Topic: Arc of Ball to make Basket

Grade Level :11 -12

Materials : Any ten minute video on players making different shots from different positions on the basketball court. (Overhead slide with the side views of different paths which the ball travels to make a score.)

Objective : To inform each student about the mathematical relationship in scoring a basket.

Mathematical Concepts :Vectors and projectile motion.

NCTM Standards : Standard 7 "Geometry from a synthetic perspective"
Standard 13 "Conceptual underpinnings of calculus"

Procedures : Get the class in the mood to think about the arc of a ball to score a basket by showing them the above mentioned 10 minute video tape on scoring. Tell the class to observe that the displacement of the ball is just a combination of vertical and horizontal displacements. And the velocity of the ball at anytime is also a combination of vertical component of the velocity and the horizontal component of the velocity. Then remind the class of how to combine vectors, (which they should have alr eady learned and know from earlier grade levels.) Inform the class that the motion of the ball as it leaves the players hands and travel to the basket is call projectile motion. Then tell the class that during projectile motion the only acceleration on the ball is due to gravity and it is vertical. (Tell them to note and always remember that there is no horizontal acceleration on the ball). Then give them an expression (or formula) for the vertical or horizontal displacement of the ball, and write it on the board.

This expression is X = Vot + (1/2)(at2) where
X - represents the vertical or horizontal displacement
Vo - represents the initial vertical or horizontal velocity
a - represents the acceleration of the ball
t - represents the time taken to make the displacement

Then show the class that the first derivative of the displacement with respect to time gives the velocity at such time. Also the second derivative of the displacement with respect to time gives the acceleration of the ball.
Then write the data which I have below on the board :

FOR VERTICAL MOTION

displacement (X) = Vot + (1/2)at2
dx/dt (velocity) = Vo + at
d2x/dt2 (acceleration) = a

FOR HORIZONTAL MOTION

displacement X=Vot+(1/2)at2=Vot {since no acceleration (a=0)}
dx/dt (velocity) = Vo + at = Vo {since a=0}
d2x/dt2 (acceleration) = a = 0 {since a=0}

Therefore if the time of travel of the ball is known and the projected velocity is known, a person could calculate the distance one would need to be at to make a basket. Now one must take into consideration that the player must jump exactly to the height of the rim to make a perfect score with the given data above. Then from all the data given ask the class what velocity must the player give the ball if he was 3.0 meters away from the net, and when jumps, the ball is at the same height of the rim. Als o the angle of projection is 450. Then after about continued 10 minutes I would ask the class to tell me their answers so I can get a range of possible answers. Then, together with the class I would solve the problem and point out where the class may h ave gone wrong.

Then for homework I would give the following questiion #55, below.

References : General College Physics Text-book page 82.

Contributors : Major - Michael Dixon
Minor - Carol Marinas