There's just a couple of points I wanna make—or more correctly, support—here.
First of all, the putative relevance of the `speed' vs. `velocity' distinction in connection with energy...
gnrail said:
Sorry but the engineer in me had to come out.
And the engineer was absolutely correct. Speed is the magnitude of the velocity vector. It follows that the difference between the scalar speed and the vector velocity is
irrelevant so far as energy is concerned, since (i) energy is a quantification of the capacity to do work, and (ii)
work is defined as the scalar product of force and the radial vector: dW =
F • d
r = m
a • d
r = m(d
v)/dt • d
r = m d
v • d
r/dt = m d
v •
v = mv(dv) (
u•u) = mv (dv) where
u is the radial unit vector and v is the scalar component—i.e., the
speed—of the velocity
v, since
u is invariant over time. What this ultimately means is that the direction of motion is irrelevant to the quantity identified as the kinetic energy, and gnrail is completely correct to use `speed' and `velocity' interchangably
in this context.
This can be shown trivially, and
is shown in pretty much every textbook on Newtonian mechanics in existence based on elementary vector calculus and simple differential equations. Since over an infinitesimal distance the work accomplished is measured as
dW =
F•d
r = mv(dv)
then taking the definite integral on both sides from points A to B yields
W =
int[mv dv]= m
int[v dv] between B, A = (1/2)m[v´ˆ2–vˆ2]
with W the work accomplished by the application of the force to the mass m between A and B, v´the velocity at B and v the velocity at A. In other words, the work done is equal to the increase in the quantity (m/2)Vˆ2, where V is the scalar magnitude of the velocity. Taking, as standard, the energy of an interaction to reflect the capacity to do work, the actual work accomplished by applying a force can thus be identified with an increase in this quanitity, which is again
defined as the kinetic energy, between the points A and B. And the point is that all you have to do to measure W is to measure the value of scalar quantities. Direction is irrelevant. If a particle is moving at speed
n in one direction at time t and speed
m in an orthogonal direction at time T, the change in kinetic energy will be exactly the same as if the particle is moving at
n at t in one direction and
m at T in the same direction. The distinction between speed and velocity in this context is a distinction which makes no difference whatever. And it's the transfer of kinetic energy to the target which results in the rearrangement of its structure, i.e., the damage done.
Second, I don't see how understanding Newtonian mechanics is going to aid students' ability to deliver effective strikes. Students know, without exception, that if they move their striking limb faster they will inflict more damage on the target,
all other things being equal. Why do they need to know about the precise mathematical form of the relationship between the speed of the striking limb and the energy increment at the point of impact, or that acceleration is the time derivative of velocity?? What they need to work on is accuracy, correct form (so that the energy delivered will be maximized because the striking surface is correct and anatomically well-supported by the alignment of skeletal structure), balance (so that the greatest amount of force can be delivered) and so on. My sense is that these kinds of considerations give them more than enough to keep them busy, without having to invoke physical abstractions and mathematical relations that very few of them are in a position to understand...
