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.

Ken

Implications:

(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".

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Physics over-simplified:

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

where

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

where

2 = two arm-swings in each total stroke-cycle.

f = frequency of total stroke-cycles (with both legs making their push).

Notes:

(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

Implications.

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