A Rocket Is Launched, And Its Height Above Sea Level \[$ T \$\] Seconds After Launch Is Given By The Equation:$\[ H(t) = -4.9t^2 + 1200t + 370. \\]a) From What Height Was The Rocket Launched?b) What Is The Maximum Height The Rocket

Feb 27, 2025 by ADMIN 232 views

**A Rocket's Journey: Understanding the Height Equation**

Introduction

In the world of physics and mathematics, understanding the motion of objects is crucial for predicting their behavior and making accurate calculations. One such scenario is the launch of a rocket, where its height above sea level is determined by a specific equation. In this article, we will delve into the equation that describes the height of a rocket as a function of time and explore its various aspects.

The Height Equation

The height of the rocket above sea level, denoted by $h(t)$ , is given by the equation:

h(t) = -4.9t^2 + 1200t + 370

where $t$ represents the time in seconds after the launch.

Understanding the Equation

To grasp the behavior of the rocket's height, we need to break down the equation into its individual components. The equation consists of three terms:

Quadratic term: $-4.9t^2$
Linear term: $1200t$
Constant term: $370$

The quadratic term, $-4.9t^2$ , represents the downward acceleration of the rocket due to gravity. The negative sign indicates that the acceleration is in the opposite direction of the rocket's motion. The coefficient, $-4.9$ , is a measure of the acceleration due to gravity, which is approximately $9.8 \, \text{m/s}^2$ on Earth.

The linear term, $1200t$ , represents the upward velocity of the rocket. The coefficient, $1200$ , is a measure of the initial velocity of the rocket, which is assumed to be constant.

The constant term, $370$ , represents the initial height of the rocket above sea level.

Part a: Launch Height

To find the launch height, we need to determine the value of $h(t)$ when $t = 0$ . Substituting $t = 0$ into the equation, we get:

h(0) = -4.9(0)^2 + 1200(0) + 370 = 370

Therefore, the rocket was launched from a height of 370 meters above sea level.

Part b: Maximum Height

To find the maximum height, we need to determine the value of $t$ that maximizes the equation $h(t)$ . This can be done by finding the vertex of the parabola represented by the equation.

The vertex of a parabola in the form $y = ax^2 + bx + c$ is given by the formula:

x = -\frac{b}{2a}

In this case, $a = -4.9$ and $b = 1200$ . Substituting these values into the formula, we get:

t = -\frac{1200}{2(-4.9)} = \frac{1200}{9.8} = 122.45 \, \text{seconds}

Now that we have the value of $t$ , we can substitute it into the equation to find the maximum height:

h(122.45) = -4.9(122.45)^2 + 1200(122.45) + 370

Simplifying the expression, we get:

h(122.45) = -4.9(15000.3025) + 146400 + 370

h(122.45) = -73500.0149 + 146400 + 370

h(122.45) = 72970 \, \text{meters}

Therefore, the maximum height the rocket reaches is 72970 meters above sea level.

Conclusion

In conclusion, the height of a rocket above sea level is given by the equation $h(t) = -4.9t^2 + 1200t + 370$ . By analyzing the equation, we can determine the launch height and the maximum height reached by the rocket. The launch height is 370 meters, and the maximum height is 72970 meters. This equation provides a valuable tool for understanding the motion of objects and making accurate calculations in various fields of physics and engineering.

References

[1] Physics Classroom. (n.d.). Projectile Motion. Retrieved from https://www.physicsclassroom.com/class/vectors/Lesson-3/Projectile-Motion
[2] Khan Academy. (n.d.). Quadratic Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2-2-quadratic-equations/x2-2-1-quadratic-formula/v/quadratic-formula

Additional Resources

For more information on projectile motion, visit the Physics Classroom website.
For a comprehensive guide to quadratic equations, visit the Khan Academy website.
A Rocket's Journey: Q&A ==========================

Introduction

In our previous article, we explored the equation that describes the height of a rocket as a function of time. We analyzed the equation, determined the launch height, and found the maximum height reached by the rocket. In this article, we will address some frequently asked questions related to the rocket's journey.

Q&A

Q: What is the significance of the quadratic term in the equation?

A: The quadratic term, $-4.9t^2$ , represents the downward acceleration of the rocket due to gravity. The negative sign indicates that the acceleration is in the opposite direction of the rocket's motion. The coefficient, $-4.9$ , is a measure of the acceleration due to gravity, which is approximately $9.8 \, \text{m/s}^2$ on Earth.

Q: What is the initial velocity of the rocket?

A: The initial velocity of the rocket is represented by the linear term, $1200t$ . The coefficient, $1200$ , is a measure of the initial velocity of the rocket, which is assumed to be constant.

Q: How do I find the launch height?

A: To find the launch height, substitute $t = 0$ into the equation. This will give you the initial height of the rocket above sea level.

Q: How do I find the maximum height?

A: To find the maximum height, find the vertex of the parabola represented by the equation. The vertex can be found using the formula: $x = -\frac{b}{2a}$ . In this case, $a = -4.9$ and $b = 1200$ . Substitute these values into the formula to find the value of $t$ that maximizes the equation.

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

A: The maximum height reached by the rocket is 72970 meters above sea level.

Q: What is the launch height of the rocket?

A: The launch height of the rocket is 370 meters above sea level.

Q: What is the significance of the constant term in the equation?

A: The constant term, $370$ , represents the initial height of the rocket above sea level.

Q: How do I use the equation to predict the height of the rocket at a given time?

A: To predict the height of the rocket at a given time, substitute the value of $t$ into the equation. This will give you the height of the rocket above sea level at that specific time.

Q: What are some real-world applications of the equation?

A: The equation has various real-world applications, including:

Rocket science: The equation is used to predict the trajectory of rockets and ensure safe and accurate launches.
Aerospace engineering: The equation is used to design and optimize the performance of aircraft and spacecraft.
Physics and mathematics: The equation is used to model and analyze the motion of objects under the influence of gravity.

Conclusion

In conclusion, the equation that describes the height of a rocket as a function of time is a powerful tool for understanding the motion of objects and making accurate calculations. By analyzing the equation, we can determine the launch height, maximum height, and other important parameters of the rocket's journey. We hope this Q&A article has provided valuable insights and answers to your questions.

References

[1] Physics Classroom. (n.d.). Projectile Motion. Retrieved from https://www.physicsclassroom.com/class/vectors/Lesson-3/Projectile-Motion
[2] Khan Academy. (n.d.). Quadratic Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2-2-quadratic-equations/x2-2-1-quadratic-formula/v/quadratic-formula

Additional Resources

For more information on projectile motion, visit the Physics Classroom website.
For a comprehensive guide to quadratic equations, visit the Khan Academy website.