Analyzing the Height of the Rock

Oct 13, 2025 by wbrum.org 109 views

Introduction

In this article, we will analyze the height of a rock thrown straight up by Sofia. The table provided shows the height of the rock in meters, $y$ , at $x$ seconds. We will use this data to understand the motion of the rock and determine the maximum height it reaches.

Understanding the Data

The table provided shows the height of the rock at different time intervals. The data is as follows:

$x$ (seconds)	$y$ (meters)
0	3
0.5	8.75
1	12
1.5	12.75
2	11
2.5

Analyzing the Height of the Rock

To analyze the height of the rock, we need to understand the relationship between the height and the time. We can see from the table that the height of the rock increases as the time increases, but it reaches a maximum height and then starts to decrease.

Finding the Maximum Height

To find the maximum height of the rock, we need to look for the highest value of $y$ in the table. From the table, we can see that the maximum height of the rock is 12.75 meters, which occurs at $x$ = 1.5 seconds.

Calculating the Velocity

To calculate the velocity of the rock, we need to find the rate of change of the height with respect to time. We can do this by finding the difference in height between two consecutive time intervals and dividing it by the time interval.

Let's calculate the velocity of the rock at $x$ = 1 second.

Velocity = (Change in height) / (Change in time) = (12 - 8.75) / (1 - 0.5) = 3.25 / 0.5 = 6.5 m/s

Calculating the Acceleration

To calculate the acceleration of the rock, we need to find the rate of change of the velocity with respect to time. We can do this by finding the difference in velocity between two consecutive time intervals and dividing it by the time interval.

Let's calculate the acceleration of the rock at $x$ = 1 second.

Acceleration = (Change in velocity) / (Change in time) = (6.5 - 6) / (1 - 0.5) = 0.5 / 0.5 = 1 m/s^2

Conclusion

In this article, we analyzed the height of a rock thrown straight up by Sofia. We used the data provided in the table to understand the motion of the rock and determine the maximum height it reaches. We also calculated the velocity and acceleration of the rock at different time intervals.

Maximum Height

The maximum height of the rock is 12.75 meters, which occurs at $x$ = 1.5 seconds.

Velocity

The velocity of the rock at $x$ = 1 second is 6.5 m/s.

Acceleration

The acceleration of the rock at $x$ = 1 second is 1 m/s^2.

Discussion

The data provided in the table shows that the height of the rock increases as the time increases, but it reaches a maximum height and then starts to decrease. This is because the rock is under the influence of gravity, which pulls it downwards.

The velocity of the rock is also affected by gravity. As the rock rises, its velocity decreases due to the downward force of gravity. When the rock reaches its maximum height, its velocity becomes zero, and then it starts to fall downwards.

The acceleration of the rock is also affected by gravity. As the rock falls downwards, its acceleration increases due to the downward force of gravity.

References

[1] Physics textbook by Halliday, Resnick, and Walker
[2] Online resources on physics and mathematics

Appendix

The following table shows the height of the rock at different time intervals.

$x$ (seconds)	$y$ (meters)
0	3
0.5	8.75
1	12
1.5	12.75
2	11
2.5

Q: What is the initial height of the rock?

A: The initial height of the rock is 3 meters, which is the height at $x$ = 0 seconds.

Q: What is the maximum height of the rock?

A: The maximum height of the rock is 12.75 meters, which occurs at $x$ = 1.5 seconds.

Q: What is the velocity of the rock at $x$ = 1 second?

A: The velocity of the rock at $x$ = 1 second is 6.5 m/s.

Q: What is the acceleration of the rock at $x$ = 1 second?

A: The acceleration of the rock at $x$ = 1 second is 1 m/s^2.

Q: Why does the rock's velocity decrease as it rises?

A: The rock's velocity decreases as it rises because of the downward force of gravity. As the rock gains height, the force of gravity pulls it downwards, causing its velocity to decrease.

Q: Why does the rock's acceleration increase as it falls?

A: The rock's acceleration increases as it falls because of the downward force of gravity. As the rock gains speed, the force of gravity pulls it downwards with increasing force, causing its acceleration to increase.

Q: What is the relationship between the rock's height and its velocity?

A: The rock's height and velocity are related in that the velocity of the rock is the rate of change of its height with respect to time. As the rock rises, its velocity decreases, and as it falls, its velocity increases.

Q: What is the relationship between the rock's height and its acceleration?

A: The rock's height and acceleration are related in that the acceleration of the rock is the rate of change of its velocity with respect to time. As the rock falls, its acceleration increases due to the downward force of gravity.

Q: What is the significance of the rock's maximum height?

A: The rock's maximum height is significant because it represents the highest point that the rock reaches before it starts to fall. This is an important concept in physics, as it helps us understand the motion of objects under the influence of gravity.

Q: How does the rock's motion relate to real-world phenomena?

A: The rock's motion is a simplified model of real-world phenomena, such as the motion of projectiles, the trajectory of a thrown ball, or the orbit of a satellite. Understanding the motion of the rock helps us understand these more complex phenomena.

Q: What are some real-world applications of the rock's motion?

A: Some real-world applications of the rock's motion include:

Projectile motion: Understanding the motion of the rock helps us design and optimize the trajectory of projectiles, such as missiles or artillery shells.
Aerodynamics: The motion of the rock helps us understand the behavior of air and gas molecules, which is essential for designing and optimizing aircraft and other vehicles.
Space exploration: The motion of the rock helps us understand the motion of satellites and other objects in space, which is essential for designing and optimizing space missions.

Conclusion

In this Q&A article, we have explored the motion of a rock thrown straight up by Sofia. We have discussed the initial height, maximum height, velocity, and acceleration of the rock, as well as the relationships between these quantities. We have also explored the significance of the rock's motion and its real-world applications.