Air resistance, dimensional analysis confusion

In summary, air resistance is a force that opposes the motion of a falling body, and it is proportional to the velocity of the body. The force equation is F=ma, but air resistance is independent of velocity. Introductory material uses v not because it is correct, but because it makes the differential equation a lot easier.
  • #1
pjordan
3
0
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!
 
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  • #2
pjordan said:
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?
 
  • #4
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.
 
  • #5
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
 
  • #6
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.
 
  • #7
pjordan said:
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).
 
  • #8
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.
 

FAQ: Air resistance, dimensional analysis confusion

What is air resistance?

Air resistance is a force that acts on objects as they move through the air. It is caused by the collision of air molecules with the surface of the object, creating a drag force that opposes the motion of the object.

How does air resistance affect objects?

Air resistance can slow down the motion of an object, making it harder for the object to move through the air. This can also cause the object to change direction or experience turbulence.

What factors affect air resistance?

The factors that affect air resistance include the surface area, shape, and speed of the object, as well as the density and viscosity of the air. Objects with larger surface area or irregular shapes will experience more air resistance, while objects with higher speeds will experience greater drag force.

What is dimensional analysis confusion?

Dimensional analysis confusion is a common mistake made by scientists and students when performing calculations involving units. It occurs when the units of measurement are not properly converted or cancelled out, leading to incorrect results.

How can dimensional analysis confusion be avoided?

To avoid dimensional analysis confusion, it is important to keep track of the units throughout the calculation and make sure they are properly converted and cancelled out. It can also be helpful to use dimensional analysis as a problem-solving tool, verifying that the units of the final answer are correct. Practice and attention to detail can also help reduce errors related to dimensional analysis.

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