The Table Shows The Approximate Height Of An Object $x$ Seconds After The Object Was Dropped. The Function $h(x) = -16x^2 + 100$ Models The Data In The Table.Height Of A Dropped

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Introduction

When an object is dropped from a certain height, it experiences a downward acceleration due to gravity. The height of the object at any given time can be modeled using a quadratic function. In this article, we will explore the physics behind the fall of a dropped object and use the given function to calculate its height at different times.

The Function h(x) = -16x^2 + 100

The function h(x) = -16x^2 + 100 represents the height of the object x seconds after it was dropped. The function is a quadratic equation, where the coefficient of x^2 is -16, the coefficient of x is 0, and the constant term is 100.

Understanding the Coefficients

  • Coefficient of x^2 (-16): This coefficient represents the acceleration due to gravity. In this case, the acceleration is -16 feet per second squared, which is the standard value for the acceleration due to gravity on Earth.
  • Coefficient of x (0): This coefficient represents the initial velocity of the object. Since the coefficient is 0, the object is dropped from rest, meaning it has no initial velocity.
  • Constant Term (100): This term represents the initial height of the object. Since the object is dropped from a certain height, the initial height is 100 feet.

Calculating the Height at Different Times

Using the function h(x) = -16x^2 + 100, we can calculate the height of the object at different times. For example, let's calculate the height of the object at x = 1, 2, 3, and 4 seconds.

Height at x = 1 Second

To calculate the height at x = 1 second, we substitute x = 1 into the function:

h(1) = -16(1)^2 + 100 h(1) = -16(1) + 100 h(1) = -16 + 100 h(1) = 84

So, the height of the object at x = 1 second is 84 feet.

Height at x = 2 Seconds

To calculate the height at x = 2 seconds, we substitute x = 2 into the function:

h(2) = -16(2)^2 + 100 h(2) = -16(4) + 100 h(2) = -64 + 100 h(2) = 36

So, the height of the object at x = 2 seconds is 36 feet.

Height at x = 3 Seconds

To calculate the height at x = 3 seconds, we substitute x = 3 into the function:

h(3) = -16(3)^2 + 100 h(3) = -16(9) + 100 h(3) = -144 + 100 h(3) = -44

So, the height of the object at x = 3 seconds is -44 feet. Since the height cannot be negative, this means that the object has hit the ground at x = 3 seconds.

Height at x = 4 Seconds

To calculate the height at x = 4 seconds, we substitute x = 4 into the function:

h(4) = -16(4)^2 + 100 h(4) = -16(16) + 100 h(4) = -256 + 100 h(4) = -156

So, the height of the object at x = 4 seconds is -156 feet. Since the height cannot be negative, this means that the object has hit the ground at x = 4 seconds.

Conclusion

In this article, we explored the physics behind the fall of a dropped object and used the given function to calculate its height at different times. We saw that the function h(x) = -16x^2 + 100 represents the height of the object x seconds after it was dropped, and we used it to calculate the height at x = 1, 2, 3, and 4 seconds. We also saw that the object hits the ground at x = 3 and x = 4 seconds.

Key Takeaways

  • The function h(x) = -16x^2 + 100 represents the height of the object x seconds after it was dropped.
  • The coefficient of x^2 (-16) represents the acceleration due to gravity.
  • The coefficient of x (0) represents the initial velocity of the object.
  • The constant term (100) represents the initial height of the object.
  • The object hits the ground at x = 3 and x = 4 seconds.

Further Reading

If you want to learn more about the physics behind the fall of a dropped object, I recommend checking out the following resources:

  • Khan Academy: Khan Academy has a comprehensive course on physics that covers the topic of falling objects.
  • Physics Classroom: The Physics Classroom is a website that provides a detailed explanation of the physics behind falling objects.
  • Wikipedia: Wikipedia has a detailed article on the physics of falling objects that covers the topic in depth.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. 10th ed. John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. 10th ed. Cengage Learning.
    The Height of a Dropped Object: Q&A =====================================

Introduction

In our previous article, we explored the physics behind the fall of a dropped object and used the given function to calculate its height at different times. In this article, we will answer some frequently asked questions about the height of a dropped object.

Q: What is the initial height of the object?

A: The initial height of the object is 100 feet. This is represented by the constant term in the function h(x) = -16x^2 + 100.

Q: What is the acceleration due to gravity?

A: The acceleration due to gravity is -16 feet per second squared. This is represented by the coefficient of x^2 in the function h(x) = -16x^2 + 100.

Q: Is the object dropped from rest?

A: Yes, the object is dropped from rest. This means that the initial velocity of the object is 0 feet per second. This is represented by the coefficient of x in the function h(x) = -16x^2 + 100.

Q: How long does it take for the object to hit the ground?

A: The object hits the ground at x = 3 and x = 4 seconds. This means that the object takes 3 seconds to hit the ground, but it also takes 4 seconds to hit the ground. This is because the function h(x) = -16x^2 + 100 is a quadratic function, and it has two solutions for x.

Q: What is the height of the object at x = 0 seconds?

A: The height of the object at x = 0 seconds is 100 feet. This is because the function h(x) = -16x^2 + 100 is a quadratic function, and it has a maximum value at x = 0.

Q: What is the height of the object at x = 5 seconds?

A: The height of the object at x = 5 seconds is -156 feet. This is because the function h(x) = -16x^2 + 100 is a quadratic function, and it has a minimum value at x = 5.

Q: Can the height of the object be negative?

A: No, the height of the object cannot be negative. This is because the height of an object is always a non-negative value.

Q: What is the significance of the function h(x) = -16x^2 + 100?

A: The function h(x) = -16x^2 + 100 is a quadratic function that represents the height of a dropped object at any given time. It is a mathematical model that describes the motion of the object under the influence of gravity.

Q: How can I use the function h(x) = -16x^2 + 100 in real-life situations?

A: The function h(x) = -16x^2 + 100 can be used to model the motion of objects under the influence of gravity in various real-life situations, such as:

  • Projectile motion: The function can be used to model the motion of projectiles, such as balls or rockets, under the influence of gravity.
  • Falling objects: The function can be used to model the motion of falling objects, such as rocks or buildings, under the influence of gravity.
  • Aerodynamics: The function can be used to model the motion of objects in the air, such as airplanes or birds, under the influence of gravity and air resistance.

Conclusion

In this article, we answered some frequently asked questions about the height of a dropped object. We saw that the function h(x) = -16x^2 + 100 represents the height of the object x seconds after it was dropped, and we used it to answer various questions about the object's motion.

Key Takeaways

  • The function h(x) = -16x^2 + 100 represents the height of the object x seconds after it was dropped.
  • The coefficient of x^2 (-16) represents the acceleration due to gravity.
  • The coefficient of x (0) represents the initial velocity of the object.
  • The constant term (100) represents the initial height of the object.
  • The object hits the ground at x = 3 and x = 4 seconds.

Further Reading

If you want to learn more about the physics behind the fall of a dropped object, I recommend checking out the following resources:

  • Khan Academy: Khan Academy has a comprehensive course on physics that covers the topic of falling objects.
  • Physics Classroom: The Physics Classroom is a website that provides a detailed explanation of the physics behind falling objects.
  • Wikipedia: Wikipedia has a detailed article on the physics of falling objects that covers the topic in depth.

References

  • Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. 10th ed. John Wiley & Sons.
  • Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. 10th ed. Cengage Learning.