A Model Rocket Is Launched With An Initial Upward Velocity Of $188 , \text{ft/s}$. The Rocket's Height $h$ (in Feet) After $ T T T [/tex] Seconds Is Given By The Equation:$h = 188t - 16t^2$Find All Values Of

Mar 10, 2025 by ADMIN 214 views

A Model Rocket's Height: Analyzing the Trajectory with a Quadratic Equation

Introduction

Model rockets have been a staple of science and engineering education for decades, providing a hands-on way for students to learn about physics and mathematics. One of the key aspects of a model rocket's flight is its trajectory, which can be described using a quadratic equation. In this article, we will explore the equation that describes the height of a model rocket as a function of time, and use it to find all values of time for which the rocket's height is a certain value.

The Equation of the Rocket's Height

The equation that describes the height of the rocket as a function of time is given by:

h = 188t - 16t^2

where $h$ is the height of the rocket in feet, and $t$ is the time in seconds.

Understanding the Equation

To understand the equation, let's break it down into its two main components: the linear term and the quadratic term.

The linear term, $188t$ , represents the initial upward velocity of the rocket. This term tells us that the rocket is moving upward at a constant rate of $188 \, \text{ft/s}$ .
The quadratic term, $-16t^2$ , represents the acceleration due to gravity. This term tells us that the rocket is slowing down as it rises, due to the force of gravity acting on it.

Finding the Time Values

We are asked to find all values of time for which the rocket's height is a certain value. To do this, we need to set the equation equal to the desired height value, and then solve for $t$ .

Let's say we want to find the time values for which the rocket's height is $100$ feet. We can set up the equation as follows:

100 = 188t - 16t^2

Solving the Quadratic Equation

To solve this quadratic equation, we can use the quadratic formula:

t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a = -16$ , $b = 188$ , and $c = -100$ .

Plugging in these values, we get:

t = \frac{-188 \pm \sqrt{188^2 - 4(-16)(-100)}}{2(-16)}

Simplifying the expression under the square root, we get:

t = \frac{-188 \pm \sqrt{35264 - 6400}}{-32}

t = \frac{-188 \pm \sqrt{28864}}{-32}

t = \frac{-188 \pm 170}{-32}

Finding the Time Values

Now that we have the two possible values of $t$ , we can plug them back into the original equation to find the corresponding height values.

For $t = \frac{-188 + 170}{-32}$ , we get:

h = 188\left(\frac{-188 + 170}{-32}\right) - 16\left(\frac{-188 + 170}{-32}\right)^2

Simplifying the expression, we get:

h = 188\left(\frac{18}{-32}\right) - 16\left(\frac{18}{-32}\right)^2

h = -\frac{188 \cdot 18}{32} - \frac{16 \cdot 18^2}{32^2}

h = -\frac{3384}{32} - \frac{16 \cdot 324}{32^2}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384}{32} - \frac{324 \cdot 16}{32 \cdot 32}

h = -\frac{3384}{32} - \frac{5184}{1024}

h = -\frac{3384<br/> # A Model Rocket's Height: A Q&A Guide

Introduction

In our previous article, we explored the equation that describes the height of a model rocket as a function of time. We used this equation to find all values of time for which the rocket's height is a certain value. In this article, we will answer some of the most frequently asked questions about the equation and its applications.

Q: What is the equation that describes the height of a model rocket?

A: The equation that describes the height of a model rocket is given by: