Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
142
shorthand, I do not want to focus on the mathematical derivation of these equations,
only on their importance.
There are three parameters, in addition to gravitational acceleration, g, that determine
the trajectory of a simple projectile, such as a ball, shot or hammer. These are the
projection speed, angle and height (Figure 4.12). For thrown objects, these three
parameters are often called release speed, angle and height; for humans, the terms takeoff
speed, angle and height are more common. I use projection speed, angle and height
to cover both types of ‘projectile’, objects and humans, and cover these three parameters
in decreasing order of importance.
Projection speed
Projection speed (v 0) is defined as the speed of the projectile at the instant of release
or take-off (Figure 4.12). When the projection angle and height are held constant,
the projection speed will determine the maximum height the projectile reaches (its
apex) and its range, the horizontal distance it travels. The greater the projection speed,
the greater the apex and range. It is common practice to resolve a projectile’s velocity
vector into its horizontal and vertical components and then to analyse these independently.
Horizontally a projectile is not subject to any external forces, as we are
ignoring air resistance, and will therefore have a constant horizontal velocity while in
the air, as in a long jump or swimming start dive. The range (R) travelled by a projectile
is the product of its horizontal projection velocity (v x0 = v cosθ) and its time of flight
(t max). That is:
R = v x0 t max
To calculate a projectile’s time of flight, we must consider the magnitude of the vertical
component of its projection velocity (v y). Vertically a projectile is subject to a constant
acceleration due to gravity (g). The magnitude of the maximum vertical displacement
(y max), flight time (t max) and range (R) achieved by a projectile can easily be determined
from v y0 if it takes off and lands at the same level (y 0 = 0). This occurs, for example, in a
football kick. In this case, the results are as follows:
2 2 2
ymax = vy0 /2g = v0 sin θ/2g
tmax =2vy0/g = 2v0 sinθ/g
2 2 R = 2v0 sinθ cosθ/g = v0 sin2θ/g
The range (R) equation shows that the projection speed is by far the most important of
the projection parameters in determining the range achieved, because the range is
proportional to the square of the release speed. Doubling the release speed would
increase the range four-fold.