My friend had a freak experience last night. He was nearly hit by a
cat falling from the top of a 9 story high rise apartment building.
We're debating how serious this would have been if it had hit him.
Let's assume the cat fell from 100 feet and it weighed 10 pounds.

1-How fast was the cat falling when it hit the ground?
2-How much force would it have applied to my friend had it hit him?
3-How much injury could this force have potentially caused?

Hello and thank you for your question.

It says here that a falling cat soon reaches a terminal velocity of 60
mph = 88 feet/sec = 27 meters/sec
The Falling Feline
http://www.geocities.com/Petsburgh/4114/righting.html

How much force the cat would apply to your friend on impact would
depend how suddenly the impact brought its motion to a halt.  For
example, if you jump off a chair with your knees slightly bent, you'll
land comfortably; if you lock your legs instead and land on your
heels, you'll likely break a bone--the difference is that in both
cases you've decelerated from the same speed to zero, but over a
greater or smaller distance.

Let's say that the cat is able to flex its legs to cushion the fall
(the way it would if landing on a flat surface), which gives it about
4 inches = 10 cm = .1 meter to decelerate from 27 m/sec to zero.

You may recall from physics that [velocity squared] = 2 * acceleration * distance
so the cat's negative acceleration = [ v * v ] / [ 2*d ] = ( 27 * 27 )/(2 * .1)
                                   = 135 m/sec-sec
For the force, F = ma
so if kitty weighs 4.4 pounds = 2 kilograms of mass
F = 2 * 135 = 270 Newtons

or more simply, since the acceleration of gravity, g = 9.8 m/sec-sec
we can say that a/g = 135/9.8 = 13.78 times the cat's weight = 60 pounds.

So your friend will feel a 60 pound force.  Probably sufficient to
dislocate a neck or shoulder, I'd say....

Search terms used
physics falling cat terminal velocity decelerate

Isn't physics fun?

Sincerely,
Richard-ga

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Request for Answer Clarification by kkrueger-ga on 19 Dec 2003 13:19 PST

Thanks for that answer. A friend of mine came up with a much different
answer using this site:

http://hyperphysics.phy-astr.gsu.edu/hbase/flobi.html#c1

Using the same parameters you used, this page comes up with 5880
newtons, or 1321 pounds of impact force. I notice there is a slight
difference in the terminal velocity you used versus what this site
uses, but that can't explain the difference.

Can you explain to me the huge discrepancy? I wish I had taken that
physics class in high school. :-) If your answer is more correct and
you can explain why, that's worth a tip. Thanks.

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Clarification of Answer by richard-ga on 22 Dec 2003 06:27 PST

Hello again:

The main reason the answers don't match is the arithmetic error in my
answer, as described in racecar-ga's comment.  I'll have a more
complete explanation for you in a day or so.  Thanks for your
patience.

-Richard-ga

What a great answer, Richard! 

In actuality, I suspect that the claw wounds caused by the flailing
cat would be worse than the force of the impact. I once had a cat
accident where the kitty fell off a roof onto me, and some mighty
impressive holes were made in my shoulder as she scrambled around
frantically, trying to right herself and reach safe ground. I hate to
think what sort of injuries I might have sustained if my pussycat had
fallen 100 feet before landing on me. Among domestic occurrences,
there are few things scarier than a terrified and angry house cat.

Also see:
 
"How cats survive falls from New York skyscrapers" Natural History
98,no.8,20-26(1989) J.M.Daimond.
 
"Why cats have nine lives," Nature 332, 586-587; April 14, 1988. Diamod, Jared. 
 
"The tail-less cat in free-fall," The Physics Teacher. 27, 620-625;
November 1989 Fredrickson, J. E.
 
"High-Rise Syndrome in Cats" by New York veterinarians W.O. Whitney
and C.J. Mehlhaff

Best regards

THX1138

richard-ga's error comes here:

so the cat's negative acceleration = [ v * v ] / [ 2*d ] = ( 27 * 27 )/(2 * .1)
                                   = 135 m/sec-sec

If you try it, you'll see that ( 27 * 27 )/(2 * .1) gives 3645 m/sec/sec, not 135.

With no air resistance, the speed of a cat after falling a distance D is
sqrt(2*g*D), where g is 32 ft/sec/sec.  For your given number of 100
feet, this gives exactly 80 ft/sec, or 54.5 mph.  There's no way the
speed can be greater than this after falling 100 feet, but air
resistance will cause the actual number to be a bit less, say 50 mph,
or 73 ft/sec.  I like richard-ga's choice of 4 inches of deceleration
distance.  With the weight you gave (10 lb) and a 4 inch stopping
distance after falling 100 feet, the force exerted is 183 lb without
air resistance, and 156 if air resistance is sufficient to reduce the
speed of the falling cat to 73 ft/sec.

How great of an injury would these forces likely cause?  Your guess is
as good as mine.  I tend to agree with pinkfreud though--watch out for
the claws.

I once read a very nice overview of the physics of a falling cat; it
can be found at http://www-x.nzz.ch/folio/archiv/2003/05/articles/tiere.html
. Unfortunatelly it is not in English !