Just asking

[anthropos]

My mum asked, "If an ant is blown off a table, does it undergo large impact when it hits the ground? Does it die?" Does it? o_O

[Pangloss]

I don't know the answer, but it reminds me of the penny-off-a-tall-building problem. Turns out the penny can't really acquire enough velocity to really do any damage, and you could probably just catch it with your bare hands. Which suggests that the ant would be ok, but on the other hand biology could play a role here.

 

I do know that I've seen ants and other small insects fall great distances and be fine afterwards.

[Glider]

No it doesn't. An ant is too small and light to accelerate much. Air is quite viscous to an ant, which is why you can blow an ant off a table with a puff of air that would barely ruffle the hairs on your arm. Air resistance would support the ant as it fell. Basically, there is no height an ant could fall from that would kill it.

[grifter]

thats, just cool, just think if we could do that, really cheap plane flights, no landing taxes, just, maybe, falling tax ?

[ecoli]

plus it has a nice exoskeleton... not brittle like calcified people bones.

[the tree]

You could blow an ant off a tall building and it wouldn't get injured, it has an extremely low terminal velocity.

[insane_alien]

unlike us humans, we suck at falling unless we have a nice big parachute. though we're probably the only things that do it for fun.

[Bignose]

Ok, thus spurred a few calcuations, since I was interested what the numbers actually said:

 

First, I used Google scholar to find the average mass of an ant. Since there are many, many species of ant, I found info on the leaf cutter ants and found that the workers were as small as 1.4 mg to 32.1 mg. Leaf cutter ant's head width also averaged between 1.4 and 1.6 mm (depeding on what colony of ants was studied).

 

So, for calculation purposes, I assumed an ant was three spheres, the total width of the head at 1.5 mm, the middle 3.0 mm, and the back section 3.0 mm. So, taking the mass of an ant as 20 mg and divind that by total volume of those three spheres, the density of this ant is 665 kg/m^3. Do ants float on water? Their density is less than water's 1000 kg/m^3, but I don't know if they float or not so I don't know if this is a good guess or not.

 

Anyway, then, for calculation purposes, I treat the entire body of the ant as a single sphere. I added up all the volume, then calculated what the radius of the sphere that has that same volume as the entire volume of the ant. So, that mens I treat the ant as a single sphere with radius 1.9 mm (or total width of 3.8 mm).

 

The question here is what terminal velocity an ant reaches. This is obviously a difficult question, since what the terminal velocity is is dependend on what the drag force is, but the drag force is dependend upon what the velocity is, meaning you h ave two unknowns (drag force and terminal velocity). Normally, you'd have to use two equations and solve for two unknowns, but correlations exist for estimatig these numbers rather than trying to solve two simulataneous equations.

 

The first correlation to use is to calculate the Archimedes number: This is a number that is independent of the terminal velocity

 

[math]Ar = \frac{d_p^3 \rho_f (\rho_p - \rho_f) g}{\mu^2} [/math]

 

[math]d_p[/math] is the diamter of the particle

[math]\rho_f[/math],[math]\rho_p[/math] is the denisty of the fluid, particle respectively. The density of air is 1.246 kg/m^3

[math]g[/math] is the acceleration due to gravity = 9.8 m/s^2

[math]\mu[/math] is the viscosity of the fluid = 1.78*10^-5 kg/(m*s)

 

Plugging these numbers in yields an Archimedes number of 1.4*10^6

 

There is a correlation between the Archimedes number and the Reynolds number at terminal velocity.

 

But first let me define what the Reynolds number is: the Reynolds number of terminal velocity is

 

[math]Re_t = \frac{U_t d_p \rho_f}{\mu} [/math]

 

where [math]U_t[/math] is the terminal velocty, and all the other terms are defined above.

 

The correlation between Ar and Re_t is:

 

[math] Re_t = (2.33 Ar^{0.018} - 1.53 Ar^{-0.016} )^{13.3} [/math]

 

Plugging in the value for Ar above, I get a terminal Reynolds number of 2299. In physics terms, the Reynolds number compares the relative magnitudes of the inertial forces versus the viscous forces in the fluids. If the Reynolds number is very high (10^5, 10^6 or higher), the inertial forces dominate. If the Reynolds number is small (1 or less) the viscous forces are very important. This is important since if inertial forces dominate, the shape of the object is the primary determiner in how much drag it will experience. If viscous forces dominate, the surface properties (like it's roughness) are the primary determiners of how much drag the object will experience.

 

I find it interesting that in this case, the Reynolds number is in that intermediate region. To put it simply, both the inertia and the viscous forces are important.

 

Then, back calculating for the terminal velocity ([math]U_t[/math]) I get a terminal velocity of 8.5 m/s

 

Compare that with the terminal velocity of an unparachuted human skydiver, which is about 56 m/s, you see that a human falls 6 1/2 times faster.

 

Because of this higher velocity, the Reynolds number for a person is much higher than that of the ant, in the tens of thousands. That high of a Reynolds number means that the inertial forces are dominant. Or, in other words, the surface properties are not important -- or in other other words, you can't significantly change your terminal vleocity whether you were wearing a wool sweater or a spandex body suit or whether you had shaved or not -- none of that matters when skydiving.

 

My calculations for the ant are probably riddled with error. Firstly, assuming a sphere is a worst-case lowest drag scenario. The fact that an ant is actually three sphere-ish segments will increase the drag a fair amount. Also, the legs and antenne have to increase the drag somewhat too. Nevertheless, I think that the calculations are insightful in that it is obvious that an ant falls significantly slower than a person does. Couple that with their exoskeleton, and I think that explains a lot of why an ant can survive that level of fall compared with people.

[Sisyphus]

Great post, Bignose. I suspect the difference between sphere and ant-shape would be very large indeed. I would be surprised if an ant broke 4 m/s.