Questions answered

QUESTION: When doing free throws, some basketball players have a very low arc and some have a high one. Some like to shoot off the backboard. Has it ever been proven which is best from a mathematical or scientific viewpoint? Also, in baseball, coming off the bat, what angle of the ball gives the most distance?

– Jerry Matten, San Diego

ANSWER: In basketball, the use of the backboard provides about a 50 percent better chance of succeeding for close-in shots (except for very tall players who can slam dunk), according to the paper “Basketball Shooting Strategies,” published in Sports Engineering.

Energy absorbed as the ball bounces off the backboard helps compensate for shooting error. As a player gets farther from the basket, the advantage of the backboard diminishes.

The merits of the overhand push shot compared with the underhand loop shot are still disputed. Underhand shots are more stable and allow a player to put more spin on the ball. Overhand shots decrease the distance to the hoop and minimize the velocity of release.

The optimal angle of release of the basketball depends on a player's position. For instance, in “Basketball Shooting Strategies,” it was determined to be 48 degrees (upward from horizontal) for a player attempting a 3-point jump shot 20 feet from the basket, releasing the ball 8 feet above the floor.

In baseball, mathematical models to determine the optimal batting angle must take into consideration nearly 30 factors. These include physical features of the ball and bat, and the spin, speed and direction of the pitched ball.

The optimal bat swing angle decreases from about 9 degrees (upward from horizontal) to 7 degrees as the pitch changes from fastball with backspin, to knuckle ball with no spin, to curveball with topspin, according to the paper “How to Hit Home Runs,” published in the American Journal of Physics.

Undercutting the ball's center with the bat also helps maximize the ball's range. Optimal undercut is about an inch. Undercutting gives the ball backspin. Because of an aerodynamic lift force, a baseball projected with a backspin will travel farther than one without. However, a baseball can be projected faster without spin. Therefore, optimal batting trades off spin and speed.

Slow curve balls with topspin can be batted farther than fastballs with backspin because a ball with initial topspin has a larger outgoing backspin. But for a given pitch type, batting range increases with pitch speed.



