So it is summer, which means I have to deal with insects. Today one has made it into my room and while I tried to work this insect was constantly in my face, which caused me waving my hands in the air. Now I have actually hit it 3 times but it kept coming back. Then I was wondering, why don't insects actually really mind when you just basically slap them with one hand? They don't fall to the ground, they surely don't die and they don't even seem injured in any way. If a giant as much bigger than me than I am compared to an insect were to just slap me, I would imagine I were at least severely injured.

Is a human hand alone - not using a wall or another hand etc. to mash - really too slow for an insect to cause damage? Or does the insects "skin", which is made out of chitin, just give the insects this super power?

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    $\begingroup$ While there is some physics here, it may be more of a biology.SE question. $\endgroup$ – Michael Brown Jul 10 '13 at 12:25
  • $\begingroup$ @Michael Brown I don't think it is rather biology. I don't need the facts about how the insect is built really, I think physics is the one to answer which effect causes the insect to be so slap resistant. If one could argue/calculate why it could only be it's chitin shell (and e.g. how much pressure it must be able to take), it would be fine. But maybe it is the effect described by Widor and which one it is only physics can answer - I think. $\endgroup$ – Jack Jul 10 '13 at 13:05
  1. Mass and material strength do not scale equally with size. This is well known, so I'll only illustrate with a few examples.

    Imagine a small cylinder of clay 1/2 inch around and 2 inches tall sitting on your desk. It can sit there a while with little deformation. Now imagine a cylinder of the same material 10 feet around and 40 feet tall. It would slump down significantly right away.

    Look at the thickness of animal legs over animal size. Evolution would have trimmed off excess material long ago. Note how much thicker elephant legs are relative to overall size than those of a dog or ant.

  2. The effects of air viscosity are much greater at the small size of a insect than they are to our human-scale experience. The air pushed in front of your hand will push the insect well before it actually hits your hand.
  3. You can stun insects by whacking them, although it usually takes something a bit faster and more rigid than your hand for something as small as a fly. I found the best way to hunt flys in a apartment in college was to whack them in mid air with a large metal spatula. This would knock them to the ground where they'd lie for a 10-20 seconds, then buzz on like nothing happend. That gives you plenty of time to dispatch them. The long handle of the spatula allowed me to move it faster than I could move my hand.

Okay, so I just did the calculation for the acceleration $a$ of the slapped insect. The answer of John Rennie, who claims that the acceleration is constant, is wrong. I just used energy and momentum conservation and calculated the elastic and inelastic case (assuming that the friction is negligible) to get an upper and lower boundary for the acceleration of the insect.

Let $M$ be the mass of the hand with which you hit an insect, $m$ the mass of the insect, $V$ the velocity of your hand just when it makes contact with the insect, $v$ the velocity of the slapped insect and finally $\tilde{v}$ the velocity of the hand after the slap.

Elastic case:

Momentum conservation: $$MV=mv+M\tilde{v}$$

Conservation of kinetic energy:

$$\frac{MV^2}{2}=\frac{mv^2}{2}+\frac{M\tilde{v}^2}{2} $$ Now I want to solve for the insect's velocity $v$: $$MV^2=mv^2+\frac{(MV-mv)^2}{M}\\ M^2V^2=Mmv^2+M^2V^2-2MVmv+m^2v^2\\ v^2-\frac{2MmV}{Mm+m^2}v=0 $$ Solving this quadratic equation (choosing the physically meaningful solution) leaves us with $$v=\frac{2MmV}{Mm+m^2}=\frac{2MV}{M+m}=\frac{2V}{1+m/M} $$

The inelastic case is much easier to calculate, since the insect and the hand would be stuck together as one object with mass $M+m$ and velocity $v$ so you can solve for $v$ just using

Momentum conservation: $$MV=(m+M)v$$ $$\Rightarrow v=\frac{V}{1+m/M}$$

Now the acceleration $a$ of the insect is it's velocity $v$ divided by the time $t$, which is the typical time from beginning to end of the collision. So the acceleration lies in between $$\frac{V}{t}\frac{1}{1+m/M} < a < \frac{V}{t}\frac{2}{1+m/M} $$ Notice that this does not equal the acceleration of the hand just before you slapped the insect, which is just $V/\tau$, where $\tau$ is the time it took to wind up for the slap (which can be "anything", at least it certainly does not have to be just the value that would make the accelerations equal - no law would demand that). So the acceleration of the insect (or anything else that got hit for that matter) depends mostly on the time $t$, but also on the mass of the object that got slapped. Consider the the two cases in which the mass of the hit thing $m$ is equal to the mass of the (your) hand $M$ (1) and the case where $m<<M$, like it would be if m really is the mass of a small insect (2):

Case (1): $$\frac{1}{2}\frac{V}{t} < a < \frac{V}{t} $$

Case (2): $$\frac{V}{t} < a < 2\frac{V}{t} $$

One friend of mine has suggested, that a measure of how much you hurt the insect might be the pressure (force divided by the area) $F/A$ you create when you hit some body. (Jumping on somebody in high heels hurts more than if this sadist had worn flip flops). So that would just be $m a/A$, where $A$ is the surface on which you hit the insect.

Now here comes the enlightening part: The force (or pressure) on the hit object is bigger if the object has smaller mass, although it doesn't change dramatically and can at the most take the value in presented in case 2. And the the created acceleration on the hit thing is approximately twice as big when you hit an insect (with mass $m<<M$) than if you were to slap something that has about the mass of your hand.

But what is still true, is that if you then look at the force (multiplying the general formula for $a$ with $m$), the force will really largely depend on $m$ (but not being just proportional) and therefore indeed be very small if you slap an insect (although, once again, the acceleration is not simply conserved).


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