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-   -   Air resistance, dimensional analysis confusion (http://www.physicsforums.com/showthread.php?t=657247)

pjordan Dec6-12 01:46 PM

air resistance, dimensional analysis confusion
 
Hi. Consider the basic eq for a falling body with air resistance

dv/dt=g-kv/m

I dont 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 Dec6-12 01:58 PM

Re: air resistance, dimensional analysis confusion
 
Quote:

Quote by pjordan (Post 4186853)
Hi. Consider the basic eq for a falling body with air resistance

dv/dt=g-kv/m

I dont 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 Dec6-12 02:12 PM

Re: air resistance, dimensional analysis confusion
 
good point, I had assumed k to be unit-less, but its units are kg/sec (http://oregonstate.edu/instruct/mth2...02/resist.html)

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

cjl Dec6-12 03:01 PM

Re: air resistance, dimensional analysis confusion
 
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 Dec6-12 03:21 PM

Re: air resistance, dimensional analysis confusion
 
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 Dec6-12 03:55 PM

Re: air resistance, dimensional analysis confusion
 
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 Dec6-12 06:22 PM

Re: air resistance, dimensional analysis confusion
 
Quote:

Quote by pjordan (Post 4186958)
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 Dec6-12 06:35 PM

Re: air resistance, dimensional analysis confusion
 
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.


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