Understanding the Height Function

Oct 13, 2025 by miodeya.com 15 views

Introduction

When a ball is thrown straight up into the air, it follows a parabolic trajectory under the influence of gravity. The height of the ball at any given time can be described by a mathematical function, known as the height function. In this article, we will analyze the height of the ball over time using the data collected from a table.

Understanding the Height Function

The height function, denoted by $h(t)$ , represents the height of the ball at time $t$ . The table provided shows the data collected over $t$ seconds, where $h(t)$ is the height of the ball, in feet. The table has two columns: $t$ and $h(t)$ . The $t$ column represents the time in seconds, and the $h(t)$ column represents the height of the ball at that particular time.

Analyzing the Data

Let's take a closer look at the data collected from the table.

t	h(t)
0	0
1	20
2	36
3	48
4	56
5	60
6	60
7	56
8	48
9	36
10	20
11	0

From the table, we can observe that the height of the ball increases as time increases, reaches a maximum value, and then decreases as time continues to increase. This is a characteristic of a parabolic trajectory under the influence of gravity.

Calculating the Maximum Height

To find the maximum height of the ball, we need to find the time at which the height is maximum. Looking at the table, we can see that the height is maximum at $t = 5$ seconds, and the corresponding height is $h(5) = 60$ feet.

Calculating the Time of Return

To find the time it takes for the ball to return to the ground, we need to find the time at which the height is zero. Looking at the table, we can see that the height is zero at $t = 10$ seconds.

Calculating the Total Time of Flight

To find the total time of flight, we need to find the difference between the time it takes for the ball to return to the ground and the time at which it was thrown. Looking at the table, we can see that the ball was thrown at $t = 0$ seconds, and it returned to the ground at $t = 10$ seconds. Therefore, the total time of flight is $10 - 0 = 10$ seconds.

Conclusion

In this article, we analyzed the height of a ball thrown straight up into the air using the data collected from a table. We calculated the maximum height, the time of return, and the total time of flight. The results show that the ball reaches a maximum height of 60 feet at $t = 5$ seconds, returns to the ground at $t = 10$ seconds, and has a total time of flight of 10 seconds.

Mathematical Representation

The height function, $h(t)$ , can be represented mathematically as a quadratic function:

h(t) = at^2 + bt + c

where $a$ , $b$ , and $c$ are constants. To find the values of $a$ , $b$ , and $c$ , we can use the data collected from the table.

Finding the Values of $a$ , $b$ , and $c$

Using the data collected from the table, we can set up a system of equations to find the values of $a$ , $b$ , and $c$ .

h(0) = 0 = a(0)^2 + b(0) + c

h(1) = 20 = a(1)^2 + b(1) + c

h(2) = 36 = a(2)^2 + b(2) + c

h(3) = 48 = a(3)^2 + b(3) + c

h(4) = 56 = a(4)^2 + b(4) + c

h(5) = 60 = a(5)^2 + b(5) + c

h(6) = 60 = a(6)^2 + b(6) + c

h(7) = 56 = a(7)^2 + b(7) + c

h(8) = 48 = a(8)^2 + b(8) + c

h(9) = 36 = a(9)^2 + b(9) + c

h(10) = 20 = a(10)^2 + b(10) + c

h(11) = 0 = a(11)^2 + b(11) + c

Solving this system of equations, we get:

a = -2

b = 20

c = 0

Therefore, the height function can be represented mathematically as:

h(t) = -2t^2 + 20t

Graphical Representation

The height function, $h(t)$ , can be represented graphically as a parabola.

Graph of $h(t)$

The graph of $h(t)$ is a parabola that opens downward. The vertex of the parabola is at $(5, 60)$ , which represents the maximum height of the ball.

Conclusion

In this article, we analyzed the height of a ball thrown straight up into the air using the data collected from a table. We calculated the maximum height, the time of return, and the total time of flight. We also represented the height function mathematically and graphically. The results show that the ball reaches a maximum height of 60 feet at $t = 5$ seconds, returns to the ground at $t = 10$ seconds, and has a total time of flight of 10 seconds.

Final Thoughts

The analysis of the height of a ball thrown straight up into the air is a classic problem in physics and mathematics. The results show that the ball follows a parabolic trajectory under the influence of gravity, and the height function can be represented mathematically and graphically. This problem is a great example of how mathematics can be used to describe and analyze real-world phenomena.

Introduction

In our previous article, we analyzed the height of a ball thrown straight up into the air using the data collected from a table. We calculated the maximum height, the time of return, and the total time of flight. We also represented the height function mathematically and graphically. In this article, we will answer some of the most frequently asked questions about the problem.

Q: What is the maximum height of the ball?

A: The maximum height of the ball is 60 feet, which occurs at $t = 5$ seconds.

Q: How long does it take for the ball to return to the ground?

A: The ball returns to the ground at $t = 10$ seconds.

Q: What is the total time of flight?

A: The total time of flight is 10 seconds.

Q: What is the height function?

A: The height function, denoted by $h(t)$ , represents the height of the ball at time $t$ . It can be represented mathematically as:

h(t) = -2t^2 + 20t

Q: What is the graph of the height function?

A: The graph of the height function is a parabola that opens downward. The vertex of the parabola is at $(5, 60)$ , which represents the maximum height of the ball.

Q: How does the height of the ball change over time?

A: The height of the ball increases as time increases, reaches a maximum value, and then decreases as time continues to increase.

Q: What is the relationship between the height of the ball and the time?

A: The height of the ball is directly proportional to the time, but with a negative coefficient. This means that as time increases, the height of the ball also increases, but at a decreasing rate.

Q: Can the height function be used to predict the height of the ball at any given time?

A: Yes, the height function can be used to predict the height of the ball at any given time. Simply plug in the value of $t$ into the height function to get the corresponding height.

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

A: The height function has many real-world applications, such as:

Calculating the trajectory of a projectile
Designing the trajectory of a satellite
Predicting the height of a building or a bridge
Calculating the time of flight of an object

Q: Can the height function be used to solve other problems?

A: Yes, the height function can be used to solve other problems, such as:

Calculating the maximum height of a roller coaster
Designing the trajectory of a rocket
Predicting the height of a wave in the ocean

Conclusion

In this article, we answered some of the most frequently asked questions about the problem of a ball thrown straight up into the air. We hope that this article has provided a better understanding of the problem and its solutions.

Final Thoughts

Additional Resources

For more information on the problem of a ball thrown straight up into the air, please refer to the following resources:

[1] "Physics for Scientists and Engineers" by Paul A. Tipler and Gene Mosca
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Physics and Engineering" by Lawrence C. Andrews

Note: The resources listed above are just a few examples of the many resources available on the topic.