Physicopoeia

φυσικοποιΐα
THINGS I WISH I'D KNOWN WHEN I STARTED TEACHING PHYSICS

THE COMPLEXITIES OF AIR RESISTANCE

At school-level, air resistance regularly gets completely ignored in questions. However, some courses try to include it in some way, since it is clearly an issue that cannot always (or even often) be safely ignored. Although it may not be explicitly stated, some qualitative understanding of the drag equation is what is commonly expected:

FD = ½CDρAv2

The danger here is that it's very easy not to realise the limitations on the use of this equation and end up applying it to situations where it doesn't apply and getting confusing results. The above equation only applies to situations where the object is causing turbulence in the fluid it's moving through, so it's fine for many everyday objects moving fairly quickly through air, for example.

An interesting comparison can be made with an equation that also sometimes crops up at school level, namely Stokes' Law:

​FD = 6πηvr

This only applies to spherical objects (hence the inclusion of radius r) in situations where viscous drag is the dominant effect (hence the inclusion of viscosity η) and the flow is perfectly laminar. So, perfectly good for things like ball bearings falling through glycerine or oil - a common school experiment which Stokes' Law can be used to analyse - or oil droplets falling slowly through air, as in Millikan's oil-drop experiment. Not so good for a tennis ball flying through the air.

​The interesting comparison is that drag force is proportional to v² in the first equation, but in Stokes' Law it's only proportional to v which can seem odd. This is related to the transition from laminar to turbulent flow, also known as going from low to high Reynolds number. Drag force is proportional to v at low Reynolds Number but approaches v² proportionality at higher Reynolds numbers. This gradual transition from linear to quadratic is quite unusual in Physics but simply serves to illustrate how oddly friction forces behave.

A nice bit of extension knowledge that relates to this is that the drag coefficient (CD in the first equation) actually turns out to be dependent on Reynolds number. This is often not obvious if you simply look up a table of typical values of CD - they often just quote a unique value for a given shape, but there should be some small print nearby about the conditions under which that value was measured. If you think about it, a smooth sphere is not going to have the same drag coefficient a low speeds - where the airflow around it would be pretty laminar - as it would at very high speeds where the flow would become turbulent.