Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
188
By similar reasoning, if you are prepared to accept that displacement of the jumper’s
centre of mass equals the area between the horizontal zero-velocity line – the time axis
of the graph – and the velocity curve from the start of the movement at 0 up to any
particular time, and, again, that areas below the time axis are negative and those above
positive, then several key points on the displacement–time graph follow.
At time B in Figure 5.12(b), the area (−A 4) under the time axis and above the vertical
velocity–time curve from 0 to B reaches its greatest negative value, so the vertical
displacement of the jumper’s centre of mass also reaches its greatest negative value
there, corresponding to a zero velocity. We can then sketch the displacement graph
in Figure 5.12(c) up to B. Please note that this is the lowest point reached by the
jumper’s centre of mass at full hip and knee flexion, before the jumper starts to rise.
At time T in Figure 5.12(b), the area (A 5) under the velocity–time curve from B to T
and above the time axis is positive so the vertical displacement becomes less negative.
Now we can sketch the vertical displacement curve from B to T in Figure
5.12(c). Note that the maximum displacement will occur after the person has left
the force plate at the peak of the jump.
Note again, as in Chapter 2, the trend of peaks or, more obviously in this case, troughs
is acceleration then velocity then displacement.
Quantitative evaluation of a force–time or acceleration–time pattern
Our volleyball coach may wish to obtain, from this acceleration–time curve, values for
the magnitude of the vertical velocity at take-off and the maximum height reached by
the player’s centre of mass. The process of obtaining velocities and displacements from
accelerations qualitatively was outlined in the previous section. The quantitative process
for doing the same thing is referred to, mathematically, as integration; it can be performed
graphically or numerically. If quantitative acceleration data are available – Figure
5.12(a) with the numbers shown on the axes if you like – we can integrate the
acceleration data to obtain the velocity–time graph, as in Figure 5.12(b), which can, in
turn, be integrated to give the displacement–time graph of Figure 5.12(c). Most quantitative
ways of doing these integrations basically involve determining areas as above, but
for increasing, small time intervals from left to right. A very slow but accurate way of
doing this is counting areas under the curves drawn on graph paper. The resulting
graphs, but with numbered axes as in Figure 5.12, would be more accurate than the
‘sketched’ qualitative ones but the shapes should be very similar if the qualitative analyst
understood well the process of obtaining the velocity and displacement patterns. Our
volleyball coach could now simply read the vertical take-off velocity (v t) for each player
from graphs such as Figure 5.12(b) at take-off, T (see Study task 2). We can calculate
the maximum height (h) reached by a player’s centre of mass by equating his or her
take-off kinetic energy, ½ m v t 2 , to the potential energy at the peak of the jump, m g h,
where m is the player’s mass and g is gravitational acceleration, so h = v t 2 /(2g).