Khaled Metrek Alajmi/11 orange

1. Collect Data:

-Measure or estimate the initial height of the basketball from the

ground (for example, the player's release height).

-Estimate the horizontal distance and maximum height reached

by the ball during the shot.

-Assume a smooth, parabolic trajectory from release to basket.

To create our basketball trajectory model, we need to gather three essential

measurements and follow one key assumption:

A. Initial Height (c = 2.2 meters)

 Definition: The height at which the basketball is released from the

player's hands

 How to Measure: Calculate the height from ground level to the point of
release

 Example Value: 2.2 meters (based on a player releasing the ball above
their head)

B. Horizontal Distance (x = 4.6 meters)

 Definition: The distance between the release point and the basketball
hoop

 How to Measure: Measure the straight-line distance from shooting

position to basket

 Example Value: 4.6 meters (standard free throw line distance)

C. Maximum Height (h = 3.5 meters)

 Definition: The highest point the basketball reaches during its flight

 How to Measure: Observe and estimate the peak height of the ball's
path

 Example Value: 3.5 meters (peak height during trajectory)

D. Parabolic Trajectory Assumption

 The ball follows a smooth, curved path

 The trajectory forms a perfect parabola

Data Summary Table:

Measurement Symbol Value

Initial Height c 2.2 meters

Horizontal Distance x 4.6 meters

 Maximum Height h 3.5 meters

2. Define the Function:

Using your data, set up a quadratic function of the form y = ax +

bx + e to model the basketball's trajectory. Here:

-y represents the height of the ball at a given horizontal position

x.

- is the initial height of the ball.

-Solve for the coefficients a and b based on maximum height and

distance to the basket

define the quadratic function for the basketball trajectory using our collected data:

Step 1: Understanding the Quadratic Function Form Our function will be y = ax² + bx + c, where:

 y is the height of the basketball

 x is the horizontal distance
 c is our initial height (2.2 meters)
 We need to find a and b using our data

Step 2: Using Our Known Points We have three key points for our parabola:

1. Starting point (0, 2.2): When x = 0, y = initial height

2. Maximum point (2.3, 3.5): Halfway through the distance
3. Endpoint (4.6, 3.05): At the basket

Step 3: Finding Coefficient 'a' To find 'a', we can use the vertex form of a parabola: y = a(x - h)² + k
where (h,k) is our maximum point (2.3, 3.5)

a = (y - k)/(x - h)² a = (2.2 - 3.5)/(0 - 2.3)² = -0.25

Step 4: Finding Coefficient 'b' Using our starting point: 2.2 = a(0)² + b(0) + 2.2 b = 1.15

Final Function Therefore, our quadratic function is: y = -0.25x² + 1.15x + 2.2

This function will create a parabola that:

 Starts at our initial height (2.2 meters)

 Reaches maximum height (3.5 meters) halfway through
 Ends at basket height (3.05 meters)
3. Graph the Function:

-Plot the quadratic function on graph paper or use graphing

software.

-Label the vertex (maximum point), x-intercepts (where the ball

hits the ground), and y-intercept (initial height).

-Draw the axis of symmetry for the parabola.

1. Y-intercept (Initial Height) The starting point of our basketball trajectory:

 Point: (0, 2.2)

 This represents where the player releases the ball
 The height of 2.2 meters matches our initial measurement

2. Vertex (Maximum Height) The peak of our parabola, calculated from our quadratic function:

 Point: (3, 4.45)

 This is the highest point the basketball reaches
 Found using vertex formula: x = -b/(2a) = -1.5/(2(-0.25)) = 3
 When x = 3, y = 4.45 meters

3. Endpoint Where the ball meets the basketball hoop:

 Point: (6, 3.05)

 Represents where the basketball should enter the hoop
 Standard basketball hoop height is 3.05 meters

4. Axis of Symmetry The vertical line that divides our parabola into mirror images:

 Located at x = 3 meters
 Passes through our vertex point (3, 4.45)
 Creates a perfectly symmetrical path on either side

The equation produces a smooth parabolic arc that:

 Starts at our initial release height of 2.2 meters

 Rises to a maximum height of 4.45 meters at x = 3
 Descends to meet the basketball hoop at 3.05 meters
4. Analyze the Graph:

-Explain how each part of the graph relates to the basketball's

flight path.

-Discuss how changes in the coefficients a, b and e would alter

the shot's path (e.g., making the shot higher, shorter, or longer).

Part 1: Understanding Each Part of the Graph

Our parabolic curve tells a complete story of the basketball's journey:

Starting Point (0, 2.2): This represents the release point where the player shoots
the ball. At 2.2 meters, this height makes sense for a typical player releasing the
ball above their head.

Rising Section (0 to 3 meters): The ball travels upward as both the forward
momentum and upward force work together. The curve's steepness shows how
quickly the ball gains height, influenced by the initial shooting force.

Maximum Height (3, 4.45): At this vertex, the ball reaches its peak height of 4.45
meters. This happens at x = 3 meters, exactly halfway through the ball's journey.
At this moment, the vertical velocity becomes zero before gravity pulls the ball
downward.

Descending Section (3 to 6 meters): The ball falls under gravity's influence,

creating a mirror image of the rising section. The symmetrical nature of our
parabola shows how the ball loses height at the same rate it gained it.

Endpoint (6, 3.05): The ball reaches the basket at regulation height (3.05
meters). The graph shows how the ball approaches the hoop at a good angle for
scoring.

Part 2: Impact of Changing Coefficients

Let's examine how changing each coefficient affects the shot:

Coefficient 'a' (-0.25):

 Making 'a' more negative (e.g., -0.35): Creates a narrower arc; the ball
rises and falls more quickly
  Making 'a' less negative (e.g., -0.15): Creates a wider arc; the ball travels
in a higher, more gradual path

 This coefficient controls how "tight" or "loose" the parabola is

Coefficient 'b' (1.5):

 Increasing 'b': Shifts the peak point to the right; the ball travels further
before reaching maximum height

 Decreasing 'b': Shifts the peak point to the left; the ball reaches maximum
height sooner

 This affects the forward momentum of the shot

Initial Height 'c' (2.2):

 Increasing 'c': Raises the entire path; useful for taller players

 Decreasing 'c': Lowers the entire path; represents a lower release point

 This reflects different player heights or shooting styles

These adjustments might be necessary for:

 Different shooting distances (free throw vs. three-point shot)

 Various player heights and shooting styles

 Compensating for defenders between the player and basket

 Different game situations requiring higher or flatter arcs