What it shows:
The impulse momentum theorem is demonstrated in a most dramatic way by breaking several boards with the blow of your fist. You need not be a karate expert to show how the force of a well executed hammer-fist strike will easily break a stack of five to eight boards. The impulse is given by

impulse = F∆t = ∆mv

The point of the demonstration is: the greater the speed, the smaller ∆t will be and thus the greater the force.

Images by Jeffrey Pike

How it works:
A detailed description of the physics of karate has appeared in a Scientific American article. ¹   We will only present the salient features here relevant to the impulse momentum theorem. First we estimate the momentum of the fist. Its effective mass is greater than just the hand—after all, you're swinging the whole arm to which your fist is attached. Resting the forearm on a scale suggests an effective mass somewhere between 2 and 4 kg. Next we use a photogate to measure the time it takes the striking fist to pass through it, establishing a velocity ²   of 10 m/sec and a momentum of 30 kg m/s.

Wood is a fairly elastic material and a 30 cm long board typically requires about a centimeter of deflection before breaking; 500 Newtons will deflect it to that extent. If we assume that a 2 cm thick board actually stops the swing of the fist, the action takes place over a distance of 2 cm + 1 cm = 3 cm and the time of this interaction will be the distance divided by the average velocity of the fist = (v_i + v_f)/2 = 7.5 m/sec. Thus Δt = 0.03 m ÷ 7.5 m/sec = 4 millisec. In this time the momentum of the fist would change by 45 kg m/s. Thus



That's equivalent to 2250 lbs! Of course the board breaks with this kind of force and does not stop the fist.

The boards are stacked on top of one another and kept slightly separate with pencils placed between them; rupture can proceed successively through the boards with each rupture involving a smaller force than if a single thick board were used. This also has the effect of the momentum of the downward-moving broken pieces of the top board helping to break the board beneath it, and so on down the stack. Thus the peak force to break, say, eight boards is less than eight times the force needed to break one board.

Setting it up:
The boards we use are cut from a so called 1×12 (nominally 3/4" by 11.5") of dry white pine. They're cut 6" wide with the grain parallel to the width. (All the references mention that the karate blow should be parallel to the grain. If that's how the karate experts do it, it's good enough for us; ignore the wise guy that always says, "oh, that's easy if you have the grain running that way...ha, ha." Regardless of the way the grain is oriented, a stack of five to eight boards looks quite impressive indeed.) Separate the two lecture benches about 10" apart and stack the boards straddling the two benches. Two sturdy Harvard chairs work well too. The lecturer should do this, and slowly build up the stack (and the tension) to heighten the anticipation in the audience.

Comments:
If this is your first time trying this demonstration, you will most likely be as timid as we were. Try to strike the top board squarely in the middle. An off-center strike might fail to break the bottom board. Start with four or five boards, which is easy, and build your confidence slowly by adding one more board with each successive try. Remember that speed, not strength, is everything, so concentrate on having maximum speed when fist meets board. Karate experts refer to this as "focusing the strike inside, or a little beyond, the object." We refer to this as "moving your fist as fast as you can." It will only hurt if you wimp out. According to the references, the hand can actually withstand forces much larger than 25,000 Newtons. Have faith in physics.

Try introducing the demonstration with the old straw-through-the-potato trick. Also, a rather fun and related demonstration is the Egg Throw, which makes the opposite point: what happens when you try to maximize ∆t.

It might also be of interest to the audience that the stomatopod (commonly known as the mantis shrimp) executes the fastest animal movement known to man. ³    It can give a karate-like blow with its claw in a millisecond and uses this talent to smash open the hard shells of clams and the like that it feeds on. It's blow is so effective that an aquarium can not be used to keep it captive ... it will smash its way through 1/4" thick glass!

1 M.S. Feld, R.E. McNair and S.R. Wilk, Scientific American 240, 150 (April 1979): "The Physics of Karate." For an analysis of the energy lost to deformation of an opponent, see J.D. Walker, Am J Phys 43, 845 (1975): "Karate strikes"
2 The references claim that karate blackbelts achieve maximum speeds of about 10 m/sec (see Walker, above) suggesting that our estimate of the speed is a bit high.
3 R.L. Caldwell and H. Dingle, Scientific American 234, 80-89 (Jan 1976): "Stomatopods"

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