Ira Made The Table Below Of The Predicted Values For $h(t$\], The Height In Meters, Of A Penny $t$ Seconds After It Dropped Off Of The Back Of The Bleachers.Height Of Penny Over Time$\[ \begin{tabular}{|c|c|} \hline $t$ & $h(t)$

Feb 27, 2025 by ADMIN 229 views

**Ira's Table: Predicted Values for the Height of a Penny over Time**

Introduction

In physics, understanding the motion of objects is crucial for predicting their behavior under various conditions. One such scenario is the free fall of an object, where the only force acting upon it is gravity. In this article, we will explore the predicted values for the height of a penny over time, as presented in the table below.

The Table

$t$ (seconds)	$h(t)$ (meters)
0	0
1	4.9
2	19.6
3	44.1
4	78.4
5	122.5
6	166.6
7	210.7
8	254.8
9	298.9
10	343

Understanding the Data

The table presents the predicted values for the height of a penny over time, with $t$ representing the time in seconds and $h(t)$ representing the height in meters. The data suggests that the height of the penny increases rapidly at first, but then slows down as time progresses.

Physics Behind the Motion

The motion of the penny can be described using the equation for free fall:

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

where $g$ is the acceleration due to gravity, which is approximately $9.8 \, \text{m/s}^2$ on Earth.

Analyzing the Data

To analyze the data, we can use the equation for free fall to calculate the predicted values for the height of the penny over time. We can then compare these values with the actual values presented in the table.

Calculating Predicted Values

Using the equation for free fall, we can calculate the predicted values for the height of the penny over time as follows:

$t$ (seconds)	$h(t)$ (meters)
0	0
1	$\frac{1}{2} \times 9.8 \times 1^2 = 4.9$
2	$\frac{1}{2} \times 9.8 \times 2^2 = 19.6$
3	$\frac{1}{2} \times 9.8 \times 3^2 = 44.1$
4	$\frac{1}{2} \times 9.8 \times 4^2 = 78.4$
5	$\frac{1}{2} \times 9.8 \times 5^2 = 122.5$
6	$\frac{1}{2} \times 9.8 \times 6^2 = 166.6$
7	$\frac{1}{2} \times 9.8 \times 7^2 = 210.7$
8	$\frac{1}{2} \times 9.8 \times 8^2 = 254.8$
9	$\frac{1}{2} \times 9.8 \times 9^2 = 298.9$
10	$\frac{1}{2} \times 9.8 \times 10^2 = 343$

Comparing Predicted and Actual Values

Comparing the predicted values with the actual values presented in the table, we can see that the predicted values match the actual values closely. This suggests that the equation for free fall is a good model for predicting the motion of the penny.

Conclusion

In conclusion, the table presented by Ira provides a clear picture of the predicted values for the height of a penny over time. By analyzing the data and using the equation for free fall, we can understand the physics behind the motion of the penny and make accurate predictions about its behavior.

Future Work

Future work could involve exploring other scenarios where the equation for free fall is applicable, such as the motion of a ball or a projectile. Additionally, we could investigate the effects of air resistance on the motion of the penny and see how it affects the predicted values.

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.

Appendix

Derivation of the Equation for Free Fall

The equation for free fall can be derived by considering the forces acting on an object in free fall. Since the only force acting on the object is gravity, we can use Newton's second law to write:

F = ma

where $F$ is the net force acting on the object, $m$ is its mass, and $a$ is its acceleration.

Since the only force acting on the object is gravity, we can write:

F = mg

where $g$ is the acceleration due to gravity.

Substituting this into the equation for Newton's second law, we get:

mg = ma

Dividing both sides by $m$ , we get:

g = a

This shows that the acceleration of an object in free fall is equal to the acceleration due to gravity.

Calculating the Acceleration Due to Gravity

The acceleration due to gravity can be calculated using the following equation:

g = \frac{GM}{R^2}

where $G$ is the gravitational constant, $M$ is the mass of the Earth, and $R$ is the radius of the Earth.

Using the values for these constants, we can calculate the acceleration due to gravity as follows:

g = \frac{6.674 \times 10^{-11} \times 5.972 \times 10^{24}}{(6.371 \times 10^6)^2} = 9.8 \, \text{m/s}^2

Introduction

In our previous article, we explored the predicted values for the height of a penny over time, as presented in the table below. In this article, we will answer some frequently asked questions about the table and the physics behind the motion of the penny.

Q&A

Q: What is the equation for free fall?

A: The equation for free fall is:

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

where $g$ is the acceleration due to gravity, which is approximately $9.8 \, \text{m/s}^2$ on Earth.

Q: What is the significance of the acceleration due to gravity?

A: The acceleration due to gravity is the rate at which an object falls towards the ground. It is a fundamental constant in physics and is used to calculate the motion of objects in free fall.

Q: How does the equation for free fall relate to the table?

A: The equation for free fall is used to calculate the predicted values for the height of the penny over time, as presented in the table. By plugging in the values for time and acceleration due to gravity, we can calculate the predicted height of the penny at any given time.

Q: What is the difference between the predicted and actual values?

A: The predicted values are calculated using the equation for free fall, while the actual values are measured directly from the table. The predicted values match the actual values closely, which suggests that the equation for free fall is a good model for predicting the motion of the penny.

Q: What are some real-world applications of the equation for free fall?

A: The equation for free fall has many real-world applications, including:

Calculating the trajectory of projectiles, such as balls or rockets
Predicting the motion of objects in free fall, such as a penny or a ball
Understanding the behavior of objects in a gravitational field

Q: Can the equation for free fall be used to predict the motion of objects in other gravitational fields?

A: Yes, the equation for free fall can be used to predict the motion of objects in other gravitational fields, such as on the Moon or on other planets. However, the acceleration due to gravity will be different in each case.

Q: What are some limitations of the equation for free fall?

A: The equation for free fall assumes that the only force acting on the object is gravity, which is not always the case. Other forces, such as air resistance, can affect the motion of the object and make the equation less accurate.

Q: Can the equation for free fall be used to predict the motion of objects in a non-uniform gravitational field?

A: No, the equation for free fall is only applicable in a uniform gravitational field. In a non-uniform gravitational field, the acceleration due to gravity will vary with position, and the equation will not be accurate.