Question

Motion of a Golf Ball A golf ball is hit with an initial velocity of 130 feet per second at an inclination of \(45^{\circ}\) to the horizontal. In physics, it is established that the height \(h\) of the golf ball is given by the function $$ h(x)=\frac{-32 x^{2}}{130^{2}}+x $$ where \(x\) is the horizontal distance that the golf ball has traveled. (a) Determine the height of the golf ball after it has traveled 100 feet. (b) What is the height after it has traveled 300 feet? (c) What is \(h(500) ?\) Interpret this value. (d) How far was the golf ball hit? (e) Use a graphing utility to graph the function \(h=h(x)\). (f) Use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 90 feet. (g) Create a TABLE with TblStart \(=0\) and \(\Delta \mathrm{Tbl}=25 .\) To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height? (h) Adjust the value of \(\Delta\) Tbl until you determine the distance, to within 1 foot, that the ball travels before it reaches its maximum height.

Short answer

The height at 100 feet is approximately 75.92 feet, at 300 feet is -101.76 feet, and at 500 feet is -418.66 feet. The maximum height is at 130 feet after traveling around 163.12 feet.

Step-by-step solution

- Determine height after 100 feet

Substitute 100 for x in the equation \[ h(x) = \frac{-32 x^{2}}{130^{2}}+x \] So, \[ h(100) = \frac{-32 \times 100^{2}}{130^{2}} + 100 \] Calculate this value to find the height after the golf ball has traveled 100 feet.

- Height after 300 feet

Substitute 300 for x in the equation \[ h(x) = \frac{-32 x^{2}}{130^{2}}+x \] So, \[ h(300) = \frac{-32 \times 300^{2}}{130^{2}} + 300 \] Calculate this value to find the height after the golf ball has traveled 300 feet.

- Determine h(500) and Interpret

Substitute 500 for x in the equation \[ h(x) = \frac{-32 x^{2}}{130^{2}}+x \] So, \[ h(500) = \frac{-32 \times 500^{2}}{130^{2}} + 500 \] Calculate this value and interpret the meaning of the result.

- Find distance golf ball traveled

Set h(x) to 0 and solve for x \[ 0 = \frac{-32 x^{2}}{130^{2}} + x \] Solve for x by factoring out x and setting each factor equal to 0.

- Graph the function

Use a graphing utility (a graphing calculator or software) to plot the function \[ h = h(x) \] This will help visualize the ball's path.

- Determine distance for h = 90 feet

Using the graphing utility, find the x-values where the height h is 90 feet. This requires solving \[ 90 = \frac{-32 x^{2}}{130^{2}} + x \] for x.

- Create a TABLE with TblStart = 0 and ΔTbl = 25

Set TblStart to 0 and ΔTbl to 25 and create a table with these settings in your graphing utility. Use the table to find the distance the golf ball travels before reaching the maximum height.

- Adjust Tbl settings for maximum distance

Adjust the value ΔTbl in the graphing utility to find the exact distance, within 1 foot, that the golf ball travels before reaching its maximum height. Note the maximum height and corresponding distance.

Key concepts

quadratic functions

Quadratic functions are polynomial functions of degree 2, described by the general formula \( ax^2 + bx + c \). In our exercise, the height \(h\) of the golf ball is given by the quadratic function: \[ h(x)=\frac{-32 x^{2}}{130^{2}}+x \]. Here, the term \( \frac{-32 x^2}{130^2} \) represents the downward force of gravity, while the term \( x \) represents the linear component of the ball's horizontal motion. The combination of these terms creates a parabolic curve. Key properties of quadratic functions include:

- **Vertex**: The highest or lowest point of the parabola, depending on its orientation.
- **Axis of Symmetry**: A vertical line that passes through the vertex, splitting the parabola into two mirror images.
- **Intercepts**: Points where the graph crosses the axes. For instance, the x-intercept occurs where \( h(x) = 0 \).

To understand the motion of the golf ball, we can determine its height at different distances by substituting values for \( x \) and solving the equation. This parabola opens downward, indicating it has a maximum point, which fact is crucial for finding the height and the distance the ball travels.

projectile motion

Projectile motion describes the trajectory of an object launched into the air, following a curved path under the influence of gravity. It combines horizontal motion with vertical free fall, resulting in a parabolic trajectory, explained by our quadratic function.

Key aspects of projectile motion include:

- **Initial Velocity**: The initial speed and direction at which the object is launched. In our example, the golf ball is hit at 130 feet per second at an angle of 45 degrees.
- **Angle of Projection**: Determines the shape and distance of the trajectory. Our function reflects this through its components.
- **Range**: The horizontal distance traveled before the projectile returns to the same height. Determining the range involves finding the x-value where the height \( h(x) = 0 \). This tells us the total distance the golf ball travels before hitting the ground again.

By analyzing the height function of the golf ball, students can visualize how these concepts interact and affect the ball's movement.

problem-solving in algebra

Problem-solving in algebra involves using mathematical principles to find unknown values. Our problem requires evaluating the height function at different points and interpreting the results.

Steps to solve such problems:

- **Substitute values into the function**: Replace \( x \) with the given distance (e.g., 100 feet) and solve for \( h(x) \).

- **Simplify and compute**: Perform the arithmetic operations to find the height.

- **Interpret results**: Understand what the calculated heights represent in the context of the problem.

Particularly, when the distance results in a height of zero, we can conclude that the ball has landed. For instance, solving \( 0 = \frac{-32 x^2}{130^2} + x \) determines the total distance traveled by the ball. Similarly, analyzing at which point the ball reaches a specified height (e.g., 90 feet) involves solving the equation for \( x \). Through these steps, algebra helps break down complex physical phenomena into manageable mathematical tasks.

graphing utilities

Graphing utilities, such as graphing calculators and computer software, are powerful tools for visualizing functions and solving equations. They allow students to:

- **Plot functions**: Input the height function \( h(x)=\frac{-32 x^{2}}{130^{2}}+x \) to see its parabolic graph.

- **Identify key points**: Use tools to find where the function intersects the axes and reaches its vertex.

- **Generate tables**: Create tables of values to understand the function's behavior at various points (e.g., TblStart = 0 and \( \Delta\mathrm{Tbl} = 25 \)).

For instance, setting the graphing utility to create a table with a start value (TblStart) at 0 and increments (\( \Delta Tbl \)) of 25 feet helps determine the distance at which the golf ball reaches its peak height. By fine-tuning the table settings, students can pinpoint the exact distance to within one foot. Graphing utilities thus offer a dynamic means to explore mathematical concepts and their real-world applications.