Whenever objects move relative to the air, the objects experience a resistive force which is in the opposite direction to the body’s velocity relative to the air. This resistive force is called air resistance. For instance, c***ider a falling sheet of paper. Relative to the air, the paper is moving downwards, and so there will be an upward resistive force on the paper.

如何计算物体在空中缓慢下落时的空气阻力

For objects which move slowly relative to the air (such as falling dust particles), the resistive force is directly proportional to the object’s velocity  relative to air. So, we can write:

The value of  depends on the shape and size of the body.

Let us c***ider what happens to a body as it falls from rest at a low speed through the air. Initially, the body’s velocity relative to the air is 0 and so there is no air resistance. As the body speeds up under the resultant downwards force, the air resistance also increases in proportion.

Eventually, the body reaches a speed where the body’s weight is exactly balanced by the air resistance. Here, the body has reached terminal velocity,  . The variation of velocity with time can be shown on a graph, as follows:

It can be shown that the terminal velocity for these cases is given by  .

例子：

Suppose a pollen grain with a mass of 3.8×10-14 kg is falling through air. If the value of the c***tant   4.0×10-11 kg s-1, find the terminal velocity.

Since  ,

注：实际上，计算并不是那么简单，还有许多其他因素也在起作用。然而，为了这个例子，我们假设影响花粉粒下落的唯一因素是重力和空气阻力，并且空气阻力也假设与花粉粒的速度成正比。

如何计算物体在空中快速下落时的空气阻力

物体在空中落得更快（例如，跳伞者）会引起大量的湍流，因此它们所经历的空气阻力要大得多。对于这些物体，空气阻力与物体相对于空气的速度的平方成正比。这些情况下的空气阻力由下式给出：

Here,  is drag coefficient,  is the density of air (typically about 1.2 kg m-3), and  is the effective cross section of the body perpendicular to the direction of the body’s velocity. Typically, takes values between 0.1 and 2.

The terminal velocity for objects moving fast in air can be given by  .

例子：

落在空中的高尔夫球的阻力系数为0.26。假设它的有效横截面积为1.4×10-3 m2，当球以20 m s-1的速度移动时，找出球上的空气阻力。

We have 

Then,