Stefano Accidentally Dropped His Sunglasses Off The Edge Of A Canyon As He Was Looking Down. The Height, H ( T H(t H ( T ], In Meters (as It Relates To Sea Level), Of The Sunglasses After T T T Seconds, Is Shown In The Table.Height Of Sunglasses

Introduction

Imagine a scenario where Stefano accidentally drops his sunglasses off the edge of a canyon as he is looking down. The height, $h(t)$, in meters (as it relates to sea level), of the sunglasses after $t$ seconds, is a crucial piece of information that can be used to determine the trajectory of the sunglasses. In this article, we will delve into the mathematical analysis of the height of the sunglasses as a function of time.

The Height Function

The height function, $h(t)$, represents the height of the sunglasses at any given time $t$. To analyze this function, we need to understand the underlying physics of the situation. Since the sunglasses are dropped from rest, the only force acting on them is gravity, which is a constant acceleration of $9.8 \, \text{m/s}^2$.

Assuming that the initial velocity of the sunglasses is zero, we can use the equation of motion under constant acceleration to derive the height function:

$$h(t) = h_0 - \frac{1}{2}gt^2$$

where $h_0$ is the initial height of the sunglasses, $g$ is the acceleration due to gravity, and $t$ is the time in seconds.

The Table of Heights

The table below shows the height of the sunglasses at various times:

Time (s)	Height (m)
0	100
1	94.1
2	87.4
3	80.1
4	72.2
5	63.7
6	54.6
7	44.9
8	34.6
9	23.7
10	12.3

Analyzing the Height Function

From the table, we can see that the height of the sunglasses decreases over time. To analyze this function further, we can use the concept of quadratic functions.

A quadratic function has the general form:

$$f(x) = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants. In our case, the height function can be written as:

$$h(t) = -\frac{1}{2}gt^2 + h_0$$

This is a quadratic function with a negative coefficient of $t^2$, indicating that the height decreases over time.

The Vertex of the Parabola

The vertex of a parabola is the point where the function changes from decreasing to increasing or vice versa. In our case, the vertex of the parabola represents the maximum height of the sunglasses.

To find the vertex, we can use the formula:

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

In our case, $a = -\frac{1}{2}g$ and $b = 0$, so the vertex is at:

$$t = -\frac{0}{2(-\frac{1}{2}g)} = 0$$

This means that the vertex of the parabola is at $t = 0$, which is the initial time when the sunglasses are dropped.

The Maximum Height

Since the vertex of the parabola is at $t = 0$, the maximum height of the sunglasses is:

$$h(0) = h_0 - \frac{1}{2}g(0)^2 = h_0$$

This means that the maximum height of the sunglasses is equal to the initial height, which is $100$ meters.

The Time of Impact

The time of impact is the time when the height of the sunglasses becomes zero. To find the time of impact, we can set the height function equal to zero and solve for $t$:

$$h(t) = 0 = h_0 - \frac{1}{2}gt^2$$

Solving for $t$, we get:

$$t = \sqrt{\frac{2h_0}{g}}$$

Substituting the values of $h_0 = 100$ meters and $g = 9.8 \, \text{m/s}^2$, we get:

$$t = \sqrt{\frac{2(100)}{9.8}} \approx 4.97 \, \text{s}$$

This means that the time of impact is approximately $4.97$ seconds.

Conclusion

In conclusion, the height of the sunglasses as a function of time can be analyzed using the concept of quadratic functions. The vertex of the parabola represents the maximum height of the sunglasses, which is equal to the initial height. The time of impact is the time when the height of the sunglasses becomes zero, which can be calculated using the equation of motion under constant acceleration.

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.

Table of Contents

1. Introduction
2. The Height Function
3. The Table of Heights
4. Analyzing the Height Function
5. The Vertex of the Parabola
6. The Maximum Height
7. The Time of Impact
8. Conclusion
9. References
   Q&A: The Height of Sunglasses =====================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to the height of sunglasses.

Q: What is the height function?

A: The height function, $h(t)$, represents the height of the sunglasses at any given time $t$. It is a mathematical function that describes the relationship between the height of the sunglasses and the time.

Q: How is the height function related to the acceleration due to gravity?

A: The height function is related to the acceleration due to gravity, $g$, through the equation:

$$h(t) = h_0 - \frac{1}{2}gt^2$$

This equation shows that the height of the sunglasses decreases over time due to the acceleration due to gravity.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point where the function changes from decreasing to increasing or vice versa. In the case of the height function, the vertex represents the maximum height of the sunglasses.

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

A: To find the vertex of the parabola, you can use the formula:

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

In the case of the height function, $a = -\frac{1}{2}g$ and $b = 0$, so the vertex is at:

$$t = -\frac{0}{2(-\frac{1}{2}g)} = 0$$

Q: What is the maximum height of the sunglasses?

A: The maximum height of the sunglasses is equal to the initial height, $h_0$. In the case of the example, the maximum height is $100$ meters.

Q: How do I find the time of impact?

A: To find the time of impact, you can set the height function equal to zero and solve for $t$:

$$h(t) = 0 = h_0 - \frac{1}{2}gt^2$$

Solving for $t$, you get:

$$t = \sqrt{\frac{2h_0}{g}}$$

Q: What is the time of impact in the example?

A: In the example, the time of impact is approximately $4.97$ seconds.

Q: Can I use the height function to predict the height of the sunglasses at any given time?

A: Yes, you can use the height function to predict the height of the sunglasses at any given time. Simply plug in the value of $t$ into the equation:

$$h(t) = h_0 - \frac{1}{2}gt^2$$

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

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

• Predicting the trajectory of projectiles
• Calculating the height of buildings and bridges
• Determining the time of impact of objects in free fall

Conclusion

In conclusion, the height of sunglasses is a complex topic that can be analyzed using mathematical functions. The height function, $h(t)$, represents the height of the sunglasses at any given time $t$. The vertex of the parabola represents the maximum height of the sunglasses, and the time of impact can be calculated using the equation of motion under constant acceleration.

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.

Table of Contents

1. Introduction
2. The Height Function
3. The Table of Heights
4. Analyzing the Height Function
5. The Vertex of the Parabola
6. The Maximum Height
7. The Time of Impact
8. Conclusion
9. References
10. Q&A: The Height of Sunglasses