Calculus IA: The Math Behind Basketball

Calculus IA: The Math Behind Basketball

Throughout this investigation I wasn’t surprised to learn that there was a great deal of math behind basketball. There’s the angle of shooting, bounce passes, chest passes, jump shots, that all involve a great deal of math and physics. The angle of shooting is crucial to making a basket, but what angle is best? Specifically, which angle is best for each type of shot; jump shot, free throw, three point shot? I’ve been playing basketball since I was five years old, and I’ve always been curious if math could help me with my shot. I will examine how shooting angle, initial velocity of shot, and release height can effect a shot.

If we look at someone shooting a ball at angle Ө at a distance d from the basket

Therefore, if we find the time it takes for the basketball to reach its maximum height and multiply it by 2, it will give us the total time of the ball in the air.

At the top of its path the velocity of the ball in the y-direction is 0 m/s , therefore starting from the top of its path falling downwards its initial velocity will be 0 m/s

Therefore the time it takes for the ball from its peak to reach the ground

Since V is a constant when we take the derivative, it doesn’t matter what the initial velocity of the shot is, the optimum angle will still be 45°. However, our calculation doesn’t account for the fact that the rim is much higher than the average person. Let’s say for example you are 2.05 m tall and from the floor to the top of the rim is 3.05 m, the ball will go through the hoop when y= 1.0 m not when y=0 m, a different angle is need to reach satisfy this.

Let’s say the initial velocity of the shot is 6m/s, which means:

(2V^2)/g = -7.35

Using the equation: d(Ө)=(2v^2)/g • sin Ө cos Ө and since the distance traveled in the y-direction is 1.0 m we get:

1 = 7.35 sin Ө cos Ө

1/7.35 = sin Ө cos Ө

2/7.35 = 2 sin Ө cos Ө Multiply by 2 to use identity

2/7.35= sin