Terminal Velocity

终端速度

When an object which is falling under the influence of gravity or subject to some other constant driving force is subject to a resistance or drag force which increases with velocity, it will ultimately reach a maximum velocity where the drag force equals the driving force. This final, constant velocity of motion is called a "terminal velocity", a terminology made popular by skydivers. For objects moving through a fluid at low speeds so that turbulence is not a major factor, the terminal velocity is determined by viscous drag. The expression for the terminal velocity is of the form

Motion analysis

运动分析

Objects moving at high speeds through air encounter air drag proportional to the square of the velocity. This quadratic drag leads to a terminal velocity of the form

Examples

示例

当一个物体在重力作用下或受到某种其他恒定驱动力作用下，若其受到的阻力或阻力力随着速度增加而增加，则最终会达到一个最大速度，此时阻力力等于驱动力。这种最终的恒定速度称为‘终端速度’，这一术语由跳伞运动员推广开来。对于在流体中以低速运动的物体，湍流不是主要因素，终端速度由粘滞阻力决定。终端速度的表达形式为：物体以高速通过空气时，会遇到与速度平方成正比的空气阻力。这种二次方阻力导致终端速度的表达形式为：

Index

Fluid friction

索引流体摩擦

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Quadratic Velocity Dependence

二次速度依赖性

For large objects moving through air, the air resistance is approximately proportional to the square of the velocity. The form of the resistance is

对于通过空气运动的大型物体，空气阻力近似与速度的平方成正比。阻力的形式为

where ρ is the air density, A the crosssectional area, and C is a numerical drag coefficient. The drag coefficient C is 0.5 for a spherical object and can reach 2 for irregularly shaped objects according to Serway. An object falling through the air will reach a terminal velocity when the drag force is equal to the weight:

This gives a terminal velocity

这给出了终端速度

Examples

示例

Vertical trajectory with quadratic drag

二次阻力的垂直轨迹

Freefall velocity with quadratic drag

自由下落时的二次阻力速度

Freefall velocity with quadratic drag for given fall distance

考虑二次阻力的自由下落速度（给定下落距离）

Freefall velocity with quadratic drag as function of time

有二次阻力的自由落体速度随时间变化

Time to peak with quadratic drag

二次阻力下的峰值时间

Peak height with quadratic drag

峰值高度（二次阻力情况）

Upward velocity vs height

向上速度与高度

Fall velocity vs height

下落速度与高度

Impact velocity

冲击速度

其中ρ是空气密度，A是横截面积，C是一个数值阻力系数。根据Serway，球形物体的阻力系数C为0.5，不规则形状物体的C可达到2。一个物体在空气中下落时，当阻力力等于重量时，将达到终端速度：

Index

Fluid friction

Reference
Serwary & Beichner
Ch 6

索引流体摩擦参考 Serwary & Beichner 第6章

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Terminal Velocity Examples

终端速度示例

Falling object 下落物体	Mass 质量	Area 面积	Terminal velocity 终端速度
Skydiver 跳伞者	75 kg	0.7 m²	60 m/s	134 mi/hr 134英里每小时
Baseball (3.66cm radius) 棒球（半径3.66厘米）	145 gm 145 克	42 cm² 42平方厘米	33 m/s	74 mi/hr 74英里/小时
Golf ball (2.1 cm radius) 高尔夫球（半径2.1厘米）	46 gm 46 克	14 cm² 14平方厘米	32 m/s	72 mi/hr 72英里/小时
Hail stone (0.5 cm radius) 冰雹（半径0.5厘米）	.48 gm .48 克	.79 cm² .79 cm²	14 m/s	31 mi/hr 31英里/小时
Raindrop (0.2 cm radius) 雨滴（半径0.2厘米）	.034 gm	.13 cm² .13 平方厘米	9 m/s	20 mi/hr 20 英里/小时

Data from Serway, Physics for Scientists and Engineers, Table 6.1. A drag coefficient C=0.5 is assumed, falling through air.

Hailstone calculation

冰雹计算

Calculation for a sphere

球体的计算

数据来自Serway的《物理学：为科学家和工程师的物理学》表6.1。假设阻力系数C=0.5，物体在空气中下落。

Index

Fluid friction

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Hailstone Terminal Velocity

石块终端速度

Contributing to the danger of large hailstones is the fact that they fall faster than small ones. That is, the terminal velocity increases with the size of the hailstone. Assuming the hailstones to be spherical and using a drag coefficient of C = 0.5 gives the following :

促成大冰雹危险的因素之一是它们比小冰雹下落得更快。也就是说，终端速度随着冰雹大小的增加而增加。假设冰雹为球形，并使用阻力系数C = 0.5，可得到以下结果：

Radius (cm) 半径（厘米）	v (km/hr) v (km/hr)（专名或术语）	v(m/s)	v(mi/hr) v(mi/hr)（专名或术语）
.01	7	1.9	4.3
0.1	22	6.1	13.7
0.2	31	8.6	19.3
0.5	49	13.6	30.5
1.0	69.5	19.3	43.2
2.0	98.3	27.3	61
3.0	120	33.4	74.8
5.0	155	43.2	96.6
10.0	220	61	136