Pondering the ants on my kitchen counter

Factual Questions
1

As I brushed yet another ant off my kitchen counter today, I wondered if the fall of three feet to the floor would harm it in any way. Would it be the equivalent of me jumping off a twenty-story buiding? If I were ant-sized (ala Honey, I Shrunk the Kids), could I jump off my counter without harm?

2

Injury depends on the amount of impact a body experiences when it collides with another body. Because the ant is light and the velocity also being broken by the air resistance it experiences as it falls, the final “impact”(function of its mass and the speed at which it hits the ground) is very low and so does not suffer any injury. The same would not be true for you if you jumped out of a 20 storeyed building. Your mass and the speed at which you would hit the ground would be extremely high. Conversely, if you were to shrink to the size of the ant and jump of the counter top, the injury would depend on what would be your mass after the “shrinking”. If you were the size of the ant but still weigh as much as you are weighing now, the imact would be quite high and the injury would most likely be the same as in the 20 storey case.

3

Part of the reason for the ant’s survival is it’s low terminal velocity. Small objects have a large surface area with respect to their volume, so they can’t fall as fast through the air as larger objects of similar density. Another factor is the ant’s strong exoskeleton, which forms a tough armor protecting its squishy insides. You don’t have that. The third, and possibly most important, factor is that small objects can endure greater stresses in proportion to their size than large objects, mainly because they don’t need so much structural support to hold up the weight of their own parts. Overall, I’d say you could survive such a fall if you were ant-size, but you might have quite a few bruises, since you lack the insect’s exoskeleton.

4

Check out this brief explanation of terminal velocity. The short answer is “no, he’ll be fine.” Because his mass is so small, his momentum when he hits the ground (mass times velocity) is going to be extremely small. Even if we assume a completely inelastic collision (that is, neither the floor nor the ant bends or flexes to take up shock) the momentum transferred is still remarkably tiny.

V[sub]term[/sub] = SQRT ( 2 * weight / C[sub]d[/sub] *air density * area)

Let’s say the ant has a frontal area of 2mm[sup]2[/sup] or 2x10[sup]-5[/sup] m[sup]2[/sup]. An ant has a mass of between one and five milligrams[sup]1[/sup]. We’ll say his weight is 2.5 x 10[sup]-5[/sup] Newtons. The density of dry air at sea level is 1.2929 kg/m[sup]3[/sup]. I’ve estimated the coefficient of drag as 0.5 (dimensionless) because I consider the ant to be a “blunt body” at very low Reynolds number flow (that means I made it up). You can substitute any number between .001 and 1.5 if you feel it would be more appropriate.

I get a terminal velocity (from formula above) of 60cm/s. That means that to fall a meter from your countertop, he might take as much as two seconds. Experience tells us this is a close (but probably not perfect) estimate. Even moving at that horrific pace, his momentum is still only (.6m/s) * (2.5 x 10[sup]-6[/sup] kg) = 1.5x10[sup]-6[/sup]. That’s a very small amount of momentum. Imagine a .5-kg weight (about a pound) moving 3mm every ten seconds; when that “smashes” into the ant, he feels the same impact he does when he hits the floor.

I think he’ll be safe.

As for you and the tall building, remember that mass is a cubic function of length, but area is only a square function. So your terminal velocity increases as the square root of a cubic function (the 1.5th power). Multiply by your mass (another cubic function) to get your momentum at impact, and you’ve got your primary dimension raised to the 4.5th power. If an ant were ten times larger (and of similar density), its impact at momentum could conceivably be ten thousand times larger. Put simply, the bigger they are, the harder they fall.

1 - that’s technically “mass”, but we’ll convert it to Newtons by multiplying by ~10: 2.5e-6 kg * 10m/s[sup]2[/sup] = 2.5e-5 N

5

Guess that settles it…I really am better off just squishing them.

6
      • I prefer a 50/50 mix of crushed roach tablets (borax) and sugar, with a bit of water mixed in. If this is safe for kids/pets I dunno, --well I guess I do know, if you have pet ants. They eat the stuff like mad for a day, then disappear totally over about the second day and don’t return for months.
        ~
7

THAT concoction I am going to try tonight! They have been roaming my kitchen for a while now. I say, let them eat cake!!! :smiley:

8

J.B.S. Haldane said

  • very broadly speaking, the smaller you are, the less you are at risk of injury from falling - you could have dropped the ant out of the window of a very tall building and it would land unharmed. Much of this is because terminal velocity is related to mass versus surface area, but mass increases out of proportion with surface area as the size of the animal increases (which I think is why elephants have large ears - their mass is quite considerable in ratio with their surface area, but the big ears increase the overall surface area so that they can fall from tall buildings unharmed).

[sub]OK, OK; they need the extra surface area to radiate body heat.[/sub]

9
      • And to speed it up, try this: when you first put the bottlecap with the poison down, they will avoid it, even if it’s right in one of their “trails”, because they know it’s unfamiliar. Soooo… what you do then is, you get a toothpick or a bread twistie, dip one end in the poison, and “draw” a line across the path leading right up to the bottle cap. You don’t have to leave a visible trail of liquid, just a scent–and the instant the next ant comes along, they find it and start eating, and then circle right there looking for the source.
        ~