Free-throw Shots According to physicist Peter Brancazio, the...

Free-throw Shots According to physicist Peter Brancazio, the key to a successful foul shot in basketball lies in the arc of the shot. Brancazio determined the optimal angle of the arc from the free-throw line to be 45 degrees. The arc also depends on the velocity with which the ball is shot. If a player shoots a foul shot, releasing the ball at a 45 -degree angle from a position 6 feet above the floor, then the path of the ball can be modeled by the function $$ h(x)=-\frac{44 x^{2}}{v^{2}}+x+6 $$ where $h$ is the height of the ball above the floor, $x$ is the forward distance of the ball in front of the foul line, and $v$ is the initial velocity with which the ball is shot in feet per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second. (a) Determine the height of the ball after it has traveled 8 feet in front of the foul line. (b) Determine $h(12) .$ What does this value represent? (c) Find additional points, and graph the path of the basketball. (d) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go moh the hoon

Free-throw Shots According to physicist Peter Brancazio, the key to a successful foul shot in basketball lies in the arc of the shot. Brancazio determined the optimal angle of the arc from the free-throw line to be 45 degrees. The arc also depends on the velocity with which the ball is shot. If a player shoots a foul shot, releasing the ball at a 45 -degree angle from a position 6 feet above the floor, then the path of the ball can be modeled by the function
$$
h(x)=-\frac{44 x^{2}}{v^{2}}+x+6
$$
where $h$ is the height of the ball above the floor, $x$ is the forward distance of the ball in front of the foul line, and $v$ is the initial velocity with which the ball is shot in feet per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second.
(a) Determine the height of the ball after it has traveled 8 feet in front of the foul line.
(b) Determine $h(12) .$ What does this value represent?
(c) Find additional points, and graph the path of the basketball.
(d) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go moh the hoon

Key Concepts

Quadratic Functions

Quadratic functions are used to model the relationship between the horizontal displacement and vertical height in projectile motion. These functions capture the effects of gravity on the projectile, producing a parabolic curve that can be analyzed to determine specific points along the trajectory. The use of quadratic equations is a powerful mathematical tool in predicting and optimizing projectile paths.

Graphical Analysis

Graphical analysis involves plotting the calculated points of a projectile’s flight path to visually represent its trajectory. By graphing the quadratic function describing the motion, one can better understand the behavior of the projectile, verify analytical results, and make adjustments to optimize performance in practical applications such as ensuring that a basketball follows the correct arc to reach a hoop.

Initial Velocity

Initial velocity is the speed and direction at which a projectile is launched. It directly influences the maximum height reached and the distance covered by the projectile. In the context of free throws, adjusting the initial velocity allows one to control the trajectory of the ball to meet specific objectives, such as clearing a target or reaching a desired height.

Optimal Launch Angle

The launch angle is a critical parameter in projectile motion that determines the shape and reach of the trajectory. An angle of 45 degrees is often considered optimal for maximizing horizontal range in a vacuum, as it balances the vertical and horizontal components of the initial velocity. This concept is integral when analyzing techniques for achieving precision in tasks like free-throw shots in basketball.

Parabolic Trajectory

The path of a projectile under the influence of gravity, when no other forces such as air resistance act on it, forms a parabola. This parabolic trajectory can be described mathematically by quadratic equations in terms of time or horizontal distance. Understanding the parabolic nature of the trajectory is crucial for predicting where and when a projectile will reach certain heights or distances.

Projectile Motion

Projectile motion is the study of objects thrown into space, where they move under the influence of gravity. This concept involves decomposing the motion into horizontal and vertical components, with the horizontal motion occurring at constant velocity and the vertical motion undergoing constant acceleration due to gravity. It forms the fundamental basis for analyzing the curved path of projectiles like basketballs in free throws.

Transcript

00:01 In this problem, we are going to use functions and the graph of functions to investigate the path of a basketball.

00:10 So our equation is representing the path of our basketball where h is the height.

00:19 So since this is h of x, our h is going to be the y equals.

00:23 So this is going to be an h equals.

00:27 And v is the initial velocity that the ball is shot at, and x is the forward distance from the line.

00:36 So that's going to be our x here and here.

00:39 So first, we're going to substitute that v in to get our equation that we'll be working with, and we'll have h of x is equal to negative 44x squared plus x plus 6, sorry, the negative 44x squared is going to have a 28 squared under it.

01:07 So that's our equation that we'll be working with.

01:10 So the first question is, what is h when x is 8? so remember that x is 8 means it's shot 8 distance in front of the line.

01:24 So what's the height when it's shot 8 distance in front of the line? so we're doing h of 8 is equal to negative 44 times 8 squared over 28 squared plus 8 plus 6.

01:43 And let's grab a calculator for that one.

01:53 So we have negative 44 times 8 squared.

01:58 So that's 64.

02:01 And let's get that in parentheses to make this accurate.

02:05 Negative 44 times 8 squared which is 64 divided by 28 squared plus 8 plus 6 so when 8 when x is 8 or h would be about 10 .4 so if you shoot the ball 8 feet in front of the line then the height of the ball would be at about 10 .4.

02:44 4 feet high.

02:50 So that would be h equals 10 .4.

02:59 Okay, next, h of 12.

03:05 H of 12 means let's shoot that 12 feet in front of the line.

03:11 I don't want to do that there.

03:12 So h of 12 would be equal to negative 44 times 12 squared over 28 squared plus 8 plus 6 oh, plus 12 plus 6, sorry.

03:32 And let's go to our calculator.

03:36 I'm going to change that 64, negative 44 times 12 squared.

03:45 12 squared would be 144 plus 12 plus 6.

03:59 And that would be at about 9 .9.

04:10 Next, we are going to take a look at what the graph of this function looks like.

04:16 And we're just going to look at the first quadrant.