Air resistance, dimensional analysis confusion

[pjordan]

Hi. Consider the basic eq for a falling body with air resistance

dv/dt=g-kv/m

I don't understand air resistance as a force, since it seems irreconcilable to the force equation F=ma. How is a force a function of velocity? I am also not sure how this equation makes sense in terms of dimensional anaysis--the right side is m/s^2, the left m/s^2+(m/s)/kg. I am apparently the only one troubled by this, as extensive googling has yeilded nothing. Thanks!

 

[berkeman]

Hi. Consider the basic eq for a falling body with air resistance

dv/dt=g-kv/m

I don't understand air resistance as a force, since it seems irreconcilable to the force equation F=ma. How is a force a function of velocity? I am also not sure how this equation makes sense in terms of dimensional anaysis--the right side is m/s^2, the left m/s^2+(m/s)/kg. I am apparently the only one troubled by this, as extensive googling has yeilded nothing. Thanks!

Does k have units?

 

[pjordan]

good point, I had assumed k to be unit-less, but its units are kg/sec (http://oregonstate.edu/instruct/mth252h/Bogley/w02/resist.html)

so F=kv would have the same dimension as F=ma. thanks!

 

[cjl]

Another problem: your proportionality is wrong. Air resistance follows a v² proportionality, so in reality, it should be:

dv/dt = g - kv²/m, in which k = ρ/2*Cd*A, where ρ is the density of the fluid, Cd is the drag coefficient (unitless), and A is the reference area.

 

[pjordan]

generally it is given as proportional to v or v^2--the quadratic relationship is usually for larger objects. Most introductory material on diff eq use v. thanks

 

[K^2]

Precisely. Drag equation can be different under different conditions. Quadratic drag is more common in practical situations, but slow motion through viscous medium will often produce linear drag.

 

[cjl]

generally it is given as proportional to v or v^2--the quadratic relationship is usually for larger objects. Most introductory material on diff eq use v. thanks

Introductory material uses v not because it is correct, but because it makes the differential equation a lot easier. Even for small objects, air resistance tends to have a v² proportionality - the relatively low viscosity of air, and high velocity objects falling through air attain make the v² relationship correct for nearly all objects in air. A linear proportionality (implying viscous-dominated drag rather than inertial) tends to happen more commonly in other fluids, especially highly viscous ones (for example, dropping a marble through corn syrup).

 

[K^2]

I missed the bit about it being specific to drag in air. Yes, with air, you are unlikely to see linear drag outside of Millikan Oil Drop, or similar setup.