four 0.40 m threads symmetrically and tie them to a 50 g mass hanger. Record the total mass m using the balance.
步骤 1：剪出一个半径为 r = 0.200 m 的塑料圆片。对称地系上四根 0.40 m 的细线，并将其系在一个 50 g 的钩码上。用天平记录总质量 m。
Step 2: Set up the camera on a tripod so that a 1.00 m vertical distance is clearly framed with the metre rule visible. Ensure good lighting and a dark background to enhance tracking contrast.
步骤 2：将手机固定在三脚架上，使 1.00 m 的竖直距离清晰地进入画幅，并保证米尺可见。确保光线充足，使用暗色背景以增强追踪对比度。
Step 3: Hold the parachute at the top of the frame and release it without spin. Allow the system to fall through the entire field of view. Repeat for a total of five trials.
步骤 3：在画幅顶端手持降落伞，无旋转地释放。让系统下坠穿过整个视场。共重复进行五次试验。
Step 4: Import each video into Tracker. Calibrate the scale using the metre rule. Track the payload mass centre frame by frame to produce a y–t graph. Fit a straight line to the linear portion of the graph; the slope gives the terminal speed vt. Record the mean vt and its standard deviation.
步骤 4：将每段视频导入 Tracker。用米尺标定比例尺。逐帧追踪负载的质心，生成 y–t 图。对图像的线性段进行直线拟合；其斜率即为终端速度 vt。记录平均 vt 及其标准偏差。
5. Data Collection | 数据收集
The table below shows the raw and derived data for one parachute configuration. The terminal speed vt was recorded for each trial, and the mean value was used to calculate Cd.
下表展示了一种降落伞配置的原始数据与导出数据。每次试验记录了终端速度 vt，并使用其平均值来计算 Cd。
| Trial | Mass m (kg) | Radius r (m) | vt (m s⁻¹) |
|---|---|---|---|
| 1 | 0.0521 | 0.200 | 1.89 |
| 2 | 0.0521 | 0.200 | 1.92 |
| 3 | 0.0521 | 0.200 | 1.86 |
| 4 | 0.0521 | 0.200 | 1.90 |
| 5 | 0.0521 | 0.200 | 1.88 |
Mean vt = 1.89 m s⁻¹, standard deviation = 0.02 m s⁻¹. The air density was taken as ρ = 1.20 kg m⁻³ and g = 9.81 m s⁻².
平均 vt = 1.89 m s⁻¹，标准偏差 = 0.02 m s⁻¹。空气密度取为 ρ = 1.20 kg m⁻³，g = 9.81 m s⁻²。
6. Data Analysis: Calculating Drag Coefficient | 数据分析：计算阻力系数
Using the working equation and the mean values:
Cd = (2 × 0.0521 kg × 9.81 m s⁻²) / (1.20 kg m⁻³ × (1.89 m s⁻¹)² × π × (0.200 m)²)
Evaluating the numerator:
Published by TutorHao | IB Physics Revision Series | aleveler.com
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