Fundamentals of Biomechanics

Let's consider a quick example of using
these facts before we examine the implications
for the best angles of projecting objects.
Great jumpers in the National
Basketball Association like Michael Jordan
or David Thompson are credited with
standing vertical jumps about twice as high
(1.02 m or 40 inches) as typical college
males. This outstanding jumping ability is
not an exaggeration (Krug & LeVeau, 1999).
Given that the vertical velocity is zero at the
peak of the jump and the jump height, we
can calculate the takeoff velocity of our elite
jumper by applying VAS. Solving for V i in
the equation:
V f 2 = Vi 2 + 2ad
0 = V i 2 + 2(–9.81)(1.002)
V i = 4.47 m/s or 9.99 mph
We select the velocity to be positive when
taking the square root because the initial
velocity is opposite to gravity, which acts in
the negative direction. If we wanted to calculate
his hang time, we could calculate the
time of the fall with SAT and double it because
the time up and time down are equal:
d = V i t + 0.5at 2
–1.02 = 0 + 0.5(–9.81)t 2
t = 0.456 s
So the total fight time is 0.912 seconds.
If you know what your vertical jump is,
you can repeat this process and compare
your takeoff velocity and hang time to that
of elite jumpers. The power of these empirical
relationships is that you can use the
mathematics as models for simulations of
projectiles. If you substitute in reasonable
values for two variables, you get good predictions
of kinematics for any instant in
time. If you wanted to know when a partic-
CHAPTER 5: LINEAR AND ANGULAR KINEMATICS 117
ular height was reached, what two equations
could you use? Could you calculate
how much higher you could jump if you increased
your takeoff velocity by 10%?
So we can see that uniformly accelerated
motion equations can be quite useful in
modeling the vertical kinematics of projectiles.
The final important point about uniformly
accelerated motion, which reinforces
the directional nature of vectors, is
that, once the object is released, the vertical
component of a projectile's velocity is independent
of its horizontal velocity. The extreme
example given in many physics
books is that a bullet dropped the same instant
another is fired horizontally would
strike level ground at the same time. Given
constant gravitational conditions, the
height of release and initial vertical velocity
uniquely determine the time of flight of
the projectile. The range or horizontal distance
the object will travel depends on this
time of flight and the horizontal velocity.
Athletes may increase the distance they can
throw by increasing the height of release
(buying time against gravity), increasing
vertical velocity, and horizontal velocity.
The optimal combination of these depends
on the biomechanics of the movement, not
just the kinematics or trajectory of uniformly
accelerated motion. The next section will
summarize a few general rules that come
from the integration of biomechanical models
and kinematic studies of projectile activities.
These rules are the basis for the Optimal
Projection Principle of biomechanics.
OPTIMAL PROJECTION
PRINCIPLE
For most sports and human movements involving
projectiles, there is a range of angles
that results in best performance. The
Optimal Projection Principle refers to the
angle(s) that an object is projected to
achieve a particular goal. This section will