Supercapes

In preparation to a Panel at CONvergence, I’m going to explore some of the oddities in the superhero dress code. More specifically, I intend to explore the physics needed for them to do what they do. My first exploration will be into the drag forces on capes.

The cape is a rather popular accessory for the comic book hero. Their aesthetic appeal clearly outweighs the tripping and strangulation hazards they impose, but do they impose other, more subtle risks? What sort of dangers do they impart to their wearer when confronted with that most notorious of villains, the stiff wind?

The drag force created by a strong breeze shouldn’t be underestimated. We’ve all had trouble walking straight on a windy day, how hard would it be if you were wearing a nice long flowing cape?

To calculate a drag force,  we can follow the following formula

Where F represents the drag force
ρ represents the density of the fluid, in this case air
v represents the velocity of the fluid
C Represents the Drag Coefficient and
A represents the surface area of the object

I’m going to assume that for dramatic purposes, most heroes will, at some point in their career, gaze down upon a city from the top of a skyscraper on a windy day. For simplicity, we’ll use the empire state building as our frame of reference.  The roof is at 381m above the ground and manhattan is pretty close to sea level, so we’ll use that for air density.  On a windy summer evening, that would yield a density of 1.14 kg/m^3. A high wind would be anything much over 65 km/h (40 mph) with a good gust speed for a tall building being around 80 km/h (50 mph).

Surface area is a bit harder since capes are so varied. I’m going to go based on a tall male hero wearing a cape that goes from his neck to a few inches off of the ground. That’s about 1.5m (5 feet) and they tend to be a truncated sector of a circle with roughly a 60 degree arc. That gives us a surface area of around 2m^2 (20 sq.ft).

Now for the hard part, the C factor. The coefficient of drag is different for each material and shape. The only way to determine this  is through experimentation. However, we can model the cape as a flag. I have found a few estimates for the C of a flag ranging from 0.05 to 0.15. I’m going to split the difference, which is terrible practice, but I don’t have a wind tunnel and 0.1 is easy to work with.

Running those numbers, we get a drag force of 730N or 160 lbf.  Now 160 pounds is a force that a normal person can fight and a strong fabric can handle, but it’s not easy. You have to lean well into that and have good traction. My conclusion is that it is unwise to find yourself balancing on a highrise in a cape.

Another question that follows directly from Batman’s brooding in the breeze, is Superman’s supercape. Superman wears a cape and travels at ludicrous speeds. He can, apparently, exceed the speed of light, but he only does that in outer space, to my knowledge, and drag isn’t an issue there. Down in the atmosphere, we’ll have to assume that he stays at speeds that won’t ignite our atmosphere or vaporize our oceans. I’m going so say, for argument’s sake, that he caps out around Mach 3.  Mach 3 is around 3600 km/h depending on elevation.

If you examine the equation above, you will notice that drag force increases by the square of velocity. At Mach 3, superman has to exert over three hundred thousand pounds on his cape just to drag it along with him. For reference, that’s roughly equivalent to holding up a blue whale.  That makes the material of his cape well outside anything you can get at a fabric store. Purely on tensile strength, it could be made of Kevlar, however that wouldn’t stand up to the rigorous flapping that would happen at even low speeds. So superman’s cape must be made of an incredible alien material with high tensile strength, coupled with low friction and be extremely flexible even at very high speeds.

Featured image from DC Comics