## Fun with Dimensional Analysis

This post is a hodge-podge of interesting applications of dimensional analysis.  It is motivated by some recent work with students simulating and analyzing projectile motion.  Consider a ball (a baseball, in our running example) falling under gravity.  Then throw the ball in a flat-earth gravity field.  Then throw it, or drop it from a tall building, considering not just gravity but also air resistance, terminal velocity, etc.  (By the way, this is all very easy to experiment with using VPython, which I mentioned a few weeks ago and am now using with some success.)

This progression of complexity leads to the problem of “getting the units right” in the relevant equations.  For example, the terminal velocity of a baseball may be estimated by equating the opposing forces of drag due to air resistance and gravity:

$F_D = \frac{1}{2}\rho v^2 C_D A = m g = W$

where $\rho$ is the air density (1.229 kg/m^3 at sea level on a standard day), $v$ is the terminal velocity (m/s), $C_D$ is the dimensionless coefficient of drag (approximately 0.3 for a baseball at these speeds), $A$ is the reference area, or cross sectional area of the baseball (0.00427 m^2), $m$ is the mass of the baseball (0.145 kg), and $g$ is the acceleration due to gravity (9.8 m/s^2).

It may not be obvious to the student, or perhaps to the interested reader, that the “units” associated with each variable multiply and cancel to yield kg m/s^2 on both sides.  Or if it is obvious, then perhaps more interesting are the situations where (1) the equation is not known ahead of time, but can in fact be derived (up to a point) simply by considering the dimensions of the quantities involved; or (2) dimensional analysis of a problem can simplify experimental design by reducing the number of knobs to turn, so to speak.

As usual, my posts are usually just jumping-off points to better writing than you will find here.  I recommend reading Ain A. Sonin’s paper, which has some cool examples of this sort of thing, as well as a very clear and well-written motivation for and explanation of the related Buckingham’s $\pi$-Theorem, which puts dimensional analysis on a more rigorous linear algebraic footing.

I can think of a few other neat applications of dimensional analysis.  An episode of Mythbusters from a year or so ago involved trying to duplicate the bus jump stunt in the movie Speed.  The guys made a 1:12 scale model of the bus and bridge… but how should they scale the 50 mph speed of the full-size bus?  (Hint: it’s not just 1/12th of 50 mph.)  This was a nice puzzle, and I was pleased to see that they got it right… or at least, they used the same speed that I came up with.

Finally, I recall first seeing the following problem in an essay in my (very old) edition of Halliday and Resnick.  Automobile gas mileage is usually expressed in units like miles per gallon.  But both miles and gallons are derived length units; we can convert miles to, say, meters, and gallons to meters^3, yielding a gas mileage in units of 1/meters^2, or the reciprocal of an area.  For example, 25 miles per gallon corresponds to (1 divided by) 0.094 mm^2.

This “reciprocal area” has a nice physical interpretation.  Suppose that instead of a large fuel tank that travels with the car, we instead lay a very thin cylindrical tube along the length of the road.  We fill this thin tube with fuel, and the car “scoops” fuel from the tube as it travels.  The car’s gas mileage is then the required area of the cross section of the tube.  That is, lower gas mileage corresponds to a larger area, and vice versa.

(So as not to leave the falling baseball example behind, we can solve the equation above for velocity and, plugging in values, find that the terminal velocity of a baseball is approximately 95 miles per hour.  A follow-up question for students: how, then, have pitchers managed to throw faster than this, at speeds exceeding 100 mph?)

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