Air Resistance: Dimensional Analysis We know that as an object passes through th

Question

Air Resistance: Dimensional Analysis
We know that as an object passes through the air, the air exerts a resistive force on it. We can get some
idea as to the form of this force by dimensional analysis. Suppose we have a spherical object of radius R
and mass m. What might the force plausibly depend on?
• It might depend on the properties of the object. The only ones that seem relevant are m and R.
• It might depend upon the object's coordinate and its derivatives: x, v, a, . . .
• It might depend on the properties of the air, such as the density, .
a. Explain why it is plausible that the force the air exerts on a sphere depends on R but implausible that
it depends on m.
b. Explain why it is plausible that the force the air exerts depends on the objects speed through it, v,
but not on its position, x, or acceleration, a.
c. Using dimensional analysis, construct a plausible form for the force that air exerts on a spherical body
moving through it.

Explanation / Answer

a.) collisions between air molecules and the surface of the object are what produces the drag force. The bigger the R; the bigger the surface area and the more collisions, so it makes sense that the force the air exerts depends on R. m doesn't seem to be important, you could have a lead ball inside a styrofoam shell and it would have the same number of collisions with air molecules as a solid styrofoam sphere. The mass is all the way through, but the collisions are just on the surface, so it is implausible that the force depends on the mass. (you might think that the fact that a lead ball flies down and a styrofoam ball wafts down disproves this, but it actually supports it - if the drag force is the same but the force of gravity is less, the dense ball will fall faster) b) collisions between molecules are ellastic so we would expect something of the form (1/2)mv^2 which is kinetic energy. Air molecules coming in bounce off with the same velocity in the opposite direction. m is the mass of the air; not the mass of the sphere. c.) force is kg m/s^2 area of sphere has dimentsions of m^2 velocity squared has units of m^2/s^2 number of collisions and kinetic energy depends on density of the air; this has units of kg/m^3 all together we get (kg/m^3)*(m^2)*(m^2/s^2) = kg m / s^2 these are units of force, Newtons Force = constant * density of air * area of sphere * (velocity of sphere)^2

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