An Object Is Thrown Upward At A Speed Of 146 Feet Per Second By A Machine From A Height Of 15 Feet Off The Ground. The Height H H H Of The Object After T T T Seconds Can Be Found Using The Equation:$h = -16t^2 + 146t + 15$1. When

Introduction

In this article, we will explore the trajectory of an object thrown upward by a machine. The height of the object can be calculated using a quadratic equation, which is a polynomial equation of degree two. The equation $h = -16t^2 + 146t + 15$ represents the height $h$ of the object after $t$ seconds. In this discussion, we will analyze the given equation, understand its components, and determine the maximum height reached by the object.

Understanding the Quadratic Equation

The given equation $h = -16t^2 + 146t + 15$ is a quadratic equation in the form of $ax^2 + bx + c$. In this equation, $a = -16$, $b = 146$, and $c = 15$. The coefficient $a$ represents the rate at which the height decreases, $b$ represents the initial velocity of the object, and $c$ represents the initial height of the object.

Analyzing the Coefficients

• Coefficient $a$: -16 - This coefficient represents the rate at which the height decreases. Since the coefficient is negative, the height of the object decreases over time.
• Coefficient $b$: 146 - This coefficient represents the initial velocity of the object. Since the coefficient is positive, the object is thrown upward with an initial velocity of 146 feet per second.
• Coefficient $c$: 15 - This coefficient represents the initial height of the object. Since the coefficient is positive, the object is thrown from a height of 15 feet off the ground.

Determining the Maximum Height

To determine the maximum height reached by the object, we need to find the vertex of the parabola represented by the quadratic equation. The vertex of a parabola is the point where the parabola changes direction, and it represents the maximum or minimum value of the function.

The x-coordinate of the vertex of a parabola can be found using the formula $x = -\frac{b}{2a}$. In this case, $a = -16$ and $b = 146$. Plugging these values into the formula, we get:

$x = -\frac{146}{2(-16)}$ $x = -\frac{146}{-32}$ $x = \frac{146}{32}$ $x = 4.5625$

The x-coordinate of the vertex represents the time at which the object reaches its maximum height. To find the maximum height, we need to plug this value into the quadratic equation:

$h = -16(4.5625)^2 + 146(4.5625) + 15$ $h = -16(20.75) + 146(4.5625) + 15$ $h = -332 + 665.4375 + 15$ $h = 348.4375$

Therefore, the maximum height reached by the object is approximately 348.44 feet.

Analyzing the Trajectory

To analyze the trajectory of the object, we can plot the quadratic equation on a graph. The graph will represent the height of the object over time.

The graph of the quadratic equation will be a parabola that opens downward, since the coefficient $a$ is negative. The vertex of the parabola represents the maximum height reached by the object, which we found to be approximately 348.44 feet.

Conclusion

In this article, we analyzed the trajectory of an object thrown upward by a machine using a quadratic equation. We determined the maximum height reached by the object and analyzed the trajectory of the object over time. The quadratic equation $h = -16t^2 + 146t + 15$ represents the height $h$ of the object after $t$ seconds, and it can be used to calculate the height of the object at any given time.

References

• [1] "Quadratic Equations." Math Open Reference, mathopenref.com/quadratic.html.
• [2] "Parabolas." Math Is Fun, mathisfun.com/algebra/parabolas.html.

Further Reading

• "Quadratic Equations: A Comprehensive Guide." Math Open Reference, mathopenref.com/quadraticguide.html.
• "Parabolas: A Visual Guide." Math Is Fun, mathisfun.com/algebra/parabolas-visual.html.

Glossary

• Quadratic Equation: A polynomial equation of degree two, in the form of $ax^2 + bx + c$.
• Vertex: The point where the parabola changes direction, and it represents the maximum or minimum value of the function.
• Parabola: A quadratic equation in the form of $ax^2 + bx + c$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.
  Frequently Asked Questions: An Object Thrown Upward =====================================================

Q: What is the initial velocity of the object?

A: The initial velocity of the object is 146 feet per second. This is the speed at which the object is thrown upward by the machine.

Q: What is the initial height of the object?

A: The initial height of the object is 15 feet off the ground. This is the height from which the object is thrown upward by the machine.

Q: What is the maximum height reached by the object?

A: The maximum height reached by the object is approximately 348.44 feet. This is the highest point that the object reaches during its trajectory.

Q: How long does it take for the object to reach its maximum height?

A: It takes approximately 4.5625 seconds for the object to reach its maximum height. This is the time at which the object reaches the vertex of the parabola represented by the quadratic equation.

Q: What is the rate at which the height decreases?

A: The rate at which the height decreases is represented by the coefficient $a$ in the quadratic equation, which is -16. This means that the height of the object decreases by 16 feet per second squared.

Q: Can the quadratic equation be used to calculate the height of the object at any given time?

A: Yes, the quadratic equation can be used to calculate the height of the object at any given time. Simply plug in the value of $t$ (time) into the equation, and you will get the height of the object at that time.

Q: What is the significance of the vertex of the parabola?

A: The vertex of the parabola represents the maximum or minimum value of the function. In this case, the vertex represents the maximum height reached by the object.

Q: Can the quadratic equation be used to model other types of motion?

A: Yes, the quadratic equation can be used to model other types of motion, such as the trajectory of a projectile or the motion of an object under the influence of gravity.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, such as:

• Modeling the trajectory of a projectile
• Calculating the maximum height reached by an object
• Determining the time it takes for an object to reach its maximum height
• Modeling the motion of an object under the influence of gravity
• Calculating the distance traveled by an object

Q: How can I use quadratic equations in my own life?

A: Quadratic equations can be used in many areas of life, such as:

• Calculating the trajectory of a golf ball or a baseball
• Determining the maximum height reached by a kite or a balloon
• Calculating the time it takes for a object to reach its maximum height
• Modeling the motion of an object under the influence of gravity
• Calculating the distance traveled by an object

Conclusion

In this article, we have answered some frequently asked questions about the object thrown upward by a machine. We have discussed the initial velocity, initial height, maximum height, and rate at which the height decreases. We have also discussed the significance of the vertex of the parabola and the real-world applications of quadratic equations.