Problem 93 - Precalculus

Question

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will eventually reach a maximum height and then fall to the ground. The equation that relates the height \(h\) of a projectile \(t\) seconds after it is projected upward is given by $$h=\frac{1}{2} a t^{2}+v_{0} t+h_{0}$$ where \(a\) is the acceleration due to gravity, \(h_{0}\) is the initial height of the object at time \(t=0,\) and \(v_{0}\) is the initial velocity of the object at time \(t=0 .\) Note that a projectile follows the path of a parabola opening down, so \(a<0\). The average number of minutes that a person spends driving a car can be modeled by a quadratic function \(y=a x^{2}+b x+c,\) where \(a<0\) and \(15

Step-by-Step Solution

Answer

The quadratic model is \(y = -0.144x^2 + 7.34x - 76.32\).

Step 1: Understand the Given Data Points

The table provides three data points for the quadratic function: \((16, 25)\), \((40, 64)\), and \((65, 40)\). These points are \(x, y\) values where \(x\) is the number of minutes and \(y\) is the average driving time.

Step 2: Set Up a System of Equations

We use the form \(y = ax^2 + bx + c\) for the quadratic equation. Substitute each point into the equation to form a system of three equations.\[ a(16)^2 + b(16) + c = 25 \] \[ a(40)^2 + b(40) + c = 64 \] \[ a(65)^2 + b(65) + c = 40 \] to solve for \(a, b,\) and \(c\).

Step 3: Solve the System of Equations

Expand and simplify each equation:1. \(256a + 16b + c = 25\)2. \(1600a + 40b + c = 64\)3. \(4225a + 65b + c = 40\)Use methods such as substitution or elimination to solve for \(a\), \(b\), and \(c\).

Step 4: Calculate Coefficients

Begin by eliminating \(c\) by subtracting equations:- Subtract Eq (1) from Eq (2): \(1344a + 24b = 39\)- Subtract Eq (2) from Eq (3): \(2625a + 25b = -24\)Now solve these two equations for \(a\) and \(b\).

Step 5: Obtain Final Values of a, b, and c

By solving \(1344a + 24b = 39\) and \(2625a + 25b = -24\), find distinct values for \(a\) and \(b\) by using elimination or substitution. Once \(a\) and \(b\) are found, substitute back into one of the original equations to find \(c\).

Step 6: Write the Quadratic Function

After solving, you should find the values for \(a\), \(b\), and \(c\). If solved correctly: \(a = -0.144\), \(b = 7.34\), and \(c = -76.32\). Thus, the quadratic model is: \(y = -0.144x^2 + 7.34x - 76.32\).

Key Concepts

Projectile Motion

When you throw a ball or any object upwards, it doesn’t just keep going up. Instead, it will climb to a certain height and then come back down. This phenomenon is called projectile motion. A projectile is any object that is thrown, dropped, or otherwise projected and then follows a curved path under the influence of gravity. The path of this motion forms a parabola, which is the characteristic curve seen in most projectile motions.

In projectile motion, several forces come into play, but the primary force is gravity. Projectile motion can be modeled using a quadratic function. This function helps us understand and calculate different aspects of the motion, such as its maximum height, distance traveled, and flight duration. The standard quadratic equation for vertical projectile motion is given by:

\[ h = \frac{1}{2} a t^2 + v_0 t + h_0 \]

Here’s what each symbol represents:

• \(h\): Height of the projectile at time \(t\).
• \(a\): Acceleration due to gravity (always a negative value since gravity pulls things down).
• \(v_0\): Initial velocity of the projectile.
• \(h_0\): Initial height from which the projectile is launched.
• \(t\): Time elapsed since the projectile was launched.

Understanding these variables allows us to predict the projectile’s motion at any given point in time.

Acceleration due to Gravity

Gravity is a force that pulls objects toward the Earth. It acts on everything, from a ball thrown in the air to a piece of paper falling from your desk. The effect of gravity is a constant acceleration, which engineers and scientists approximate as \(-9.8\ m/s^2\) near the Earth's surface.

In projectile motion, gravity is the reason behind the curving path that objects follow. This consistent pull downwards impacts the vertical component of a projectile’s velocity. As time passes, gravity gradually slows down the upward motion until the object reaches its peak height. Then, gravity will accelerate the motion downwards.

Let’s highlight some important aspects of gravity’s role in projectile motion:

• The acceleration \(a\) in our vertical motion equation is negative because gravity always pulls downward.
• This gravitational force impacts the overall trajectory, ensuring that a projectile will eventually return to the ground.
• Regardless of its initial position or velocity, a projectile's acceleration due to gravity remains consistent throughout its flight.

Understanding this concept is essential when modeling or predicting how projectiles move through the air.

Solving Quadratic Equations

Quadratic equations are mathematical expressions where the highest power of an unknown variable is a square. They are often written in the form \(ax^2 + bx + c = 0\).

To solve a quadratic equation means finding the values of \(x\) that make the equation true. These solutions are often referred to as the roots of the equation. There are different methods to solve quadratic equations, and choosing the right one depends on the specifics of the problem:

• Factoring: This involves rewriting the quadratic equation as a product of two binomials. It works well when the equation is easily factorable.
• Quadratic Formula: This is a universal method used to solve any quadratic equation. The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: This method rewrites the equation so that one side forms a perfect square trinomial, making it easy to solve for \(x\).

In problems involving things like projectile motion, we often use these methods to find specific times when the projectile reaches a certain height or when it hits the ground again. Solving quadratic equations efficiently requires practice and a good understanding of each method’s suitability for different scenarios.