A Ball Is Thrown Directly Upward From A Height Of 2 Ft With An Initial Velocity Of 20 Ft/sec. The Function $s(t)=-16t^2+20t+2$ Gives The Height Of The Ball, In Feet, $t$ Seconds After It Has Been Thrown. Determine The Time At Which

Feb 27, 2025 by ADMIN 232 views

**A Ball Thrown Upward: Analyzing the Motion with Mathematics**

Introduction

In this article, we will delve into the world of mathematics and explore the motion of a ball thrown upward from a height of 2 ft with an initial velocity of 20 ft/sec. The function $s(t)=-16t^2+20t+2$ represents the height of the ball, in feet, $t$ seconds after it has been thrown. Our goal is to determine the time at which the ball reaches its maximum height and returns to the initial height of 2 ft.

Understanding the Function

The given function $s(t)=-16t^2+20t+2$ is a quadratic function, which represents a parabola when graphed. The coefficient of the $t^2$ term, -16, represents the acceleration due to gravity, which is acting downward. The coefficient of the $t$ term, 20, represents the initial velocity of the ball, which is acting upward. The constant term, 2, represents the initial height of the ball.

Finding the Maximum Height

To find the maximum height of the ball, we need to find the vertex of the parabola represented by the function $s(t)=-16t^2+20t+2$ . The vertex of a parabola is given by the formula $t=-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the $t^2$ and $t$ terms, respectively.

import math

# Define the coefficients of the function
a = -16
b = 20

# Calculate the time at which the ball reaches its maximum height
t_max = -b / (2 * a)
print("The ball reaches its maximum height at t =", t_max, "seconds")

Calculating the Maximum Height

Now that we have found the time at which the ball reaches its maximum height, we can substitute this value into the function $s(t)=-16t^2+20t+2$ to find the maximum height.

# Define the function
def s(t):
    return -16 * t**2 + 20 * t + 2

# Calculate the maximum height
max_height = s(t_max)
print("The maximum height of the ball is", max_height, "ft")

Finding the Time at Which the Ball Returns to the Initial Height

To find the time at which the ball returns to the initial height of 2 ft, we need to set the function $s(t)=-16t^2+20t+2$ equal to 2 and solve for $t$ .

# Define the function
def s(t):
    return -16 * t**2 + 20 * t + 2

# Set the function equal to 2 and solve for t
from sympy import symbols, Eq, solve

t = symbols('t')
eq = Eq(s(t), 2)
solution = solve(eq, t)

print("The ball returns to the initial height at t =", solution[0], "seconds")

Conclusion

In this article, we have analyzed the motion of a ball thrown upward from a height of 2 ft with an initial velocity of 20 ft/sec. We have used the function $s(t)=-16t^2+20t+2$ to represent the height of the ball, in feet, $t$ seconds after it has been thrown. We have found the time at which the ball reaches its maximum height and returns to the initial height of 2 ft. The results of our analysis provide valuable insights into the motion of the ball and can be used to predict the behavior of similar objects in the future.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Vertex Form of a Quadratic Function" by Purplemath
[3] "Solving Quadratic Equations" by Mathway

Appendix

The following is a list of the Python code used in this article:

import math
from sympy import symbols, Eq, solve

# Define the coefficients of the function
a = -16
b = 20

# Calculate the time at which the ball reaches its maximum height
t_max = -b / (2 * a)

# Define the function
def s(t):
    return -16 * t**2 + 20 * t + 2

# Calculate the maximum height
max_height = s(t_max)

# Set the function equal to 2 and solve for t
t = symbols('t')
eq = Eq(s(t), 2)
solution = solve(eq, t)
```<br/>
**A Ball Thrown Upward: Q&A**
==========================

**Q: What is the initial velocity of the ball?**
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A: The initial velocity of the ball is 20 ft/sec.

**Q: What is the height of the ball at time t=0?**
---------------------------------------------

A: The height of the ball at time t=0 is 2 ft.

**Q: What is the acceleration due to gravity?**
---------------------------------------------

A: The acceleration due to gravity is -16 ft/sec^2.

**Q: What is the maximum height of the ball?**
---------------------------------------------

A: The maximum height of the ball is 22 ft.

**Q: At what time does the ball reach its maximum height?**
---------------------------------------------------

A: The ball reaches its maximum height at t=0.5 seconds.

**Q: What is the time at which the ball returns to the initial height of 2 ft?**
-------------------------------------------------------------------

A: The ball returns to the initial height of 2 ft at t=1.25 seconds.

**Q: How can I calculate the time at which the ball reaches its maximum height?**
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A: To calculate the time at which the ball reaches its maximum height, you can use the formula t=-b/(2a), where a and b are the coefficients of the t^2 and t terms, respectively.

**Q: How can I calculate the maximum height of the ball?**
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A: To calculate the maximum height of the ball, you can substitute the time at which the ball reaches its maximum height into the function s(t)=-16t^2+20t+2.

**Q: How can I calculate the time at which the ball returns to the initial height of 2 ft?**
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A: To calculate the time at which the ball returns to the initial height of 2 ft, you can set the function s(t)=-16t^2+20t+2 equal to 2 and solve for t.

**Q: What is the significance of the vertex of the parabola represented by the function s(t)=-16t^2+20t+2?**
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A: The vertex of the parabola represented by the function s(t)=-16t^2+20t+2 represents the maximum height of the ball.

**Q: How can I use the function s(t)=-16t^2+20t+2 to predict the behavior of similar objects in the future?**
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A: You can use the function s(t)=-16t^2+20t+2 to predict the behavior of similar objects in the future by substituting different values of t into the function and analyzing the resulting values of s(t).

**Q: What are some real-world applications of the function s(t)=-16t^2+20t+2?**
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A: Some real-world applications of the function s(t)=-16t^2+20t+2 include modeling the motion of projectiles, predicting the trajectory of a thrown ball, and analyzing the behavior of objects under the influence of gravity.

**Q: How can I modify the function s(t)=-16t^2+20t+2 to model different types of motion?**
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A: You can modify the function s(t)=-16t^2+20t+2 to model different types of motion by changing the coefficients of the t^2 and t terms, or by adding additional terms to the function to represent other forces acting on the object.

**Q: What are some common mistakes to avoid when working with the function s(t)=-16t^2+20t+2?**
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A: Some common mistakes to avoid when working with the function s(t)=-16t^2+20t+2 include forgetting to include the negative sign in front of the t^2 term, or substituting the wrong value of t into the function.