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.

如何計算物體在空中緩慢下落時的空氣阻力

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  ,

註：實際上，計算並不是那麼簡單，還有許多其他因素也在起作用。然而，為了這個例子，我們假設影響花粉粒下落的唯一因素是重力和空氣阻力，並且空氣阻力也假設與花粉粒的速度成正比。

如何計算物體在空中快速下落時的空氣阻力

物體在空中落得更快（例如，跳傘者）會引起大量的湍流，因此它們所經歷的空氣阻力要大得多。對於這些物體，空氣阻力與物體相對於空氣的速度的平方成正比。這些情況下的空氣阻力由下式給出：

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：

落在空中的高爾夫球的阻力繫數為0.26。假設它的有效橫截面積為1.4×10-3 m2，當球以20 m s-1的速度移動時，找出球上的空氣阻力。

We have 

Then,