Relative Speed and Strength Calculator: Animals Scaled to Human Size (Running, Flying, Jumps, Falls)

Free relative speed and strength calculator: turn an animal's speed into body lengths per second and scale it to human size (a tiger beetle's 2.5 m/s is about 790 km/h for a person), for runners, birds, insects and swimmers. Also how much animals lift for their weight (dung beetles, ants), how high they jump (fleas, squirrels, cats) and why mice and cats survive falls, with the fair square-cube and Froude comparisons.

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Animal or thing

Speed

Body length

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The maths

body lengths per second \( \displaystyle n = \frac{v}{L}, \qquad v_{\text{scaled}} = n\,L_{\text{you}} \)

fair (Froude) speed \( \displaystyle \frac{v^2}{gL} = \frac{v_{\text{fair}}^2}{g L_{\text{you}}} \;\Rightarrow\; v_{\text{fair}} = v\sqrt{\frac{L_{\text{you}}}{L}} \)

take-off speed \( \displaystyle v_0 = \sqrt{2gh}\ \text{(up)}, \qquad v_0 = \sqrt{g\,d}\ \text{(long jump at 45°)}, \qquad a = \frac{v_0^2}{2s} \)

strength (square-cube law) \( \displaystyle \frac{F}{W} \propto \frac{L^2}{L^3} = \frac{1}{L}, \qquad x_{\text{fair}} = x\left(\frac{M}{M_{\text{you}}}\right)^{1/3} \)

terminal and impact speed \( \displaystyle v_t = \sqrt{\frac{2mg}{\rho\, C_d A}}, \qquad v(h) = v_t\sqrt{1 - e^{-2gh/v_t^2}} \)

How the relative speed, strength, jump and fall calculator works

Small animals look astonishingly fast and strong when you measure them in body lengths. A tiger beetle runs about 2.5 m/s, which is 125 of its own 2 cm body lengths every second. A person doing 125 body lengths a second would be going about 790 km/h (490 mph). A flea jumps about 90 times its body length; scaled up to a person, that is a jump of about 160 m.

Those headline numbers are fun but misleading, because physics does not scale in a straight line. This page shows the straight-line number and, next to it, what physics says.

Speed: the Froude number

For running and walking, engineers compare animals of different sizes with the Froude number v²/(gL). Two animals with the same Froude number move in a “dynamically similar” way (it is why a toddler has to trot to keep up with a walking adult). Matching it gives the fair speed v√(L_you/L): for the tiger beetle about 84 km/h, still far beyond any human sprinter.

Jumps: everyone jumps about the same height

As far back as the 1680s Giovanni Borelli noticed that animals of very different sizes jump roughly the same absolute height. Muscle energy grows with muscle mass, and so does the weight to lift, so the height a body can reach stays about the same. What changes is the acceleration: a flea has only a fraction of a millimetre to push off, so it needs hundreds of g and uses a spring (resilin) instead of plain muscle.

Falls: the square-cube law

Double an animal's length and its weight goes up 8 times but its area only 4 times, so its terminal speed (the speed at which air drag balances weight) goes up by about √2. A mouse tops out near 35 km/h and usually walks away from any fall; a cat spreads out and tops out near 100 km/h; a person near 190 km/h; a horse far faster. J. B. S. Haldane made the point in his 1926 essay On Being the Right Size. The “like a person falling from” number shows how hard each impact is in human terms.

Strength: why ants and beetles look so strong

Muscle force depends on the muscle's cross-section (length squared), but body weight depends on volume (length cubed). Shrink an animal and its strength falls more slowly than its weight, so the load it can lift as a multiple of its own weight goes up. The fair comparison scales the multiple by the cube root of the mass ratio: a 0.15 g dung beetle pulling 1,141 times its weight is like a 75 kg person pulling about 14 times their weight. Still extraordinary, but not six buses.

Flying and swimming

The Speed tab also has birds, bats, insects, fish and other swimmers. Anna's hummingbird reaches about 385 body lengths a second in its courtship dive, faster for its size than a peregrine falcon's famous stoop (about 260) or a fighter jet. The Froude comparison is meant for walking and running, so for flyers and swimmers the page shows the body-length numbers on their own.

Questions

What can I type?: Speeds like 2.5 m/s, 60 mph, 8 ft/s, 100 km/h; lengths like 20 mm, 1.75 m, 5 ft 9 in; masses like 4 kg, 20 g, 10 lb.
How accurate are the animal numbers?: They are commonly reported values, rounded. Real animals vary a lot with species, size, temperature and how the measurement was made, so treat them as typical examples and type your own numbers if you have better ones.
Is it true that cats survive falls from great heights?: Cats often survive falls that seem impossible, because their terminal speed is modest and they turn over and spread out. Falls are still dangerous for cats: keep windows and balconies safe.
What is the drag area?: C_d×A is the drag coefficient times the area facing the air. For a new animal, a rough guess is 0.3 times the body length squared; the presets use values that give the usually quoted terminal speeds.

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