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.