## 11 October 2013

### The Power of an Axe Kick!

My last post examined a punch from my karate instructor, Mr Abercrombie.  In this post, I examine his axe kick, which involves bringing the leg up as high as possible, followed by a fast downward kick.  The knee bends very little, which makes it appear as if an axe is swinging downward.  Check out the video below (click on the image for a larger view).

Mr Abercrombie begins by executing what is called a crescent kick.  Instead of swinging across as with a crescent kick, he brings his foot straight down.  Mr Davis holds the board, which acts as victim in this demonstration.  The image below shows Mr Abercrombie's foot at the apex of its motion (click on the image for a larger view).
His foot passes through the board at about 10 m/s (36 kph or 22 mph).  His foot's acceleration through the board is nearly 15 times the acceleration due to gravity.  I included acceleration vectors in the video so that you may see how the magnitude and direction of his foot's acceleration changes during the kick.  You'll note the downward pointing acceleration as the foot rise, which is necessary to slow the foot down.  A great deal of potential energy is stored at the instant seen in the above photo.  Once he brings his foot down, the acceleration begins to turn to the left in the movie.  That is because his foot moves along the arc of a circle during the hit, which means centripetal acceleration must be present.  At the point of contact, his foot delivers about 2200 N (500 lb) force to the board, breaking it easily.

I modeled Mr Abercrombie's leg as a sequence of rigid rods.  Knowing the percentage each segment comprises of a man's body mass, I can get a reasonable estimate of the moment of inertia of Mr Abercrombie's left leg.  During the explosive kick through the board, Mr Abercrombie's leg has a rotational kinetic energy of over 200 J (almost 150 ft lb).  In the roughly one-eighth of a second it takes for his foot to go from rest at the apex to breaking the board, more than 1500 W (just over 2 hp) of power is required.  Though an athlete like Mr Abercrombie is capable of large power outputs for only a very short length of time, I am very impressed that the power output of such a kick exceeds two horsepower!

Given that the body is only about 20% efficient in its energy conversions, Mr Abercrombie probably burned about 1000 J (almost 750 ft lb) internally just to move his leg from its apex to the board.  That amount of energy represents about a quarter of a nutritional Calorie.

Did you ever imagine so much power in a kick?