Katie R.
asked • 10/28/24write down a definition for the function f(x) that gives the height of the ball as a function of its horizontal distance x from the shooter
If a Basketball player is 6.6 feet tall and is throwing from the free throw line (13.75 feet from basket). Basket stands at 10 feet tall and ball is thrown upwards at 18 feet per second. How would I be able to use the equation to find the x-coordinate of the vertex? How could function notation help set up an equation that sets up the perfect free throw shot(not hitting backboard or rim?
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1 Expert Answer
Maxx A. answered • 10/29/24
Tutor
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To begin with, write down what we know given this problem and note what we are looking for.
Given:
Player's height = 6.6 feet (when the ball is released)
Basket Height = 10 feet
Distance from the Free Throw Line to the Basket = 13.75 feet
Initial Vertical Velocity of Ball = 18 feet per second
Initial Horizontal Distance = 0 feet (from position of player) to 13.75 feet (the basket)
Gravity (since we are dealing with a constant force) = -9.8 meters per second squared (convert to feet per second squared = -32 feet per second squared then we take half of this value based on the below equation)
What we want to find:
x-coordinate of the vertex = let's define this as t
Define the function:
Let h(t) represent the height of the ball over time
h(t) = -16t^2 + v_ot + h_o
Where:
-16t^2 is the effect of gravity converted to feet per second squared
v_o is our given velocity in feet per second = 18 feet per second; t is our variable
h_o is our given height = 6.6 feet
Find time where ball reaches basket
We know that the ball reaches the basket at 10 feet high; so our h(t) defined in the function is 10. Now, find t
10 = -16t^2 + 18t + 6.6
-10 -10
-16t^2 + 18t - 3.4 = 0
Use the quadratic formula (use this formula since it will give us our values of t...plus when you have a decimal constant it is impossible to just use standard factorization!)
t = -b +- square root(b^2 - 4ac) / 2a
Where:
a = -16
b = 18
c = -3.4
My favorite thing to say...plug and chug!
t = -18 +- square root(18^2 - 4*-16*-3.4) / 2*-16
t = -18 +- square root(324 - 217.6) / -32
t = 18+-square root(106.4) / -32
Note: we are looking for the highest point...so all we care about in this equation is -b / 2a
So, are t_v = -b / 2a
where: t_v is our time where the ball reaches the highest point
t_v = -18 / 2 * -16
t_v = -18 / -32
t_v approx. = 0.5625 seconds.
So, in order to find our speed it would be x(t) = horizontal speed * t
Where:
x(t) is our x-coordinate or amount in feet needed
horizontal speed is the time in feet/second it reaches the basket
t = 0.5625 seconds or the time it takes to reach basket from the given distance
Brenda D.
tutor
Isn’t that gravity supposed to be 32 feet per second and it gets divided by 2 to give the average of the velocity change since the time the ball or projectile is launched? Just noticing because if you look up the direct conversion of 9.8 meters per second squared it will say 32.17 feet per second squared. It also comes from one of the standard quadratics used for a projectile in motion h = -1/2gt^2 +vt + h0, where g is the acceleration due to gravity. In this typical equation the (-1/2)(32) yields -16. I do agree with your equation just had a question the statement about gravity.
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10/29/24
Maxx A.
Hi Brenda, Yes thanks for that correction! I missed the addition of breaking down the proper conversion to feet per second 😅
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10/29/24
Brenda D.
tutor
You very are welcome. I have to go back and review my own answers and comments because of typos and battles with the autocorrect.
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10/30/24
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Julius N.
To create a function that models the basketball’s height as a function of its horizontal distance \( x \) from the shooter, let's break down the problem into key parts. Step 1: Define Known Values and Constraints 1. Shooter's Height: The basketball player is 6.6 feet tall. So, the ball’s initial height when thrown, \( h_0 \), is 6.6 feet. 2. Basket Height: The hoop is 10 feet tall. 3. Free Throw Line Distance: The free throw line is 13.75 feet from the basket, so we need the ball’s height at \( x = 13.75 \) to be 10 feet. 4. Initial Velocity (Upwards): The ball is thrown with an initial vertical speed of 18 feet per second. Step 2: Setting Up the Function for the Ball’s Height We can model the height of the ball as a function of horizontal distance using a parabolic function in the form: \[ f(x) = ax^2 + bx + c \] However, it’s often easier to think of height in terms of time \( t \) first and then translate it to horizontal distance. Using a Projectile Motion Equation Since we are dealing with projectile motion, let’s use the vertical motion formula: \[ h(t) = -\frac{1}{2}gt^2 + v_0t + h_0 \] where: - \( g \) is the acceleration due to gravity (approximately 32 feet per second squared), - \( v_0 \) is the initial vertical velocity (18 feet per second), - \( h_0 \) is the initial height (6.6 feet). Substituting these values, we get: \[ h(t) = -16t^2 + 18t + 6.6 \] This equation gives us the height of the ball over time \( t \), but we want the height in terms of the horizontal distance \( x \). Relating Horizontal Distance \( x \) to Time \( t \) Assuming the ball is thrown at a constant horizontal speed \( v_x \), the horizontal distance \( x \) can be expressed as: \[ x = v_x t \] Thus, \( t = \frac{x}{v_x} \). We can substitute \( t = \frac{x}{v_x} \) into our height equation \( h(t) \) to express it as a function of \( x \) rather than \( t \). To solve for \( v_x \), we would need the launch angle, which isn't provided. However, we can determine the required shape of the parabola based on the vertex and the final height at \( x = 13.75 \) feet to create a function that ensures the ball reaches the correct height at the basket. Finding the Vertex and Optimizing the Shot 1. Using Function Notation to Set the Height at 13.75 feet: Define \( f(x) \) such that \( f(13.75) = 10 \) and adjust \( f(x) \) based on realistic projectile motion constraints. 2. Determining Vertex for Maximum Height: The vertex \( x \)-coordinate of a parabolic function \( f(x) = ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). This allows us to set up the peak of the shot appropriately. Function notation, \( f(x) = ax^2 + bx + c \), lets us easily substitute \( x \)-values to model the shot height at any point along the free throw line.10/29/24