I've been thinking about the physics of swinging my arms (or
torso-chest-shoulders) from side to side when skating -- how it can help my
forward-motion power. Below is an over-simplified presentation of the
physics, with some implications for traveling farther in less time.
(a) sudden quick acceleration of the arms is good. A critical measure of
upper-body effectiveness is the maximum sideways speed attained, relative to
the hips. Serious racers might consider doing special training of the
muscles and joints doing the swinging: training focused on high acceleration
and velocity (? "plyometric" ?)
(b) longer range of sideways motion is good: the further out to one side I
can hold my arms before starting the sudden quick acceleration, the more
power I get from my arm-swing -- because it gives more distance to attain a
higher maximum velocity.
(c) swinging _both_ arms is better than swinging only one. Adding the move
of the torso - chest - shoulders should help too.
(d) higher frequency turnover of the overall stroke-cycle with the legs
could be helpful.
These implications are all subject to "other things being equal". Of course
often they're not: like at high speeds lots of sideways arm motion might
also increase aerodynamic drag. Like higher-frequency turnover might
increase arm-power, but decrease the (more important) leg-power. The
trade-offs get complicated.
Obvious way to miss out on some extra power: Synchronize arm-swing simply
and smoothly with leg-push. Attaining higher acceleration and velocity (a)
requires that upper-body timing be "de-synchronized" from leg-push.
"smoothly" is another way of saying "lower-acceleration".
Moving portions of the mass of the body above (but not including) the hip
joints can add power to the push thru the skates into the pavement in the
form of "reactive" or "inertial" force, by Newton's Third Law "every action
has an equal and opposite reaction". Even though this force is largely
sideways, the magic of the skate can convert it into forward motion by the
"inclined plane" principle.
The physical Work from swinging the arms (or torse) once, from one side to
the other is:
Work = h * m * a * s
h = efficiency factor (less than 100%) of transmitting this power into
effective forward motion.
m = mass of upper body part being moved.
a = rate of acceleration of this upper body part (assumed to be constant
magnitude, but positive during the first half and negative in the second
half of the motion)
s = total side-to-side distance
The justification for this formula can be seen as Work = Force * Distance,
where Force = m * a, and Distance = s / 2 for the acceleration part, plus
another s / 2 for the negative-acceleration. Or it can be seen as the
Kinetic Energy at maximum Velocity = square root of 2 * a * s -- but with
the benefit received twice, first from acceleration, then from
de-celeration. (But this doubling of the effective work comes only if the
de-celeration is timed when the skate-push is aimed in the opposite
direction from the acceleration. If any de-celeration starts while the
skate-push is still aimed in the original direction, then the reactive force
from it _reduces_ the total effective work.)
Power = 2 * f * h * m * a * s
2 = two arm-swings in each total stroke-cycle.
f = frequency of total stroke-cycles (with both legs making their push).
(1) Interesting that with reactive force sideways you can "win both ways",
and gain both in the acceleration toward one side and in the de-celeration
toward the other side. If you get the timing right.
(2) The physics of the leg-push is different from this ("direct push" force
versus "reactive") -- with different constraints -- and with very different