📚 Determining the Drag Coefficient of a Round Parachute: Application Problem Techniques | 测定圆形降落伞阻力系数：应用题技巧
In IB Physics, the drag force acting on a parachute is a classic application of mechanics and experimental data analysis. This article focuses on the techniques for determining the drag coefficient of a round parachute through typical exam-style problems, emphasising linearisation, graphical analysis, and uncertainty evaluation.
在 IB 物理中，作用在降落伞上的阻力是力学与实验数据分析的经典应用。本文重点介绍通过典型考题测定圆形降落伞阻力系数的技巧，着重阐述线性化、图形分析和不确定度评估。
1. Understanding Drag Force and Drag Coefficient | 理解阻力与阻力系数
Drag force is the resistive force experienced by an object moving through a fluid, such as air. For a parachute, this force opposes gravity and is crucial for safe descent. The drag coefficient Cd is a dimensionless number that quantifies the drag per unit area, dependent on the shape and surface texture of the object.
阻力是物体在流体（如空气）中运动时受到的阻碍力。对降落伞而言，阻力与重力方向相反，对安全下降至关重要。阻力系数 Cd 是一个无量纲数，量化单位面积所受的阻力，取决于物体的形状和表面纹理。
In IB problems, you are often required to calculate Cd from experimental data or to predict the behaviour of a parachute system. A solid grasp of the underlying physics is essential before tackling any calculation.
在 IB 考题中，通常要求从实验数据中计算 Cd，或预测降落伞系统的行为。在动手计算之前，牢固掌握背后的物理原理至关重要。
2. The Drag Equation for a Round Parachute | 圆形降落伞的阻力方程
The magnitude of the drag force Fd is given by:
阻力 Fd 的大小由下式给出：
Fd = ½ Cd ρ A v²
where ρ is the density of air (approx. 1.2 kg m⁻³ at sea level), A is the cross-sectional area of the parachute, and v is the instantaneous speed. For a round parachute, the area A is the projected circular area πr², where r is the radius when the parachute is fully inflated.
其中 ρ 为空气密度（海平面约 1.2 kg m⁻³），A 为降落伞的横截面积，v 为瞬时速率。对于圆形降落伞，面积 A 是完全充气后的投影圆面积 πr²，r 为半径。
The factor ½ appears from the dynamic pressure ½ρv². Cd typically ranges from 1.0 to 1.5 for a round parachute, but its exact value depends on the fabric porosity and shape.
因子 ½ 来自动压 ½ρv²。圆形降落伞的 Cd 通常在 1.0 到 1.5 之间，但精确值取决于织物的透气性和形状。
3. Terminal Velocity and Force Balance | 终端速度与力平衡
When a parachutist reaches terminal velocity vt, the net force is zero. The drag force equals the weight:
当跳伞者达到终端速度 vt 时，合力为零。阻力等于重力：
½ Cd ρ A vt² = m g
This equation is the cornerstone for determining Cd. By rearranging, we obtain:
该方程是求解 Cd 的基础。重排后得到：
vt = √(2 m g / (Cd ρ A))
In many experiments, vt is measured for a known mass m, and Cd is then calculated. Alternatively, Cd can be extracted from the slope of a suitable graph.
在许多实验中，测量已知质量 m 对应的终端速度 vt，然后计算 Cd。另外，也可从合适图像的斜率中提取 Cd。
Always note that m includes the mass of the parachute and any attached payload. Use the correct gravitational field strength g (9.81 N kg⁻¹ or 10 N kg⁻¹ as specified).
务必注意 m 包括降落伞及其载荷的质量。使用正确的重力场强度 g（9.81 N kg⁻¹ 或题目指定的 10 N kg⁻¹）。
4. Experimental Set-up for Determining Cd | 测定阻力系数的实验装置
A typical experiment involves dropping a parachute-payload system from a height and recording its motion. Motion sensors, video analysis, or light gates can be used to determine the terminal velocity.
典型实验从一定高度释放降落伞—负载系统，并记录其运动。运动传感器、视频分析或光门可用于确定终端速度。
To minimise error, the drop height must be sufficient for terminal velocity to be reached. The area A is measured by flattening the parachute and measuring the diameter of the circle, then calculating A = π (d/2)².
为减小误差，释放高度必须足够高以达到终端速度。面积 A 的测量方法是将降落伞展平，测量圆形的直径，然后计算 A = π (d/2)²。
Common IB data show terminal speed vt as a function of mass m. The drag coefficient can be found by plotting a graph and using the gradient.
常见的 IB 数据给出终端速度 vt 与质量 m 的函数关系。可通过拟合图像并利用斜率求出阻力系数。
5. Data Collection and Variables | 数据收集与变量
Independent variable: total mass m (varied by adding weights to the payload).
Dependent variable: terminal velocity vt (measured after the parachute stops accelerating).
Controlled variables: parachute size (area A), air density ρ, release height, and aerodynamic shape.
自变量：总质量 m（通过向载荷添加重物改变）。
因变量：终端速度 vt（在降落伞不再加速后测量）。
控制变量：降落伞尺寸（面积 A）、空气密度 ρ、释放高度和气动外形。
It is advisable to record multiple trials for each mass and calculate mean vt and its uncertainty. A table should include columns for mass m, terminal speed vt (mean ± absolute uncertainty), and calculated vt².
建议对每个质量进行多次试验，计算平均 vt 及其不确定度。数据表应包括质量 m、终端速度 vt（平均值 ± 绝对不确定度）以及计算得到的 vt² 等列。
6. Linearising the Equation for Graphical Analysis | 方程的线性化与图像分析
To determine Cd graphically, we rearrange the force balance equation into a straight-line form. From ½ Cd ρ A vt² = m g, we can write:
为了通过图像确定 Cd，我们将力平衡方程重排成直线形式。由 ½ Cd ρ A vt² = m g，可得：
vt² = (2 g / (Cd ρ A)) × m
Thus, a graph of vt² (y-axis) against m (x-axis) should yield a straight line through the origin, with gradient = 2 g / (Cd ρ A).
因此，以 vt² 为 y 轴，m 为 x 轴作图像，应得到一条过原点的直线，斜率 = 2 g / (Cd ρ A)。
Alternatively, if m is kept constant and A is varied, a graph of vt² against 1/A gives a straight line. You can also plot m against vt², which is equally valid. The key is to identify the expression that produces a linear relationship.
或者，若保持 m 不变而改变 A，作 vt² 对 1/A 的图像将得到一条直线。也可作 m 对 vt² 的图像，这同样有效。关键是找到能产生线性关系的表达式。
In IB style problems, you may be asked to ‘suggest a graph that would verify the relationship’ and then to extract Cd from the gradient.
在 IB 风格的题目中，你可能会被要求“提出一个能够验证该关系的图像”，然后从斜率中提取 Cd。
7. Interpreting the Gradient and Intercept | 解释斜率和截距
For the graph of vt² vs m, the gradient k is:
对于 vt² 对 m 的图像，斜率 k 为：
k = 2 g / (Cd ρ A)
Rearranging gives: Cd = 2 g / (k ρ A). Ensure all units are SI: m in kg, vt² in m² s⁻², A in m², ρ in kg m⁻³, g in m s⁻². A consistent unit check is a quick way to avoid errors.
重排得：Cd = 2 g / (k ρ A)。确保所有单位均为国际单位制：m 用 kg，vt² 用 m² s⁻²，A 用 m²，ρ 用 kg m⁻³，g 用 m s⁻²。进行单位一致性检查是避免错误的快捷方法。
If the line does not pass through the origin, there may be a systematic error, such as an unaccounted air resistance component or a zero-offset in mass measurement. Commenting on this in an exam can gain you analysis marks.
若直线不过原点，则可能存在系统误差，例如未考虑的额外空气阻力或质量测量中的零偏移。在考试中对此加以评论可为你赢得分析分值。
8. Common IB-Style Application Problems | 常见 IB 风格应用题
Problem 1: A round parachute of diameter 1.0 m carries a payload of 5.0 kg. The terminal velocity is recorded as 4.0 m s⁻¹. Given ρ = 1.2 kg m⁻³ and g = 9.8 m s⁻², estimate Cd.
问题 1：一个直径 1.0 m 的圆形降落伞携带 5.0 kg 的载荷。终端速度记录为 4.0 m s⁻¹。已知 ρ = 1.2 kg m⁻³，g = 9.8 m s⁻²，估算 Cd。
Solution approach: compute A = π × (0.50)² = 0.785 m². Then use m g = ½ Cd ρ A vt² → 5.0 × 9.8 = ½ × Cd × 1.2 × 0.785 × (4.0)². Solve for Cd ≈ 1.3.
解题思路：计算 A = π × (0.50)² = 0.785 m²。然后使用 m g = ½ Cd ρ A vt² → 5.0 × 9.8 = ½ × Cd × 1.2 × 0.785 × (4.0)²。解得 Cd ≈ 1.3。
Problem 2: A student measures vt for various masses and plots vt² vs m. The slope is found to be 15.0 m² s⁻² kg⁻¹. If A = 0.50 m² and ρ = 1.2 kg m⁻³, find Cd.
问题 2：某学生测量了不同质量对应的 vt，并作 vt² 对 m 的图像。斜率为 15.0 m² s⁻² kg⁻¹。若 A = 0.50 m²，ρ = 1.2 kg m⁻³，求 Cd。
Solve: 15.0 = 2 × 9.8 / (Cd × 1.2 × 0.50) → Cd = (19.6) / (15.0 × 0.60) ≈ 2.18. A quick reality check: is this Cd realistic for a parachute? Values above 2 may indicate a very porous canopy or error; typical round parachute Cd ≈ 1.2–1.5. This kind of critical thinking impresses IB examiners.
求解：15.0 = 2 × 9.8 / (Cd × 1.2 × 0.50) → Cd = (19.6) / (15.0 × 0.60) ≈ 2.18。快速检验：此 Cd 对降落伞来说现实吗？大于 2 的值可能表明伞衣透气性很高或存在误差；典型的圆形降落伞 Cd ≈ 1.2–1.5。这种批判性思维能给 IB 考官留下深刻印象。
9. Dealing with Uncertainties and Error Propagation | 处理不确定度与误差传播
Uncertainty analysis is a vital component of IB Physics internal assessment and exam questions. When calculating Cd from measured quantities, you must propagate uncertainties.
不确定度分析是 IB 物理内部评估和考试题的关键组成部分。在根据测量量计算 Cd 时，必须进行误差传播。
If Cd = 2mg / (ρ A vt²), the fractional uncertainty can be approximated as:
若 Cd = 2mg / (ρ A vt²)，相对不确定度可近似为：
ΔCd/Cd ≈ Δm/m + Δg/g + Δρ/ρ + ΔA/A + 2(Δvt/vt)
Note the factor of 2 for vt because it appears squared. Always express the final answer with an appropriate number of significant figures reflecting the uncertainty.
注意 vt 前的因子 2，因其以平方形式出现。最终答案的有效数字应反映不确定度。
Common IB pitfalls: forgetting to convert diameter to radius in A, using cm² instead of m², and misidentifying the slope units. A systematic table of errors is highly recommended in write-ups.
常见的 IB 易错点：忘记将直径转化为半径求 A，单位用 cm² 而非 m²，以及斜率单位识别错误。在实验报告中强烈建议列出系统的误差表格。
10. Tips for Answering Extended Response Questions | 解答拓展题的技巧
IB extended response questions often ask you to describe the experimental procedure, justify the linearisation, and evaluate the results.
IB 拓展题经常要求描述实验步骤，论证线性化方法，并评估结果。
Structure your answer logically: state the force balance, derive the linear equation, identify axes, explain how to obtain Cd from the gradient, and discuss assumptions (e.g. constant ρ, no horizontal wind, parachute fully open).
逻辑清晰地组织答案：陈述力平衡，推导线性方程，确定坐标轴，解释如何通过斜率求 Cd，并讨论假设（如 ρ 恒定，无横向风，降落伞完全展开）。
If a problem provides a graph with a non-zero intercept, explain that this might indicate an initial acceleration phase not captured or an additional constant force (like a small upthrust). Linking the intercept to physical meanings demonstrates deeper understanding.
如果题目给出带有非零截距的图像，可解释这可能表明未记录的初始加速阶段，或存在额外的恒定力（如微小浮力）。将截距与物理含义联系起来体现了更深的理解。
Lastly, always check the order of magnitude of your calculated Cd. A result of 0.1 or 10 should prompt you to re-examine your working. This numerical common sense is a hallmark of a high-level physics student.
最后，务必检查计算出的 Cd 数量级。结果为 0.1 或 10 时，应促使你重新检查计算过程。这种数字常识是高水平物理学生的标志。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply