PROJECTILE MOTION
Basketball is a game full of passing and shooting. When a player passes or shoots a basketball the ball follows along a parabolic arc.
vertical Yf = Yi + VoT + 1/2gT2
horizontal X = VoT
Yf is the height reached by a ball after some amount of time
Yi is the initial height of the ball leaving a players hand
X is the distance away form the basket or another player
Vo is the velocity of the ball leaving a players hand
T is the amount of time the ball is in the air or the amount of time it takes the ball to travel from a player to the basket or to another player.
Another factor that affects the trajectory of the ball is the angle that the ball leaves the players hand.
With all of these factors known along with the given equations a physicist could determine the velocity and angle that a basketball would have to be shot at to make a shot some distance away from the basket

WHAT IS THE PURPOSE OF BACKSPIN?
The only forces acting on the ball after it leaves the players hand is the force due to the acceleration of gravity and angular momentum (L), backspin on the basketball.
Fg = mball(g)
L = Iw
Backspin is not placed on a basketball to change its path in the air because the air resistance on the ball is so small that air resistance is negligible so backspin could not physically affect the path of the ball. Instead backspin is placed on the ball so that when the ball comes into contact with the rim the ball will bounce off the rim softer than without spin. This happens because when the spinning ball rubs against the rim there is an opposing frictional force that transfers the energy due to friction into heat. Therefore the ball will have less energy after it hits the rim than before making the rebound of the ball softer. Once the ball hits the rim the backspin on the ball switches direction and the ball bounces off at a lower angle creating a better chance for the ball to go in.

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