## 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 outline some general rules for optimal projections that can be easily applied by coaches and teachers. These optimal angles are "rules of thumb" that are consistent with the biomechanical research on projectiles. Finding true optimal angles of projection requires integration of descriptive studies of athletes at all ability levels (e.g., Bartlett, Muller, Lindinger, Brunner, & Morriss, 1996), the laws of physics (like uniformly accelerated motion and the effects of air resistance; see chapter 8), and modeling studies that incorporate the biomechanical effects of various release parameters. Determining an exact optimal angle of projection for the unique characteristics of a particular athlete and in a particular environment has been documented using a combination of experimental data and modeling (Hubbard, de Mestre, & Scott, 2001). There will be some general trends or rules for teaching and coaching projectile events where biomechanical research has shown that certain factors dominate the response of the situation and favor certain release angles.

In most instances, a two-dimensional point-mass model of a projectile is used to describe the compromise between the height of release and the vertical and horizontal components of release velocity (Figure 5.7). If a ball was kicked and then landed at the same height, and the air resistance was negligible, the optimal angle of projection for producing maximum horizontal displacement would be 45°. Forty-five degrees above the horizontal is the perfect mix of horizontal and vertical velocity to maximize horizontal displacement. Angles above 45° create shorter ranges because the extra flight time from larger vertical velocity cannot overcome the loss in horizontal velocity. Angles smaller than 45° cause loss of flight time (lower vertical velocity) that cannot be overcome by the larger horizontal velocity. Try the activity below to explore optimal angles of projection.

Figure 5.7. The three variables that determine the release parameters of a projectile in two-dimensions: height of release, and the horizontal and vertical velocities of release.

Activity:Angles of Projection

Use a garden hose to water the grass and try various angles of projection of the wa-ter.The air resistance on the water should be small if you do not try to project the water too far. Experiment and find the angle that maximizes the distance the water is thrown. First see if the optimal angle is about 45°, when the water falls back to the height that it comes out of the hose. What happens to the optimal angle during long-distance sprinkling as the height of release increases?

Note how the optimal angle of projection changes from 45° when the height of release is above and below the target.

Before we can look at generalizations about how these factors interact to apply the Optimal Projection Principle, the various goals of projections must be analyzed. The mechanical objectives of projectiles are displacement, speed, and a combination of displacement and speed. The goal of an archer is accuracy in displacing an arrow to the target. The basketball shooter in Figure 5.7 strives for the right mix of ball speed and displacement to score. A soccer goalie punting the ball out of trouble in his end of the field focuses on ball speed rather than kicking the ball to a particular location.

When projectile displacement or accuracy is the most important factor, the range of optimal angles of projection is small. In tennis, for example, Brody (1987) has shown that the vertical angle of projection (angular "window" for a serve going in) depends on many factors but is usually less than 4°. The goal of a tennis serve is the right combination of displacement and ball speed, but traditionally the sport and its statistics have emphasized the importance of consistency (accuracy) so as to keep the opponent guessing. In a tennis serve the height of projection above the target, the net barrier, the spin on the ball, the objective of serving deep into the service box, and other factors favor angles of projection at or above the horizontal (Elliott, 1983). Elite servers can hit high-speed serves 3° below the horizontal, but the optimal serving angle for the majority of players is between 0 and 15° above the horizontal (Elliott, 1983; Owens & Lee, 1969).

This leads us to our first generalization of the Optimal Projection Principle. In most throwing or striking events, when a mix of maximum horizontal speed and displacement are of interest, the optimal angle of projection tends to be below 45°. The higher point of release and dramatic effect of air resistance on most sport balls makes lower angles of release more effective. Coaches observing softball or baseball players throwing should look for initial angles of release between 28 and 40° above the horizontal (Dowell, 1978). Coaches should be able to detect the initial angle of a throw by comparing the initial flight of the ball with a visual estimate of 45° angle (Figure 5.8). Note that there is a larger range of optimal or desirable angles that must accommodate differences in the performer and the situation. Increasing the height of release (a tall player) will tend to shift the optimal angle downward in the range of angles, while higher speeds of release (gifted players) will allow higher angles in the range to be effectively used. What do you think would happen to the optimal angles of release of a javelin given the height of release and speed of approach differences of an L5-disabled athlete compared to an able-bodied athlete?

There are a few exceptions to this generalization, which usually occur due to the special environmental or biomechanical conditions of an event. In long jumping, for example, the short duration of takeoff on the board limits the development of vertical velocity, so that takeoff angles are usually between 18 and 23° (Hay, Miller, & Can-terna, 1986; Linthorne et al., 2005). In the standing long jump, jumpers prefer slightly higher takeoff angles with relatively small decreases in performance (Wakai & Linthorne, 2005). We will see in chapter 8 that the effect of air resistance can quickly become dominant on the optimal release parameters for many activities. In football place-kicking, the lower-than-45° generalization applies (optimal angles are usually between 25 and 35°), but the efficient way the ball can be punted and the tactical importance of time during a punt make the optimal angle of release about 50°. With the wind at the punter's back he might kick above 50°, while using a flatter kick against a wind. The backspin put on various golf shots is another example of variations in the angle of release because of the desirable effects of spin on fluid forces and the

Figure 5.8. Coaches can visually estimate the initial path of thrown balls to check for optimal projection. The initial path of the ball can be estimated relative to an imaginary 45° angle. When throwing for distance, many small children select very high angles of release that do not maximize the distance of the throw.

Figure 5.8. Coaches can visually estimate the initial path of thrown balls to check for optimal projection. The initial path of the ball can be estimated relative to an imaginary 45° angle. When throwing for distance, many small children select very high angles of release that do not maximize the distance of the throw.

bounce of the ball. Most long-distance clubs have low pitches, which agrees with our principle of a low angle of release, but a golfer might choose a club with more loft in situations where he wants higher trajectory and spin rate to keep a ball on the green.

The next generalization relates to projectiles with the goal of upward displacement from the height of release. The optimal angle of projection for tasks emphasizing displacement or a mix of vertical displacement and speed tends to be above 45°. Examples of these movements are the high jump and basketball shooting. Most basketball players (not the giants of the NBA) release a jump shot below the position of the basket. Considerable research has shown that the optimal angle of projection for basketball shots is between 49 and 55° (see Knudson, 1993). This angle generally corresponds to the arc where the minimum speed may be put on the ball to reach the goal, which is consistent with a high-accuracy task. Ironically, a common error of beginning shooters is to use a very flat trajectory that requires greater ball speed and may not even permit an angle of entry so that the ball can pass cleanly through the hoop! Coaches that can identify appropriate shot trajectories can help players improve more quickly (Figure 5.9). The optimal angles of release in basketball are clearly not "high-arc" shots, but are slightly greater than 45° and match the typical shooting conditions in recreational basketball.

The optimal angle of projection principle involves several generalizations about

Figure 5.9. The optimal projection angles for most basketball jump shots are between 49 and 55° above the horizontal (hatched). These initial trajectories represent the right mix of low ball speed and a good angle of entry into the hoop. Novice shooters (N) often choose a low angle of release. Skilled shooters (S) really do not shoot with high arcs, but with initial trajectories that are in the optimal range and tailored to the conditions of the particular shot.

Figure 5.9. The optimal projection angles for most basketball jump shots are between 49 and 55° above the horizontal (hatched). These initial trajectories represent the right mix of low ball speed and a good angle of entry into the hoop. Novice shooters (N) often choose a low angle of release. Skilled shooters (S) really do not shoot with high arcs, but with initial trajectories that are in the optimal range and tailored to the conditions of the particular shot.

desirable initial angles of projection. These general rules are likely to be effective for most performers. Care must be taken in applying these principles in special populations. The biomechanical characteristics of elite (international caliber) athletes or wheelchair athletes are likely to affect the optimal angle of projection. Kinesiology professionals should be aware that biome-chanical and environmental factors interact to affect the optimal angle of projection. For example, a stronger athlete might use an angle of release slightly lower than expected but which is close to optimal for her. Her extra strength allows her to release the implement at a higher point without losing projectile speed so that she is able to use a slightly lower angle of release. Professionals coaching projectile sports must keep up on the biomechanical research related to optimal conditions for their athletes.

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### Responses

• osman
What is the principle of optimal projection?
7 years ago
• mewael
How the do fluid forces affect the optimal projection angles?
4 months ago