I agree with Rod Cross's first two comments^{1} on my “Decluttering our thinking with the life-changing magic of twiddle,”^{2} including the short time window within which the two balls must be dropped (a point also noted by Feinberg^{3}). In Cross's third comment, he states that using the proposed dimensionless ratio

to estimate the height difference is physically not even “approximately right,” as it neglects air's fluid nature and thereby incorrectly assumes that the air in front of the ball is accelerated to the ball's speed. Cross states, with some justification, that I did not elaborate. In support of his argument, he provides a momentum analysis.

In this response, I follow Cross in considering momentum but show that the *R*_{drag} approach is fine. Then, by approximating early and often to avoid power-series expansions, I refine the approach to explain the correction factors that convert the unvarnished *R*_{drag} estimate of a 10-m height difference into the actual height difference of 80 cm. (The following analysis assumes that drag modifies free fall only slightly.)

So first, by considering momentum, I twiddle estimate the drag force *F* on a ball with cross-sectional area *A* moving at speed *v* through a fluid of density *ρ*—assuming that the ball accelerates all the swept-out air to a speed ∼*v* and leaves the rest of the air undisturbed. Over a time *t*, the ball travels a distance *vt*, sweeping out a volume *vtA* and a mass of fluid $mfl\u223c\rho vtA$. This fluid gains, and the ball loses, a momentum $\Delta P\u223cmflv\u223c\rho v2tA$. The momentum-loss *rate* $\Delta P/t$ is the drag force *F*, and so, $F\u223c\rho v2A.$ Thus, even neglecting the fluid nature of the air, the assumption that the ball accelerates the swept-out fluid to the ball's speed gives a drag force that depends correctly on *v*, *ρ*, and *A*.

But air is indeed a fluid and flows around the ball. The result is that some of the air behind the ball, which the ball had just accelerated to a speed ∼*v*, pushes the ball on its backside and hurries it along—that is, some of the air behind the ball returns momentum to the ball. All this complicated physics is captured by the drag coefficient *c*_{d}. Including *c*_{d} and the conventional factor of 1/2 that rides with it makes the drag force exact:

Thus, for the only moderately streamlined shape of a sphere ($cd\u22480.5$), the returned momentum contributes a physical factor of 1/4. (The twiddle estimate of *F* that ignored the drag coefficient implicitly used *c*_{d} = 2, which is reasonable for bluff objects like a flat plate oriented with its large surface perpendicular to the flow.)

This drag force reduces the ball's momentum and speed and therefore its fall distance (imagine the ball racing against a drag-free, freely falling ball, with the race ending when the freely falling ball lands). The drag-induced fractional momentum loss in an unvarnished twiddle analysis is

where $tff$ is the free-fall time and *v* is the impact speed. The ball's fractional momentum loss is also its fractional speed loss Δ*v*/*v*; furthermore, the fall time is $tff\u223ch/v$. With these substitutions, the fractional loss in speed is $\u223cRdrag$:

Thus, the returned momentum contributes to Δ*P*/*P* and Δ*v*/*v* a physical factor of *c*_{d}/2—the same factor as it contributed to *F*.

The changing fall speed contributes several further factors that I call geometric. The first geometric factor is hidden in the fall time *t*_{ff}, which is actually 2*h*/*v* (because the average speed over the fall is *v*/2). So, correcting the fall time contributes a factor of 2 to Δ*v*/*v*.

The second geometric factor arises because *F* varies. Thus, the momentum change $\Delta P\u223cFtff$, which is the numerator of Eq. (3), should really be $\u27e8Ft\u27e9tff$, where $\u27e8\xb7\u27e9$ denotes the time average over the journey and $\u27e8Ft\u27e9$ is the time average of the time-varying drag force *F _{t}*. Because $Ft\u221dvt2$, where

*v*is the fall speed after a time

_{t}*t*and $vt\u221dt$ (in the small-drag approximation), $Ft\u221dt2$. Thus, $\u27e8Ft\u27e9=F/3$, where

*F*is the maximum drag force, at the end of the race. So, the second geometric factor, and the final one for Δ

*v*/

*v*, is 1/3. Including the physical factor of

*c*

_{d}/2 and the first geometric factor of 2 gives

The fractional speed loss is now correct.^{4}

Converting it to the fractional height loss Δ*h*/*h* contributes the third and final geometric factor. The height loss Δ*h* is

where Δ*v _{t}* is the drag-induced speed loss due after the ball has fallen for a time

*t*. Converting the fractional speed loss Δ

*v*/

*v*in Eq. (5) to the absolute speed loss Δ

*v*gives

_{t}where $mtair$ is the mass of air swept out after a time *t*. In the small-drag approximation, $vt\u221dt$. Furthermore, $mtair$ is proportional to *y _{t}*, the fall distance after a time

*t*; this distance is itself proportional to

*t*

^{2}(in the small-drag approximation). Thus, $\Delta vt\u221dt3$, its time average is 1/4th of its final value Δ

*v*(at

*t*=

*t*

_{ff}, when the drag-free ball lands).

The fractional height loss picks up this factor of 1/4:

Because *h*/*t*_{ff} is the average speed *v*/2 and it is in the denominator, a factor of 2 appears; thus

In summary, the complicated physics of fluids contributes a factor of *c*_{d}/2 ≈ 1/4; the geometric factors combine to contribute a factor of 1/3. The overall result is a factor of ≈1/12. It converts the unvarnished twiddle estimate of 10 m into the actual height difference of 80 cm.

A twiddle analysis omits geometric factors. However, even before I calculated these factors, they seemed unlikely to conspire enough to save Galileo's 5-cm claim, whose rescue requires a factor of 1/200. Alas, all these factors physical and geometrical could not put Galileo's estimate together again.

As a final, philosophical point, the line separating physical from geometric factors is more porous than I have pretended because these geometric factors arise from a physical effect (the varying fall speed). The porosity is probably inevitable. Here, Cross and I, and our readers, can all perhaps agree with Galileo: The book of Nature is indeed “written in the language of mathematics.”^{5}

## References

Cross's calculations for the ball's loss in speed are consistent with a factor of 1/4 rather than the 1/3 given in Eq. (5). The discrepancy lies in the differing boundary conditions. Galileo, like this analysis, compared two balls falling until the first ball hits the ground—thus, falling for the same time. Cross analyzed two balls falling for the same distance. Thus, Cross's scenario gives the higher-drag ball a little extra time in which to accelerate. By an analysis similar to the one given here, the fractional speed gained can be shown to be (1/12)*c*_{d} *R*_{drag}, giving a net fractional loss of (1/4)*c*_{d} *R*_{drag}—in agreement with Cross's calculations.

*The Philosophy of the Sixteenth and Seventeenth Centuries*, edited by