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.....or breaching the swish horizon...
two basketballs will just fit inside a 18" diameter hoop. a basketball dropped from straight above the center of hoop (a right angle to hoop) sees the hoop at it's largest opening as opposed to a basketball that is comming into the hoop at an acute angle. the less the approaching basketball is to 90degrees of the center of the hoop the smaller the opening the basketball will see. the lowest angle 45degrees for which a basketball can approach the hoop and go through the hoop is the angle made that just reachs the 9" diameter of the basketball that is perpendicular to the rim of the hoop on the side of the hoop from which the basketball is approaching. so a basketball that approachs the center of the hoop that achieves an angle of approach at the leading edge of the rim of less than 90degrees to the center of the hoop and more than the least acute angle mentioned must go in the hoop. ideally the approaching basketball should reach the least forward velocity of the fall point of the arc no later than the 90degree point to the center of the hoop or no sooner than its 45degree point at the leading edge of the hoop. (note: the 45 degree angle has its vertex at the center of the hoop as does the 90 degree angle).
so you have 45 degrees to play with from the 90degree point at the center of the hoop and the 45degree point (9" above) of the leading rim.
essenstially if you aim the center of the basketball toward dead center of the hoop and 5 inches (or more) above the leading edge of the rim with a hair more than the minimum velocity it takes to reach that point above the rim then you will have a swish.
you also can be off center in your aim by about 3" or so once the ball reaches the hoop since a basketball has about a 9" diameter and the hoop is 18" in diameter. you will find that your horizontal aim accuracy is more important than the vertical aim accuracy as long a you get 5 inches above the leading edge of the rim but the minimal velocity factor will prove as important as the horizontal aim accuracy is.
according to Bill Walton in his book "The Tao of the Jump Shot" "the first published article on the physics of basketball was made in 1981 by Brooklyn College physics professor Peter Brancasio.
Brancasio made a computer study using the laws of physics to analyze the different trajectories and launching speeds neccessary to make a successful basketball shot. of all the trajectories that result in a successful shot and there are an indefinate number of them - one arc stands out. Brancasio calls this arc the minimum force or minimum speed trajectory. when a shooter is able to launch a shot at this angle the shooter will sense that the shot feels effortless. when the ball is released at minimum force the number of arcs that can result in a successful shot increase dramatically. the corresponding launching speed for this shot (which is the lowest launching speed for the given distance) provides the widest range of launching angles for a successful shot."
cheers tim | 
