Why a squirrel would never die from falling, no matter how high it falls.
An aerospace engineering student explains it.
Imagine a squirrel. A fluffy tailed rodent, weighing just over 300 grams, and with a silhouette very similar to that of a human being, what would happen if it were to fall from an arbitrary height?
We all know that there is a force, the force of gravity (which is not really a force, but we will overlook this), which attracts objects to the Earth according to their mass. In fact, the speed with which they are attracted to the Earth does not depend on their mass, but the intensity with which they fall do depends on it: a hammer and a feather would fall at the same time… if it were not for the aerodynamic resistance.
When an object is in free fall within a fluid (such as the air in the atmosphere), two opposing forces act on it: gravity and aerodynamic resistance.
The first one pushes it downwards (we will take it as negative) and the second one upwards, here we have the equations for both forces:
Thanks to Newton’s first law we know that the net force acting on the object is the sum of the two previous ones, and thanks to Newton’s second law we know that the resulting acceleration on the object will be that net force divided by the object’s mass:
We come to the conclusion that, when both forces are exactly equal, the resulting acceleration on the object will be zero, and therefore its speed will stabilize. This is possible due to the fact that the aerodynamic resistance grows with the speed squared, that is to say, the greater the speed the much greater the aerodynamic resistance will be, so it will fight to avoid the speed increase. This means that no matter how high it falls, the speed will never exceed a certain value, known as terminal velocity. Solving for velocity in the above equation:
The rigorous way to solve this equation would be through integration, but in order not to bore the reader we will make some reasonable assumptions. First, the force of gravity can be reasonably well approximated as 9.8 times the mass of the object (mass times acceleration), since the acceleration of gravity is approximately constant for the path that our poor squirrel will take. We will also take an average air density value, a surface and masses suitable for a squirrel, and a reasonable drag coefficient:
A squirrel weighs 330g, the underside of its body has an area of approximately 500cm², the air density at sea level is 1,225, and the drag coefficient… well, we’d have to put the squirrel in a wind tunnel, and it’s going to suffer enough, so we’ll try to estimate that:
Here we have the coefficients for different shapes, so we can get an idea. It turns out that the coefficient of a human being in free fall is about 1.2, and if we look at the table, it makes a lot of sense: a human being is practically a flat surface, more like the side of a cube than a pyramid or any other of the shapes. And with a squirrel, the picture is similar.
If we do the math (and having changed the units correctly), the result gives us 10.28 m/s, about 23 mph.
The reason for this is because a squirrel has a large area/mass ratio. This means that gravity does not pull on it with too much force but relatively large aerodynamic resistance will be generated. To get an idea, the terminal speed of a skydiver in a belly-to-earth (i.e., face down) free fall position is about 54 m/s (120 mph).
The fact is that this is such a low terminal velocity, that it is reached in the first 3 seconds of the fall, so for a squirrel it is the same to fall from the top of a pine tree as from the stratosphere: in both cases it will hit the ground at the same speed.
And, yes. For those of you who were wondering, a squirrel is certainly capable of surviving a crash at that speed. I’ll leave you with a quote to ponder:
“To the mouse and any smaller animal gravity presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal’s length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force”
On Being the Right Size, from J. B. S. Haldane
Bonus
For those who were uneasy with the simplifications (constant gravity acceleration, constant density…), I integrated everything by computer and the result was very similar: 10.29 m/s. Here is the graph of speed over time:
