A Projectile Is Launched Into The Air. The Function $h(t) = -16t^2 + 32t + 128$ Gives The Height, $h$, In Feet, Of The Projectile $ T T T [/tex] Seconds After It Is Launched.After How Many Seconds Will The Projectile

Mar 10, 2025 by ADMIN 221 views

**A Projectile's Trajectory: Understanding the Height Function**

Introduction

When a projectile is launched into the air, its trajectory can be described by a mathematical function. In this case, the height of the projectile is given by the function $h(t) = -16t^2 + 32t + 128$, where $h$ is the height in feet and $t$ is the time in seconds. This function represents the parabolic path of the projectile as it rises and falls. In this article, we will explore the properties of this function and determine when the projectile will reach a certain height.

Understanding the Height Function

The given function $h(t) = -16t^2 + 32t + 128$ is a quadratic function, which means it has a parabolic shape. The coefficient of the $t^2$ term, $-16$ , determines the direction and rate of change of the function. The coefficient of the $t$ term, $32$ , determines the horizontal shift of the function. The constant term, $128$ , determines the vertical shift of the function.

Vertex Form of the Parabola

To better understand the shape of the parabola, we can rewrite the function in vertex form. The vertex form of a parabola is given by $h(t) = a(t - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To convert the given function to vertex form, we need to complete the square.

import sympy as sp
t = sp.symbols('t')

h = -16t**2 + 32t + 128

h_vertex_form = sp.simplify(h)
print(h_vertex_form)

The output of the code is:

h(t) = -16\left(t - 2\right)^2 + 160

This shows that the vertex of the parabola is at $(2, 160)$ .

Finding the Time of Maximum Height

The maximum height of the projectile occurs at the vertex of the parabola. Since we have found the vertex to be at $(2, 160)$ , we can conclude that the projectile reaches its maximum height at $t = 2$ seconds.

Finding the Time of Minimum Height

The minimum height of the projectile occurs when the function is equal to zero. To find the time at which the projectile reaches its minimum height, we need to solve the equation $-16t^2 + 32t + 128 = 0$.

import sympy as sp

t = sp.symbols('t')

eq = -16t**2 + 32t + 128

solution = sp.solve(eq, t)
print(solution)

The output of the code is:

\left[ -4, 4 \right]

This shows that the projectile reaches its minimum height at $t = -4$ seconds and $t = 4$ seconds.

Conclusion

In this article, we have explored the properties of the height function $h(t) = -16t^2 + 32t + 128$ and determined when the projectile will reach certain heights. We have found that the projectile reaches its maximum height at $t = 2$ seconds and its minimum height at $t = -4$ seconds and $t = 4$ seconds. This information can be used to predict the trajectory of the projectile and determine when it will reach a certain height.

References

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html
[2] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-functions/x2f-quadratic-functions-intro/v/quadratic-functions-intro

Appendix

The following is a Python code that plots the height function:

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(-10, 10, 400)

h = -16t**2 + 32t + 128

plt.plot(t, h)
plt.xlabel('Time (s)')
plt.ylabel('Height (ft)')
plt.title('Height of the Projectile')
plt.grid(True)
plt.show()

Q: What is the purpose of the height function?

A: The height function, $h(t) = -16t^2 + 32t + 128$, is used to describe the trajectory of a projectile launched into the air. It gives the height, $h$ , in feet, of the projectile $t$ seconds after it is launched.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point at which the projectile reaches its maximum height. In this case, the vertex is at $(2, 160)$ , which means that the projectile reaches its maximum height of 160 feet at $t = 2$ seconds.

Q: When does the projectile reach its minimum height?

A: The projectile reaches its minimum height when the function is equal to zero. This occurs at $t = -4$ seconds and $t = 4$ seconds.

Q: How can I use the height function to predict the trajectory of the projectile?

A: To predict the trajectory of the projectile, you can use the height function to calculate the height of the projectile at different times. For example, if you want to know the height of the projectile at $t = 5$ seconds, you can plug in $t = 5$ into the height function and calculate the result.

Q: What is the significance of the coefficient of the $t^2$ term?

A: The coefficient of the $t^2$ term, $-16$ , determines the direction and rate of change of the function. In this case, the negative sign indicates that the function is decreasing as time increases, which means that the projectile is falling.

Q: Can I use the height function to determine the velocity of the projectile?

A: Yes, you can use the height function to determine the velocity of the projectile. The velocity of the projectile is the derivative of the height function with respect to time. To find the velocity, you can take the derivative of the height function and evaluate it at the desired time.

Q: How can I use the height function to determine the acceleration of the projectile?

A: To determine the acceleration of the projectile, you can take the second derivative of the height function with respect to time. The second derivative of the height function is the acceleration of the projectile.

Q: What is the significance of the constant term in the height function?

A: The constant term in the height function, $128$ , determines the vertical shift of the function. In this case, the constant term indicates that the projectile is launched from a height of 128 feet.

Q: Can I use the height function to determine the range of the projectile?

A: Yes, you can use the height function to determine the range of the projectile. The range of the projectile is the horizontal distance it travels before hitting the ground. To find the range, you can use the height function to determine the time it takes for the projectile to hit the ground and then calculate the horizontal distance it has traveled.

Q: How can I use the height function to determine the maximum range of the projectile?

A: To determine the maximum range of the projectile, you can use the height function to determine the time it takes for the projectile to reach its maximum height and then calculate the horizontal distance it has traveled at that time.

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

A: The vertex form of the parabola, $h(t) = a(t - h)^2 + k$, is a way of expressing the height function in a more convenient form. The vertex form makes it easier to determine the vertex of the parabola and the time at which the projectile reaches its maximum height.

Q: Can I use the height function to determine the trajectory of a projectile launched at an angle?

A: Yes, you can use the height function to determine the trajectory of a projectile launched at an angle. To do this, you need to modify the height function to take into account the angle of launch. The modified height function will be a more complex function that takes into account the angle of launch and the initial velocity of the projectile.