Oldfist wrote:
Let m(t) be the mass as a function of time t and let v(t) be the velocity as a function of time t, then the momentum as a
function of time t is p(t) = m(t)*v(t). Further let ' represent one derivative with respect to time, i.e. the instantaneous time
rate of chang of a quantity, then
F1 + F2 + ... + Fn = p'(t) = [m(t)*v(t)] ' = m'(t)*v(t) + v'(t)*m(t) [by the product rule for derivatives]
This form of Newton's 2nd Law is needed to compute the force, for example, on a rocket whose total mass is changing
because it is burning its onboard fuel. So, to describe the force on a constant mass m (or the acceleration produced by a
force on the constant mass m) the simplified version is:
F = m*v'(t) = m*a(t), where the acceleration (the instantanteous time rate of change of the velocity) is a(t) = v'(t).
If we take the (over simplified) example of the 1-dim motion of an object in the gravitational field of the earth (e.g. hold
your pencil up over the floor and then release it) then the acceleration is also constant, i.e. a(t) = -9.8 m/s^2.
However, the acceleration can also increase and a familiar example is driving your car.
Case 1. a(t) = 0
Suppose you are driving your car down the street at a constant rate v(t) = 30 mi/hr, and so your acceleration a(t) = 0.
Case 2. a(t) = c (constant number) != 0
Now, suppose you enter a zone in which the speed limit is 40 mi/hr, and you uniformly press down on the accelerator
producing a constant, nonzero acceleration which gradually and uniformly increases you velocity to 40 mi/hr.
Case 3. a(t) = nonconstant function of time
Now, a key thing to notice for our striking application is that the acceleration may be itself increasing and this occurs when
"punch it" into passing gear, that is, instead of uniformly pressing down on the accelerator, you floor it. In this situation you
experience what is called the "jerk" and that is a nonzero time rate of change of the acceleration itself.
Summarizing, if s(t) is the position as a function of time t, then
v(t) = s'(t), the velocity function, i.e. the instantanteous time rate of change of the position
a(t) = v'(t), the acceleration function, i.e. the instantanteous time rate of change of the velocity
j(t) = a'(t), the jerk function, i.e. the instantanteous time rate of change of the acceleration
F = m*a(t), if we look at this simplified case of Newton's 2nd Law in which the jerk is nonzero and positive, then that
means that the time rate of change of the acceleration is positive, and hence in this case the acceleration is increasing,
which means that the force is increasing, when the jerk is nonzero and positive. This is the interesting case that applies to
the striking situation.
Energy analysis is of course good stuff, but it's not the only approach. One can start with the above, F = m*a(t) = m*(dv/dt) and approximate it for small delta t , yielding the impulse-momentum theorem:
F*(delta t) = delta (m*v) = m*(delta v), which says the impulse is equal to the change in momentum. So, we can use consevation of momentum instead of conservation of energy to compute F in newtons or pounds. Similar to:
http://www.fas.harvard.edu/%7Escdiroff/ ... eBlow.html
Either approach allows us to compute a quantity at impact, however, neither by itself explains why one method of striking is "harder" or more effective than another. An analysis of the jerk which occurs before impact, partially describes a key feature of the why. In order to do an experimental analysis we would need a very sophisticated, high speed motion capture camera. This type of analysis is standard stuff for sports scientists, but is nontrivial even for the experts. Just because one may have only a high school or non calculus-based understanding of physics and can't see how to do it doesn't mean that it can't be done.
Tony, BTW, this is classical mechanics which we can apply using simplifying assumptions and that the strike has been made. The probabilistic situation you are interested in, in which evasion occurs, and which isn't really random, would require a statistical mechanics type approach, and yes the math would be different.
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....but I really whacked it in on the bodybuilder