Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
Figure 4.13 Effect of projection angle on shape of parabolic trajectory for a projection speed of 15 m/s and zero projection
height. Projection angles: 30° – continuous black curve, flight time 1.53 s, range 19.86 m, maximum height 2.87 m;
45° – blue curve, flight time 2.16 s, range 22.94 m, maximum height 5.73 m; 60° – dashed black curve, flight time
2.65 s, range 19.86 m, maximum height 8.60 m.
144
Projection height
The equations relating to projection speed (page 142) have to be modified if the projectile
lands at a height higher or lower than that at which it was released. This is
the case with most sports projectiles, for example in a shot put, a basketball shot or a
long jump. For a given projection speed and angle, the greater the projection
height (y 0), the longer the flight time and the greater the range and maximum height.
The maximum height is the same as above but with the height of release added, as
follows:
2 2
ymax = y0 + v0 sin θ/2g
2 2 tmax = v0 sinθ/g + (v0 sin θ + 2gy0) 1/2 /g
2 2 2 2 1/2 R = v0 sin2θ/2g + v0 cosθ(sin θ + 2gy0v0 ) /g
The equations for the time of flight (t max) and range (R) appear to be much more
complicated than those for zero release height. The first of the two terms in each of
these relates, respectively, to the time and horizontal distance to the apex of the trajectory.
The value of these terms is exactly half of the total values in the earlier equations.
The second terms relate to the time and horizontal distance covered from the apex to
landing. By setting the release height (y 0) to zero, and noting that cosθsinθ = cos2θ/2,
you will find that the second terms in the equations for the time of flight and range
become equal to the first terms and that the equations are then identical to the earlier
ones for which the release height was zero.