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Fig. 3. The four phases of the bounce of the body. Same subjects and speeds as in
Fig. 1. In (A–F) the
trend of the Ep–Ek transduction,
Rint(t) (black), is illustrated with the
simultaneous changes in gravitational potential energy,
Ep, and kinetic energy,
Ek=Ekv+Ekf
(green), normalized to oscillate between zero and one. The colors in the
Ep curve distinguish the fractions of the step where the
vertical force exerted on the ground is greater than body weight (red), and
lower than body weight (blue). The continuous Ep line
indicates the contact phase whereas the dotted Ep line
(light-blue) indicates the aerial phase (not present in A and B). The four
phases correspond to the vertical displacement during the upward acceleration
Sce,up (red) and deceleration Sae,up
(blue), and the downward acceleration Sae,down (blue) and
deceleration Sce,down (red). The vertical dotted lines are
drawn through the two peaks of Ek and encompass the
fraction of the step where the
Ep–Ek transduction occurs, as
indicated by the increment of the Rint(t) curve.
Note that, particularly in the old subject, the transduction of
Ek into Ep during the lift (increment
Rint,up of Rint(t), below
crossing of the interrupted lines) is smaller than the transduction of
Ep into Ek during the downward
displacement (increment Rint,down of
Rint(t), above crossing of the interrupted
lines). This asymmetry is accompanied by a ballistic lift (almost nil in the
old subject) smaller than the ballistic fall. In the horizontal tracts of the
Rint(t) curve no transduction occurs between
Ep and Ek and muscle-tendon units
absorb Ep and Ek simultaneously (phase
β), and increase Ep and Ek
simultaneously (phase  ). Whereas most of β is confined within
Sce,down, usually extends beyond
Sce,up within a large fraction of
Sae,up due to a continuing increase of
Ek. Also this asymmetry is larger in the old subject. In
summary, the following features of the landing–take-off asymmetry of the
bounce of the body are shown: (i) a peak of Ek greater
during the fall than during the lift, (ii) a transduction of
Ep into Ek during the fall greater
than the transduction of Ek into Ep
during the lift (i.e.
Rint,down>Rint,up), and (iii) a
simultaneous decrement of Ek and Ep
after landing shorter than the simultaneous increment of
Ek and Ep before and after take-off
(i.e. β<). All these deviations from an elastic bounce are
larger in the old subject.
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1 because the aerial phase is
often nil (e.g. Fig. 1A,B).
With increasing speed an aerial phase of progressively greater extent takes
place during the step and the fraction of vertical displacement during contact
decreases. The decrement is less during the lift than during the descent (i.e.
during the aerial phase the lift is smaller than the fall). This is
particularly true in the old subjects who maintain contact with the ground for
almost the whole of the lift (see also light-blue dotted lines in
Fig. 3). It must be pointed out
here that heel-strike and toe-off, i.e. start and end of foot contact, do not
properly describe landing and take-off of the bouncing system. As described in
the text, landing and take-off, in a physical sense, coincide with the
instants where vertical force becomes greater than body weight (after heel
strike) and falls below body weight (before toe-off). The correct
landing–take-off asymmetry of the bouncing system is described in
Fig. 3.
13 km h–1
(Table 1) indicating an average
force during the brake greater than during the push, which is qualitatively
consistent with the force–velocity relation of muscle. This suggests
that work done by muscle within the muscle-tendon units at low speeds is
progressively substituted, with increasing speed, with elastic energy storage
and recovery by tendons (see text). On average, the ratio
tpush/tbrake is greater in the old
subjects indicating a less elastic behavior
(Table 2). Asterisks denote
statistically significant difference (P<0.05). Other indications
as in Fig. 2.