How fast does a speedster like The Flash or Dash Parr from The Incredibles have to go to run on water? As we saw from other water-walkers like the basilisk lizard and the western and Clark’s grebes, any large creature wanting to run on water needs to generate the necessary vertical impulse by hitting the water hard, pushing off against the cavity that creates, and pulling their foot up before the cavity collapses around it.
Using basilisk lizards as our guide, we can build a simplified hydrodynamic model (following Glasheen and McMahon and Minetti et al.) to describe this process and predict a speedster’s necessary speed. If we assume our runner removes their foot before the cavity collapses, we have a relatively simple relation to satisfy, namely: the vertical impulse from the slap combined with the vertical impulse from the push, or stroke, must equal or exceed the impulse from the runner’s weight:
(Impulse from slap) + (Impulse from stroke) >= Impulse from runner’s weight
The impulse from the runner’s weight is relatively straightforward. It depends on the runner’s mass, gravity, and the time it takes the runner to complete a step. The other two terms are a bit more complicated and require some approximations. One is that we’ll treat the runner’s foot like a circular disk – this makes it easier to figure out the drag while the runner pushes against the water. Ultimately, the model requires five variables (four, if we assume that we’re on Earth):
– the runner’s mass
– the area of the runner’s foot
– the depth the runner’s foot reaches underwater
– the time it takes the runner to take one step
– the acceleration due to gravity
I will spare you the math, but I’ve created an online calculator (now with English or metric versions) with the model, so you can follow along with my math or play around with your own numbers.
Click through to see how fast a human has to go to run on water.
So how fast would The Flash have to run? Barry Allen is grown man, roughly 75 kg in mass, with a foot area of about 314 cm^2. We can assume that he pushes his leg about 0.15 m into the water with each step. The best human sprinters run with a step time of 0.2 – 0.26 seconds, but Barry’s a metahuman, so we’ll give him the benefit of the doubt and say that he can take a step in 0.15 seconds. (Let’s be honest, he’s probably capable of faster than that!)
To keep from sinking, The Flash would have to strike his feet against the water at about 37 m/s. It’s a little tough to say exactly how that would translate into forward speed. Both basilisks and grebes strike the water at a higher speed than their forward velocity. Since their feet are parallel to the surface when they strike, the slap phase only gives them vertical impulse. Their forward velocity comes from the stroke phase where they can push off against the water. This suggests that a runner who generates a lot of their vertical impulse during the slap phase will be able to get more forward velocity out of the stroke phase because they can afford to push forward off the cavity instead of mostly up. That’s consistent with what we observe in the lizards and grebes; the grebe gets more of its impulse from the slap and its forward velocity is a larger percentage of its foot impact velocity compared to the basilisk.
Using the lizards and birds as our guide, we can estimate that The Flash, who gets about 45% of his necessary vertical impulse from slapping, will have a forward velocity of about 27 m/s or 98 kph. That’s a lot faster than any human has ever run – Usain Bolt has managed about 44.7 kph – but it’s not that fast. In CW’s The Flash TV show, his team estimates that he must run 650 miles per hour, or 1050 kph, to run on water. That is way faster than necessary!
How about Dash Parr, though? Dash is about 10 years old, so he’s a lot smaller than Barry. That means he has less mass to keep afloat (about 32 kg), but it also means that he has smaller feet (154 cm^2) and shorter legs (0.1 m foot depth). For the same stride rate, that means that Dash has to hit the water at 47 m/s, about 25% faster than Barry. It also means that Dash gets a tad more oomph from his slap (~46%) and runs across the water at 128 kph, about 30% faster than Barry has to go.
That’s totally doable for a superhero, but what about us regular humans? Sadly, our large mass and small feet won’t let us run on water like The Flash or Dash, but there are ways to bend the rules. One is to reduce gravity – this was the subject of an Ig Nobel prize-winning study by Minetti et al. The researchers put fins on volunteers, suspended them from a harness to reduce their effective weight, and got them to run in place in a pool. They found that fin-augmented humans could run on water in gravity about 20% of Earth’s.
Another technique is to increase a runner’s effective foot area without making them bother to lift the foot out of the water. In essence, a human can run on water across over-sized lily pads. In their study, Lothman and Ruina accomplished this with plywood pads laid out in a pool. The pads were buoyant enough to stay afloat at the water surface but would sink if a person stood still on them. But by running quickly from one to the next, their test subject was able to successfully run across water.
So, there you have it: speedsters can run on water without breaking autobahn speed limits and the rest of us can cheat. Be sure to check out the online calculator and play with the model yourself. And join me tomorrow for my special webcast with Professor Tadd Truscott and Randy Hurd, who inspired this foray into water-walking!
ETA: I updated the calculator so that there is now an English unit version as well as a metric unit version.
Research credits:
– Glasheen and McMahon, “A hydrodynamic model of locomotion in the basilisk lizard”, 1996.
– Minetti et al., “Humans running in place on water at simulated reduced gravity,” 2012.
– Lothman and Ruina, “Humans can run on water using big instantly-changable shoes,” 2012.
(Image credits: Disney, CW/DC Comics, Lothman and Ruina, source, and The Splash Lab)