Falling Cats and Dimensional Analysis

24 10 2010

I recently read an interesting story about cats falling from high buildings. Here are some statistics:

  • Of the 115 cats who were brought into the Medical Centre having sustained a fall of between two and thirty two stories 90% survived;
  • 10% of the cats which fell between 2-6 stories died;
  • Only 5% of the cats which fell between 7-32 stories died – the doubling of the survival rate as the height increased can be accounted for by the effects of terminal velocity.

I don’t want to comment on the use of statistics but on the surprising fact that a cat seems to have reasonable chances of surviving a fall from an arbitrary height. We know that this is not the case for humans. Why? Well, cats are obviously more agile than humans but there is a clear benefit of being small.

According to the story source, a cat has half the terminal velocity of a human (100 km/h vs. 200 km/h). To make the analysis simpler I will only speculate what would happen if a cat had the same mass of a human. If we take typical masses of 75 kg and 4 kg and consider that mass is proportional to the volume (m~L^3), a cat would need to grow 2.7 times in size (L) in order to weigh 75 kg. Here I’m using L to refer to a linear size, for instance the length or height of the cat.

Now I will write down how some interesting magnitudes vary with L and the increase in those magnitudes if  L grows by a factor of 2.7:

  • Terminal velocity: L^0.5  (x1.6)
  • Kinetic energy at terminal velocity: L^4   (x53)
  • Force necessary to break a bone: L^2   (x7.3)
  • Force of a muscle: L^2   (x7.3)

The kinetic energy at terminal velocity increases by a factor of 53! If we assume that when the cat makes contact with the ground the kinetic energy is absorbed by a constant force F acting on a squashing distance x, we can write: F = Ek/x ~ (L^4)/L = L^3.  This force grows faster than the resistance of the bones or muscular force which are proportional to L^2. Therefore, bigger animals will have more trouble surviving falls at terminal velocity.

I’ve made loads of assumptions but, hey, this is what this blog is about!


Modeling the World

24 10 2010

Hey, this is a new blog in which I plan to post on creating simple mathematical models of several aspects of the world we live in. My original idea was to blog on heavyweight modeling for engineering and machine learning. However, I came to realize that this would ask a lot of effort from my part and it would probably be slightly boring. Therefore I will mainly blog on more lighthearted modeling on random things that I find interesting.

I target one or two posts per week. However, given my tendency towards optimism I’m a bit wary that the posting frequency could be lower. Let’s see how this works!