A Rocket Is Launched From A Tower. The Height Of The Rocket, $y$ In Feet, Is Related To The Time After Launch, $x$ In Seconds, By The Given Equation. Using This Equation, Find The Maximum Height Reached By The Rocket, To The Nearest

Introduction

The study of motion and the behavior of objects under various forces is a fundamental aspect of physics. In this article, we will explore the motion of a rocket launched from a tower, and use a given equation to determine the maximum height reached by the rocket. The equation that relates the height of the rocket, $y$ in feet, to the time after launch, $x$ in seconds, is a quadratic equation that can be used to model the motion of the rocket.

The Equation of Motion

The equation of motion for the rocket is given by:

$$y = -16x^2 + 128x$$

where $y$ is the height of the rocket in feet, and $x$ is the time after launch in seconds. This equation is a quadratic equation, which means that it can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Understanding the Equation

To understand the equation, let's break it down into its individual components. The first term, $-16x^2$, represents the downward motion of the rocket, where the negative sign indicates that the rocket is moving downward. The second term, $128x$, represents the upward motion of the rocket, where the positive sign indicates that the rocket is moving upward.

Finding the Maximum Height

To find the maximum height reached by the rocket, we need to find the vertex of the parabola represented by the equation. The vertex of a parabola is the point where the parabola reaches its maximum or minimum value. In this case, we are interested in finding the maximum height, so we need to find the vertex of the parabola.

Using the Vertex Formula

The vertex of a parabola can be found using the vertex formula, which is given by:

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

where $a$ and $b$ are the coefficients of the quadratic equation. In this case, $a = -16$ and $b = 128$, so we can plug these values into the formula to find the x-coordinate of the vertex.

Calculating the x-Coordinate of the Vertex

Plugging in the values of $a$ and $b$ into the vertex formula, we get:

$$x = -\frac{128}{2(-16)}$$

Simplifying the expression, we get:

$$x = -\frac{128}{-32}$$

$$x = 4$$

Finding the y-Coordinate of the Vertex

Now that we have found the x-coordinate of the vertex, we can plug this value into the equation of motion to find the y-coordinate of the vertex. Plugging in $x = 4$ into the equation, we get:

$$y = -16(4)^2 + 128(4)$$

Simplifying the expression, we get:

$$y = -16(16) + 128(4)$$

$$y = -256 + 512$$

$$y = 256$$

Conclusion

The maximum height reached by the rocket is 256 feet, to the nearest foot. This is the point where the rocket reaches its maximum height, and then begins to fall back down to the ground.

Discussion

The equation of motion for the rocket is a quadratic equation that can be used to model the motion of the rocket. The vertex of the parabola represents the maximum height reached by the rocket, and can be found using the vertex formula. In this case, the maximum height reached by the rocket is 256 feet, to the nearest foot.

Applications

The study of motion and the behavior of objects under various forces is a fundamental aspect of physics. The equation of motion for the rocket can be used to model the motion of other objects, such as projectiles and vehicles. Understanding the motion of these objects is crucial in fields such as engineering, physics, and mathematics.

Future Work

In the future, it would be interesting to explore other types of motion, such as circular motion and rotational motion. These types of motion are also governed by quadratic equations, and can be used to model the behavior of objects in various situations.

References

• [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Glossary

• Quadratic equation: An equation of the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• Vertex: The point where a parabola reaches its maximum or minimum value.
• Vertex formula: A formula used to find the x-coordinate of the vertex of a parabola, given by $x = -\frac{b}{2a}$.

Keywords

• Rocket motion
• Quadratic equation
• Vertex
• Vertex formula
• Maximum height
• Physics
• Mathematics
  

Introduction

In our previous article, we explored the motion of a rocket launched from a tower, and used a given equation to determine the maximum height reached by the rocket. In this article, we will answer some of the most frequently asked questions about the motion of the rocket, and provide additional information to help you better understand the topic.

Q: What is the equation of motion for the rocket?

A: The equation of motion for the rocket is given by:

$$y = -16x^2 + 128x$$

where $y$ is the height of the rocket in feet, and $x$ is the time after launch in seconds.

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

A: The maximum height reached by the rocket is 256 feet, to the nearest foot.

Q: How do you find the vertex of the parabola?

A: The vertex of a parabola can be found using the vertex formula, which is given by:

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

where $a$ and $b$ are the coefficients of the quadratic equation.

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

A: The vertex of the parabola represents the maximum height reached by the rocket, and is the point where the rocket reaches its maximum height.

Q: Can you explain the concept of quadratic equations?

A: A quadratic equation is an equation of the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. Quadratic equations can be used to model the motion of objects, such as projectiles and vehicles.

Q: How do you use the vertex formula to find the x-coordinate of the vertex?

A: To use the vertex formula, you need to plug in the values of $a$ and $b$ into the formula, and then simplify the expression to find the x-coordinate of the vertex.

Q: Can you provide an example of how to use the vertex formula?

A: Let's say we have a quadratic equation of the form $y = -16x^2 + 128x$. To find the x-coordinate of the vertex, we can plug in the values of $a$ and $b$ into the vertex formula:

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

$$x = -\frac{128}{2(-16)}$$

$$x = -\frac{128}{-32}$$

$$x = 4$$

Q: What is the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, we can plug in the x-coordinate of the vertex into the equation of motion:

$$y = -16(4)^2 + 128(4)$$

$$y = -16(16) + 128(4)$$

$$y = -256 + 512$$

$$y = 256$$

Q: Can you explain the concept of maximum height?

A: The maximum height reached by an object is the highest point it reaches before it begins to fall back down to the ground. In the case of the rocket, the maximum height is 256 feet.

Q: How do you find the maximum height reached by an object?

A: To find the maximum height reached by an object, you need to find the vertex of the parabola represented by the equation of motion.

Q: Can you provide an example of how to find the maximum height reached by an object?

A: Let's say we have a quadratic equation of the form $y = -16x^2 + 128x$. To find the maximum height reached by the object, we can find the vertex of the parabola by using the vertex formula:

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

$$x = -\frac{128}{2(-16)}$$

$$x = -\frac{128}{-32}$$

$$x = 4$$

Then, we can plug in the x-coordinate of the vertex into the equation of motion to find the y-coordinate of the vertex:

$$y = -16(4)^2 + 128(4)$$

$$y = -16(16) + 128(4)$$

$$y = -256 + 512$$

$$y = 256$$

Q: What is the significance of the maximum height reached by an object?

A: The maximum height reached by an object is an important concept in physics, as it can be used to model the motion of objects in various situations.

Q: Can you provide additional resources for learning about the motion of objects?

A: Yes, there are many resources available for learning about the motion of objects, including textbooks, online tutorials, and educational videos.

Q: What are some common applications of the motion of objects?

A: The motion of objects has many common applications, including the design of roller coasters, the development of video games, and the study of the motion of celestial bodies.

Q: Can you provide an example of how the motion of objects is used in real-world applications?

A: Yes, the motion of objects is used in many real-world applications, including the design of roller coasters. For example, the motion of a roller coaster car can be modeled using a quadratic equation, which can be used to determine the maximum height reached by the car.

Q: What are some common misconceptions about the motion of objects?

A: There are many common misconceptions about the motion of objects, including the idea that an object will always fall straight down to the ground. However, the motion of an object is often more complex, and can be influenced by factors such as gravity, friction, and air resistance.

Q: Can you provide an example of how the motion of objects is affected by gravity?

A: Yes, the motion of an object is affected by gravity, which is a force that pulls objects towards the center of the Earth. For example, the motion of a ball thrown upwards is affected by gravity, which causes the ball to fall back down to the ground.

Q: What are some common tools used to model the motion of objects?

A: There are many common tools used to model the motion of objects, including quadratic equations, graphs, and computer simulations.

Q: Can you provide an example of how to use a quadratic equation to model the motion of an object?

A: Yes, let's say we have a quadratic equation of the form $y = -16x^2 + 128x$. This equation can be used to model the motion of a rocket, where $y$ is the height of the rocket in feet, and $x$ is the time after launch in seconds.

Q: What are some common challenges associated with modeling the motion of objects?

A: There are many common challenges associated with modeling the motion of objects, including the need to account for factors such as friction, air resistance, and gravity.

Q: Can you provide an example of how to account for friction in a model of the motion of an object?

A: Yes, let's say we have a quadratic equation of the form $y = -16x^2 + 128x$. To account for friction, we can add a term to the equation that represents the force of friction, such as $-0.1x$.

Q: What are some common applications of the motion of objects in engineering?

A: The motion of objects has many common applications in engineering, including the design of roller coasters, the development of video games, and the study of the motion of celestial bodies.

Q: Can you provide an example of how the motion of objects is used in the design of roller coasters?

A: Yes, the motion of objects is used in the design of roller coasters to create a thrilling experience for riders. For example, the motion of a roller coaster car can be modeled using a quadratic equation, which can be used to determine the maximum height reached by the car.

Q: What are some common challenges associated with modeling the motion of objects in engineering?

A: There are many common challenges associated with modeling the motion of objects in engineering, including the need to account for factors such as friction, air resistance, and gravity.

Q: Can you provide an example of how to account for friction in a model of the motion of an object in engineering?

A: Yes, let's say we have a quadratic equation of the form $y = -16x^2 + 128x$. To account for friction, we can add a term to the equation that represents the force of friction, such as $-0.1x$.

Q: What are some common applications of the motion of objects in physics?

A: The motion of objects has many common applications in physics, including the study of the motion of celestial bodies, the study of the motion of particles, and the study of the motion of waves.

Q: Can you provide an example of how the motion of objects is used in the study of the motion of celestial bodies?

A: Yes, the motion of objects is used in the study of the motion of celestial bodies to understand the behavior of planets, stars, and galaxies. For example, the motion of a planet can be modeled using a quadratic equation, which can be used to determine the planet's orbit.

Q: What are some common challenges associated with modeling the motion of objects in physics?

A: There are many common challenges associated with modeling the motion of objects in physics, including the need to account for factors such as friction, air resistance, and gravity.

Q: Can you provide an example of how to account for friction in a model of the motion of an object in physics?

A: Yes, let's say we have a quadratic equation of the form