A Ball Is Thrown From A Height Of 110 Feet With An Initial Downward Velocity Of $6 , \text{ft/s}$. The Ball's Height $h$ (in Feet) After $ T T T [/tex] Seconds Is Given By The Following Equation:$h = 110 - 6t -

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Introduction

When a ball is thrown from a certain height, it follows a predictable path under the influence of gravity. The motion of the ball can be described using mathematical equations that take into account the initial velocity, acceleration due to gravity, and time. In this article, we will explore the physics behind the motion of a ball thrown from a height of 110 feet with an initial downward velocity of 6 ft/s.

The Equation of Motion

The height hh (in feet) of the ball after tt seconds is given by the following equation:

h=110−6t−12gt2h = 110 - 6t - \frac{1}{2}gt^2

where gg is the acceleration due to gravity, which is approximately 32 ft/s^2.

Understanding the Components of the Equation

Let's break down the components of the equation to understand how they contribute to the motion of the ball.

  • Initial Height: The initial height of the ball is 110 feet, which is the starting point of the motion.
  • Initial Velocity: The initial velocity of the ball is 6 ft/s, which is the downward velocity at the start of the motion.
  • Acceleration Due to Gravity: The acceleration due to gravity is 32 ft/s^2, which is the rate at which the ball's velocity increases downward due to gravity.
  • Time: The time tt is the independent variable that determines the height of the ball at any given moment.

Graphing the Equation

To visualize the motion of the ball, we can graph the equation using a graphing calculator or a computer algebra system. The graph will show the height of the ball as a function of time.

import numpy as np
import matplotlib.pyplot as plt

def h(t):
return 110 - 6t - 0.532*t**2



t = np.linspace(0, 2, 100)



h_values = h(t)



plt.plot(t, h_values)
plt.xlabel('Time (s)')
plt.ylabel('Height (ft)')
plt.title('Height of the Ball as a Function of Time')
plt.grid(True)
plt.show()

Analyzing the Graph

The graph shows the height of the ball as a function of time. We can see that the ball starts at a height of 110 feet and then begins to fall downward due to gravity. The graph is a parabola that opens downward, indicating that the ball's height decreases as time increases.

Finding the Time of Maximum Height

To find the time at which the ball reaches its maximum height, we can take the derivative of the equation with respect to time and set it equal to zero.

dhdt=−6−32t=0\frac{dh}{dt} = -6 - 32t = 0

Solving for tt, we get:

t=−632=−316t = -\frac{6}{32} = -\frac{3}{16}

However, this value of tt is not physically meaningful, as it corresponds to a time before the ball is thrown. Therefore, we can conclude that the ball reaches its maximum height at t=0t = 0 seconds.

Finding the Maximum Height

To find the maximum height of the ball, we can substitute t=0t = 0 into the equation of motion.

h=110−6(0)−12(32)(0)2=110h = 110 - 6(0) - \frac{1}{2}(32)(0)^2 = 110

Therefore, the maximum height of the ball is 110 feet.

Conclusion

In this article, we have explored the physics behind the motion of a ball thrown from a height of 110 feet with an initial downward velocity of 6 ft/s. We have used the equation of motion to analyze the ball's height as a function of time and have found the time at which the ball reaches its maximum height. We have also graphed the equation to visualize the motion of the ball.

Introduction

In our previous article, we explored the physics behind the motion of a ball thrown from a height of 110 feet with an initial downward velocity of 6 ft/s. We used the equation of motion to analyze the ball's height as a function of time and found the time at which the ball reaches its maximum height. In this article, we will answer some frequently asked questions about the motion of the ball.

Q: What is the initial velocity of the ball?

A: The initial velocity of the ball is 6 ft/s, which is the downward velocity at the start of the motion.

Q: What is the acceleration due to gravity?

A: The acceleration due to gravity is 32 ft/s^2, which is the rate at which the ball's velocity increases downward due to gravity.

Q: What is the maximum height of the ball?

A: The maximum height of the ball is 110 feet, which is the initial height of the ball.

Q: When does the ball reach its maximum height?

A: The ball reaches its maximum height at t=0t = 0 seconds, which is the time at which the ball is thrown.

Q: How does the ball's height change over time?

A: The ball's height decreases over time due to the acceleration due to gravity. The graph of the ball's height as a function of time is a parabola that opens downward.

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

A: The equation of motion for the ball is:

h=110−6t−12gt2h = 110 - 6t - \frac{1}{2}gt^2

where gg is the acceleration due to gravity, which is approximately 32 ft/s^2.

Q: How can I graph the equation of motion?

A: You can graph the equation of motion using a graphing calculator or a computer algebra system. The graph will show the height of the ball as a function of time.

Q: What is the significance of the time of maximum height?

A: The time of maximum height is the time at which the ball reaches its maximum height. This is an important concept in physics, as it helps us understand the motion of objects under the influence of gravity.

Q: Can I use this equation to model other types of motion?

A: Yes, you can use this equation to model other types of motion, such as the motion of a projectile or the motion of a pendulum. However, you will need to modify the equation to account for the specific characteristics of the motion.

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

A: This equation has many real-world applications, such as:

  • Modeling the motion of projectiles, such as bullets or rockets
  • Modeling the motion of pendulums, such as clock pendulums
  • Modeling the motion of objects under the influence of gravity, such as falling objects or objects in orbit

Conclusion

In this article, we have answered some frequently asked questions about the motion of a ball thrown from a height of 110 feet with an initial downward velocity of 6 ft/s. We have also discussed the significance of the time of maximum height and the real-world applications of the equation of motion.