Michelle's Lab Book - Calculating the deformation energy of the impact
NOTE: If you don't follow the algebra here, don't worry. Algebra isn't difficult, but it does take a while to get used to it.
To work out the maximum kinetic energy of my hand I use the formula:
To estimate the mass of my hand I've used data about the masses of different parts of the human body, which I found in a biomechanics paper (Zatsiorsky, Selujanov, 'The mass and inertia characteristics of the main segments of the human body'. Biomechanics YIII-B.) pp. 1151-1159, (1983)).
Body part
% of total body mass (female)
Head
6.68
Trunk
42.57
Upper Arm
2.55
Forearm
1.38
Hand
0.56
Thigh
14.78
Calf
4.81
Foot
1.29
As we've already seen, Chris aims to maximise the mass involved in the strike by putting as much of his body as possible behind it. I'd rather underestimate my ability to break the board than overestimate it, so I've only included the mass of my hand and forearm in case I get it wrong. My total body mass is 65 kg, so according to these values the combined mass of my hand and forearm is 1.3 kg.
Putting the values for my maximum speed and the estimated mass into the equation for kinetic energy, the maximum kinetic energy of my hand is 63.7 J. It's clear that if I could hit the board at the right point in my swing (when the kinetic energy is at its maximum) and convert all that energy into breaking the board, I'd be able to break the board. However, a large fraction of the energy of the collision will go into the kinetic energy of the board and my hand after the impact. To work out how much energy will actually go into breaking the board I need to use the conservation laws: conservation of energy and conservation of momentum.
Conservation of energy says that the kinetic energy of my hand before the break is equal to the energy that goes into breaking the board (called the deformation energy) and the kinetic energy of the hand and the board after the strike (assuming that the energy lost as heat and sound is so small that it can be ignored).
This can be written as:
Where M= the mass of my hand and forearm, Vo = velocity of hand at point of impact, m = the mass of board, V = velocity of hand and board after impact.
The above equation also assumes that the hand and the board travel at the same speed immediately after the impact. This is a big assumption, but it makes the mathematics much easier, and will hopefully get a reasonably accurate result!
The law of conservation of momentum says the momentum of the hand before the impact is equal to the momentum of the board and the hand after the impact. This is written as:
We need to work out the deformation energy (the energy that goes into breaking the board). Rearranging equation 1 to make deformation energy the subject we have:
Rearranging equation 2 gives:
Substituting this back into equation 1 gives:
This simplifies to:
The mass of the board, m = 0.518kg. As before, the mass of my hand and forearm, M = 1.3 kilograms, and the speed of my strike, Vo = 10 m/s
So the amount of deformation energy generated by my strike would be:
