Sometimes a cat will fall out of a window or balcony—a byproduct, no doubt, of a cat doing cat things. If you have a house cat, your feline's propensity for aerial shenanigans probably doesn't worry you that much: Cats falling from low windows can use their righting reflex to land on their feet like nothing even happened. Fall from a higher story, though—in what researchers call high-rise syndrome—and they may not be as lucky.
But here's the weird part: Cats falling from super high floors can survive. A RadioLab episode on this falling cat issue states that cats falling between five and nine stories are the ones most likely to be injured. Fall from a higher story, though, and your odds of survival are better.
But how? How is it possible for a cat falling from a higher height have a greater chance of survival? The answer depends on two things: air resistance and apparent weight.
When an object falls, there are essentially two forces acting on it. There is the downward force of gravity that depends on both the gravitational field (9.8 N/kg on Earth) and the mass of the object. The other force is the air resistance force. The air resistance is a force that increases with the speed of the object and always pushes in the opposite direction as the motion of the object. If I assume the object is only moving in the downward direction, I can write the sum of the forces in the y-direction (vertical direction) as:
For the air resistance force, ρ is the density of air, A is the area of the object, and C is a coefficient that depends on shape. You can see that when the object first starts falling, the velocity would be zero such that the total force would just be -m*g (like a free falling object). One more note: I can estimate the cross sectional area of a cat, but the drag coefficient will be a bit more difficult. When the speed increases to the point where the net force is zero (the gravitational force and the air resistance force balance), then the object will move at a constant speed—we call this the terminal velocity.
But how do you find the velocity of a cat as it falls from a building? It's not such a simple problem, since the net force changes as it falls. Really, the only way to find the speed is to create a numerical calculation. With a numerical calculation, the motion is broken into many small steps of time. During each of these time steps, the forces are approximately constant so that the motion can be calculated. The smaller the time steps, the better the calculation. However, the more time steps you have the more calculations you need. Really, the only feasible way to do this is with a computer program.
I don't know all the values, so I'm going to estimate some quantities (like the mass and area of a cat), but here is my numerical model for a cat falling from a crazy tall building (100 meters). You can click the "pencil" icon to see and change the code—don't worry, you won't mess it up. Oh, I also added an object with no air resistance just for comparison.
Notice that the red curve (the no air resistance object) has a parabolic curve—this is what you would expect for constant acceleration. However, the cat stops increasing in speed as time goes on because of the increase in the air resistance.
But this plot doesn't explain why intermediate falling distances are more dangerous than higher falling distances. How about a plot of impact speed vs. starting height of the falling cat? Again, you can click the "pencil" to look at the code and change it if you like.
From this plot you can see the simple result that a higher fall results in a greater impact speed. So air resistance alone can't explain why a medium fall is more dangerous than a high fall.
We need to look at something else. What about apparent weight? Your apparent weight is not the magnitude of the gravitational force. Instead, it is the magnitude of the force acting against the gravitational force. Suppose you are in a stationary elevator and you push the down button. For a brief moment, the elevator accelerates downward and you feel a little bit lighter. Of course, your weight didn't change: It was just that the force of the floor pushing up on you decreased in magnitude. This is your apparent weight.
When a cat first jumps (or falls) from a tall window, there is no air resistance force and nothing pushing up against the gravitational force. For a short period of time, the cat will feel weightless. It is during this weightless period that the cat's instincts kick in. The cat will use its (super powers to rotate into a feet down position)[https://youtu.be/RtWbpyjJqrU] and brace for landing. This cat instinct is a good idea for normal falling distances; with the feet down, the cat is ready to land and make another move. But for super high falls wit high impact speeds, landing legs-down will just make things worse. If the cat didn't think it was falling, it would be in a different falling position that might not be good for landing but could produce a softer impact. In a way, lower apparent weight is bad for cat landings.
So, there could be two things that a cat's survivability depend on. First, the impact speed. Higher impact speeds are bad because the cat will hit the ground faster. Second, there is the apparent weight at impact. Lower apparent weights are bad because the cat will be in a position to land on the feet instead of spread out and relaxed. I have arbitrarily made a survivability score that is the sum of the impact velocity (multiplied by some factor) and the inverse of the impact acceleration (multiplied by some factor). So you have two competing factors—but with opposite relationships to the starting height. This means there should be some height that minimizes this survivability. Here is a plot of this score for different falling heights.
This has a minimum survivability score around 18 meters. That's about six stories—right smack in the middle of the height range where experts predict cats are most likely to get hurt. OK, I'm pretty happy with the way this turned out. Go ahead and make your own cat falling score and see what happens!
