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Surface Area to Volume Ratio

November 24, 2012

Years ago, I read a book that changed the way that I thought about physics: “Cat’s Paws and Catapults”(http://www.powells.com/biblio/61-0393319903-0). This book, among many other things, introduced me to the concept of surface-area to volume ratio (SA:V), and its ramifications in terms of flying creatures (which is, inherently, a problem of fluid mechanics). In essence, it pointed out that when a geometric shapes (a sphere, cube, other) increases in its size, it’s surface area grows at a slower rate than its volume (and mass if it is of the same density). This means that among two geometric shapes of identical proportion, but different absolute size, the larger will suffer less wind resistance for its relative size, than the smaller shape (see Figure 1). In the natural world, as animals increase in size, the same phenomenon tends to occur.

20121122-144950.jpg

Image 1: Surface are to volume ratios of two identically proportioned rectangular prisms of different unit size.

What does this mean for swimmers? It might mean that larger swimmers will likely have a lower SA:V than smaller ones, which would mean that their amount of drag per unit of mass wouold be lower than that of a smaller swimmer. The difference, in terms of magnitude could be substantial, as we can see from Figure 1, the 6′ tall rectangle is 10% taller than the 5’4″ tall rectangle, but it’s SA:V is approximately 10% lower. Also, if they have the same body density, while the larger swimmers mass is 37% higher, their surface area only increases by 23%. A quick back-of-the-envelope calculation would suggest that only a 24% increase in power generation capacity would be needed to produce faster swimming speeds from that extra 37% of weight – not to mention that advantages of longer limbs, bigger hands, and bigger feet. Putting this into context, table 1 compares USA Swimming 18&under national qualification times with world record times in the mens and womens 1500 meters, showing only an 10% increase in velocity to go from just making Junior nationals to setting a world record for the men, and only ~9% to make the same jump for women.

Table 1: USA Junior National qualifying times, velocities, and world record time and velocity for the 1500 meter freestyle.

Gender USA Jr Nat JN Vel (m/s) Word Record WR Vel (m/s) Velocity Inc.
Male 15:59 1.56 14:31 1.72 10.22%
Female 16:06 1.46 15:42 1.59 8.92%

This is of course, speculative, and not backed by specific quantitative evidence of swimmer SA:V and power production values – actual SA:V ratios of successful competitive swimmers have never, to my knowledge been recorded. However, if only looking at the large differences in SA:V that may occur in relatively common size changes, it suggests that consideration of the effects of such things as SA:V should be made when holding up our fastest swimmers as models of swimming efficiency.

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One Comment
  1. KIC permalink

    Thought provoking as always, Rob. My takeaway is that adjusting for SA:V, I kicked butt on Friday! Too bad race results aren’t adjusted that way. 😉

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