Determining the Drag Coefficient of a Round Parachute: Formula Derivation | 测定圆形降落伞的阻力系数：公式推导

📚 Determining the Drag Coefficient of a Round Parachute: Formula Derivation | 测定圆形降落伞的阻力系数：公式推导

Understanding fluid resistance is essential in IB Physics, especially when designing experiments to measure aerodynamic forces. This article walks through the derivation of the drag coefficient for a round parachute, starting from the fundamental drag equation and applying the condition of terminal velocity. We then discuss how such a derivation forms the basis of a reliable experimental investigation—often found in Internal Assessment (IA) work—and examine the practical steps, data handling, and uncertainty analysis required to obtain a meaningful value for C_d.

理解流体阻力是 IB 物理的重要内容，尤其在设计实验测量空气动力时尤为关键。本文将从基本阻力方程出发，应用终端速度条件，逐步推导圆形降落伞的阻力系数。我们将展示这一推导如何构成一项可靠的实验探究（常见于内部评估 IA），并讨论获得有意义的 C_d 值所需的实际步骤、数据处理和不确定度分析。

1. The Drag Force Equation | 阻力方程

For an object moving through a fluid, the resistive force opposing motion is described by the drag equation. The magnitude of this drag force F_d is proportional to the fluid density ρ, the cross‑sectional area A presented to the flow, and the square of the object’s speed v. The proportionality constant is the dimensionless drag coefficient C_d, which encapsulates the shape and surface characteristics of the object.

物体在流体中运动时，阻碍运动的阻力由阻力方程描述。阻力大小 F_d 与流体密度 ρ、迎流截面积 A 以及物体速度 v 的平方成正比，比例系数为无量纲的阻力系数 C_d，它反映了物体的形状和表面特征。

F_d = ½ ρ v² A C_d

For a round parachute, A is typically taken as the projected area of the canopy when it is fully inflated—i.e. the area of the circular disc projected onto a plane perpendicular to the motion. This area is calculated using the nominal diameter D (or radius r) of the canopy.

对于圆形降落伞，A 通常取伞衣完全张开时的投影面积，即投影到垂直于运动方向的平面上的圆盘面积。该面积由伞衣的名义直径 D（或半径 r）计算得出。

A = π (D/2)² = π D² / 4

The drag coefficient C_d is not a universal constant; it depends on the Reynolds number and the canopy’s porosity, shape, and fabric oscillation. For a typical round parachute, C_d values range approximately from 0.8 to 1.5. Determining it experimentally provides a deeper insight into how geometry affects aerodynamic efficiency.

阻力系数 C_d 并非普适常数；它取决于雷诺数以及伞衣的透气性、形状和织物摆动。典型圆形降落伞的 C_d 值约在 0.8 至 1.5 之间。通过实验测定该系数，可以更深入地理解几何形状如何影响气动效能。

2. Forces on a Parachute in Free Fall | 自由落体中的降落伞受力

When a parachute with a suspended payload is released, two primary vertical forces act on the system: the weight pulling downward and the aerodynamic drag opposing the motion. Immediately after release, the velocity is small, and the drag is negligible, causing the system to accelerate downward under gravity. As speed increases, so does the drag force.

带有悬挂载荷的降落伞被释放后，系统主要受到两个纵向力的作用：向下的重力和阻碍运动的空气阻力。刚释放时速度很小，阻力可忽略，系统在重力作用下向下加速。随着速度增加，阻力也随之增大。

The net force F_net acting on the parachute‑payload combination of total mass m is therefore:

因此，作用在总质量为 m 的降落伞-载荷组合上的净力 F_net 为：

F_net = mg – ½ ρ v² A C_d

Here, g is the acceleration due to gravity. The sign convention assumes downward direction as positive. This equation forms the starting point for both the dynamics of the fall and the derivation of terminal velocity.

此处 g 为重力加速度。以向下为正方向。该方程既是描述下落动力学的基础，也是推导终端速度的出发点。

3. Terminal Velocity Condition | 终端速度条件

As the parachute continues to accelerate, the drag force eventually becomes equal in magnitude to the weight. At this instant, the net force reduces to zero and the system stops accelerating. It then continues to descend at a constant speed known as the terminal velocity v_t.

随着降落伞持续加速，阻力最终会与重力大小相等。此时净力为零，系统停止加速，并以恒定速度下落，该速度称为终端速度 v_t。

mg = ½ ρ v_t² A C_d

This equilibrium condition is the key to determining C_d experimentally. By measuring the terminal velocity v_t, the total mass m, the fluid density ρ, and the reference area A, the drag coefficient can be isolated. The equation also reveals that for a given parachute, a higher mass leads to a higher terminal velocity, which is consistent with everyday observation.

这一平衡条件是实验测定 C_d 的关键。通过测量终端速度 v_t、总质量 m、流体密度 ρ 和参考面积 A，即可将阻力系数分离出来。该方程还表明，对于给定的降落伞，质量越大，终端速度越高，这与日常观察相符。

4. Deriving an Expression for Drag Coefficient | 推导阻力系数表达式

Rearranging the terminal velocity equation gives the direct formula for the drag coefficient of a round parachute:

整理终端速度方程，可直接得出圆形降落伞阻力系数的表达式：

C_d = 2mg / (ρ v_t² A)

Substituting A = π D² / 4 yields an expression in terms of the canopy diameter D:

代入 A = π D² / 4，可得到用伞衣直径 D 表达的式子：

C_d = 8mg / (π ρ v_t² D²)

It is crucial to recall that this derivation assumes the parachute has already reached terminal velocity, the air density ρ is constant (the fall is at low altitude), and the area A remains constant (the canopy is fully inflated). In practice, these assumptions are reasonably valid for drops from modest heights in still air.

必须记住，这一推导假设降落伞已到达终端速度、空气密度 ρ 恒定（在低空下落）且面积 A 保持不变（伞衣完全张开）。在实际中，对于在静止空气中从中等高度下落的实验，这些假设是合理成立的。

The equation above provides a straightforward route to calculate C_d once v_t is measured. Notice that the result is independent of the fall distance, which simplifies both data collection and error propagation.

上述方程提供了在测得 v_t 后直接计算 C_d 的简明途径。注意，其结果与下落距离无关，这简化了数据收集和误差传递。

5. Experimental Setup for a Round Parachute | 圆形降落伞实验装置

A typical IB Physics experiment to determine the drag coefficient involves dropping a model round parachute from a known height and recording its motion. The parachute can be constructed from lightweight fabric or plastic sheet, with strings attached to a small mass hanger carrying slotted masses. A metre rule or video tracking software (such as Tracker) can be used to measure the descent.

一个典型的 IB 物理实验使用模型圆形降落伞从已知高度释放，并记录其运动。降落伞可用轻质布料或塑料薄膜制作，系绳连接至一个加载槽码的小质量挂钩。下落过程可以用米尺或视频追踪软件（如 Tracker）测量。

Key equipment: A round parachute of measured diameter D, a mass set, a high‑speed camera or smartphone, a reference scale in the background, and a thermometer/barometer to determine air density ρ.
关键器材：已知直径 D 的圆形降落伞、砝码组、高速相机或智能手机、背景参考标尺，以及用于确定空气密度 ρ 的温度计/气压计。
Procedure overview: The parachute is released from a height sufficient to reach terminal velocity. The central portion of the fall, where velocity becomes constant, is analysed to extract v_t.
步骤概览：从足以达到终端速度的高度释放降落伞。分析下落的中段（速度恒定的部分），提取 v_t。

If video tracking is used, the position data can be plotted and the slope of the linear region of a position–time graph gives the terminal velocity. Alternatively, an ultrasonic motion sensor can capture real‑time displacement.

若使用视频追踪，可将位置数据绘图，位置–时间图线线性段的斜率即为终端速度。也可使用超声波运动传感器实时采集位移。

6. Data Collection and Key Measurements | 数据收集与关键测量

To compute C_d reliably, you need to measure five quantities: total mass m, canopy diameter D, air density ρ, and terminal velocity v_t. Each requires careful technique to minimise uncertainty.

为可靠地计算 C_d，需要测量五个量：总质量 m、伞衣直径 D、空气密度 ρ 和终端速度 v_t。每一项都需要仔细操作以减小不确定度。

Quantity	How to measure	Typical unit
Mass m	Electronic balance (include parachute, strings, payload)	kg
Diameter D	Measure the flattened canopy diameter; average several orientations	m
Air density ρ	Use ρ = pM/(RT) from pressure p and temperature T, or assume ρ ≈ 1.2 kg m⁻³ at 20°C and 101 kPa	kg m⁻³
Terminal velocity v_t	Video analysis: plot vertical position vs time, find slope of linear segment	m s⁻¹

For greater accuracy, the drop should be filmed against a plain background with good lighting. A calibration stick of known length placed vertically in the camera’s plane of motion converts pixel displacements into real distances.

为获得更高精度，应在简易背景下良好光照中拍摄下落过程。在相机运动平面内放置已知长度的竖直标定杆，可将像素位移转换为实际距离。

7. Sample Calculation and Data Analysis | 示例计算与数据分析

Consider a round parachute with diameter D = 0.60 m, total mass m = 0.120 kg. The air density during the experiment is ρ = 1.18 kg m⁻³. Video tracking yields a terminal velocity v_t = 1.65 m s⁻¹. We first compute the reference area:

考虑一个直径 D = 0.60 m 的圆形降落伞，总质量 m = 0.120 kg。实验时空气密度 ρ = 1.18 kg m⁻³。视频追踪得出终端速度 v_t = 1.65 m s⁻¹。首先计算参考面积：

A = π (0.60 m)² / 4 = 0.2827 m²

Insert the values into the C_d equation:

将数值代入 C_d 方程：

C_d = (2 × 0.120 kg × 9.81 m s⁻²) / (1.18 kg m⁻³ × (1.65 m s⁻¹)² × 0.2827 m²)

C_d ≈ 2.354 / 0.908 ≈ 2.59

A value of 2.59 appears rather high for a round parachute. This suggests either the area may be smaller than assumed due to incomplete inflation, or the canopy porosity is low, or the parachute is heavily loaded. In IA investigations, students must discuss the physical reasons behind the computed value and compare with published data.

对于圆形降落伞，2.59 明显偏高。这可能意味着伞衣未能完全张开而导致实际面积小于假设值，或者伞衣透气性很低，又或者载荷很重。在 IA 探究中，学生必须讨论计算值背后的物理原因，并与已发表数据进行比较。

Multiple trials with different masses can produce a set of C_d values. Plotting 2mg/(ρ A) versus v_t² should yield a straight line through the origin with slope equal to C_d, providing a graphical verification of the relationship.

使用不同质量进行多次试验可得到一组 C_d 值。以 2mg/(ρ A) 对 v_t² 作图应得到一条通过原点的直线，斜率即为 C_d，从而图示验证这一关系。

8. Evaluating Uncertainties and Suggestions | 不确定性评估与改进

Every measurement carries an uncertainty that propagates into the final C_d. The fractional uncertainty in C_d can be estimated by combining the relative uncertainties of m, ρ, v_t, and D. Since v_t appears as a squared term, its contribution is doubled.

每个测量值都带有不确定度，这些不确定度会传递到最终的 C_d。C_d 的相对不确定度可通过联合 m、ρ、v_t 和 D 的相对不确定度来估算。由于 v_t 以平方形式出现，其贡献将加倍。

ΔC_d/C_d ≈ Δm/m + Δρ/ρ + 2 (Δv_t/v_t) + 2 (ΔD/D)

Common sources of uncertainty include the parallax error when reading the scale, fluctuations in the parachute’s orientation causing variations in projected area, and the assumption that the velocity is perfectly constant during the analysed interval. Using higher frame‑rate recording and ensuring the drop occurs in still air can minimize these effects.

常见的不确定度来源包括读取标尺时的视差、降落伞朝向波动导致投影面积变化，以及假设在分析区间内速度完全恒定。使用更高帧率的拍摄并确保在静止空气中下落，可以尽量减少这些影响。

Repeated trials (at least five) for each mass setting allow calculation of mean v_t and standard deviation, improving the reliability of the final C_d. Students should also discuss systematic errors, such as the added mass of the strings or the possible drag on the payload itself.

对每个质量设置进行多次重复试验（至少五次），可计算平均 v_t 和标准差，提高最终 C_d 的可靠性。学生还应讨论系统误差，例如系绳的附加质量或载荷本身可能受到的阻力。

9. Application and Typical C_d Values | 应用与典型阻力系数值

In real parachute design, the drag coefficient is a critical performance parameter. Round parachutes typically have C_d between 0.75 and 1.2, while more advanced ram‑air parachutes can achieve C_d over 2.0 due to their wing‑like shape. The experimental values found in a school laboratory may deviate from these ranges because of scale effects and imperfect canopy inflation.

在实际降落伞设计中，阻力系数是一项关键性能参数。圆形降落伞的 C_d 通常在 0.75 至 1.2 之间，而更先进的翼型降落伞由于其类似机翼的形状，C_d 可超过 2.0。学校实验室测得的实验值可能因尺度效应和伞衣充气不完全而偏离这些范围。

The derived formula C_d = 2mg/(ρ v_t² A) demonstrates a direct proportionality between mass and the square of terminal velocity when other factors are constant. This is why heavy cargo parachutes descend faster than lightly loaded ones of the same size. Understanding this relationship helps engineers to size parachutes for specific landing speeds.

推导出的公式 C_d = 2mg/(ρ v_t² A) 表明，在其他因素不变时，质量与终端速度的平方成正比。这就是为什么相同尺寸的降落伞在承载较重的货物时下降更快。理解这一关系有助于工程师为特定的着陆速度来设计降落伞尺寸。

Extending the investigation, students can explore how C_d varies with Reynolds number by changing the parachute size or the fluid medium (e.g. water). Such extensions connect the IA to the broader topic of fluid dynamics and provide rich opportunities for evaluation.

扩展探究时，学生可通过改变降落伞尺寸或流体介质（例如水）来研究 C_d 如何随雷诺数变化。这样的扩展将 IA 与更广泛的流体动力学主题联系起来，并为评估提供了丰富机会。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Determining the Drag Coefficient of a Round Parachute: Formula Derivation | 测定圆形降落伞的阻力系数：公式推导