While Hiking, Marek Throws A Rock Off A Cliff That Is Roughly 10 Meters Above A Lake. This Situation Is Modeled By The Equation H = − 4.9 T 2 + 10 H = -4.9t^2 + 10 H = − 4.9 T 2 + 10 , With H H H Representing The Height In Meters And T T T The Time In Seconds. If

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Introduction

While hiking, Marek throws a rock off a cliff that is roughly 10 meters above a lake. This situation is modeled by the equation h=−4.9t2+10h = -4.9t^2 + 10, with hh representing the height in meters and tt the time in seconds. In this article, we will delve into the mathematical world of projectile motion and explore the physics behind Marek's thrown rock.

Understanding the Equation

The equation h=−4.9t2+10h = -4.9t^2 + 10 is a quadratic equation that describes the height of the rock as a function of time. The coefficient of the t2t^2 term, -4.9, represents the acceleration due to gravity, which is approximately 4.9 meters per second squared. The constant term, 10, represents the initial height of the rock.

Breaking Down the Equation

To better understand the equation, let's break it down into its individual components.

  • Height (h): The height of the rock is represented by the variable hh. As the rock falls, its height decreases, and as it rises, its height increases.
  • Time (t): The time is represented by the variable tt. As the rock falls, time increases, and as it rises, time decreases.
  • Acceleration due to gravity: The coefficient of the t2t^2 term, -4.9, represents the acceleration due to gravity. This value is negative because the rock is accelerating downward.
  • Initial height: The constant term, 10, represents the initial height of the rock.

Solving for Time

To find the time it takes for the rock to hit the lake, we need to solve for tt when h=0h = 0. We can do this by setting the equation equal to zero and solving for tt.

−4.9t2+10=0-4.9t^2 + 10 = 0

We can rearrange the equation to isolate the t2t^2 term:

4.9t2=104.9t^2 = 10

Dividing both sides by 4.9, we get:

t2=104.9t^2 = \frac{10}{4.9}

Taking the square root of both sides, we get:

t=±104.9t = \pm \sqrt{\frac{10}{4.9}}

Since time cannot be negative, we take the positive square root:

t=104.9t = \sqrt{\frac{10}{4.9}}

Calculating the Time

Now that we have the equation for tt, we can calculate the time it takes for the rock to hit the lake.

t=104.9t = \sqrt{\frac{10}{4.9}}

Using a calculator, we get:

t≈1.02t \approx 1.02

Conclusion

In this article, we explored the physics of a thrown rock using the equation h=−4.9t2+10h = -4.9t^2 + 10. We broke down the equation into its individual components and solved for time when h=0h = 0. We found that the time it takes for the rock to hit the lake is approximately 1.02 seconds.

Further Exploration

This problem can be further explored by considering the following questions:

  • What happens if the rock is thrown from a different height?
  • What happens if the rock is thrown with a different initial velocity?
  • How does the acceleration due to gravity affect the motion of the rock?

By exploring these questions, we can gain a deeper understanding of the physics behind projectile motion and the mathematical equations that describe it.

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

The following is a list of formulas and equations used in this article:

  • h=−4.9t2+10h = -4.9t^2 + 10
  • t=±104.9t = \pm \sqrt{\frac{10}{4.9}}
  • t=104.9t = \sqrt{\frac{10}{4.9}}

Q: What is the equation h=−4.9t2+10h = -4.9t^2 + 10 used for?

A: The equation h=−4.9t2+10h = -4.9t^2 + 10 is used to model the height of a rock thrown from a cliff as a function of time. It takes into account the acceleration due to gravity and the initial height of the rock.

Q: What is the significance of the coefficient -4.9 in the equation?

A: The coefficient -4.9 represents the acceleration due to gravity, which is approximately 4.9 meters per second squared. This value is negative because the rock is accelerating downward.

Q: How do you solve for time when h=0h = 0?

A: To solve for time when h=0h = 0, we need to set the equation equal to zero and solve for tt. We can do this by rearranging the equation to isolate the t2t^2 term and then taking the square root of both sides.

Q: What is the time it takes for the rock to hit the lake?

A: Using the equation t=104.9t = \sqrt{\frac{10}{4.9}}, we can calculate the time it takes for the rock to hit the lake. This value is approximately 1.02 seconds.

Q: What happens if the rock is thrown from a different height?

A: If the rock is thrown from a different height, the equation h=−4.9t2+10h = -4.9t^2 + 10 will change to reflect the new initial height. For example, if the rock is thrown from a height of 20 meters, the equation would be h=−4.9t2+20h = -4.9t^2 + 20.

Q: What happens if the rock is thrown with a different initial velocity?

A: If the rock is thrown with a different initial velocity, the equation h=−4.9t2+10h = -4.9t^2 + 10 will change to reflect the new initial velocity. However, the acceleration due to gravity will remain the same.

Q: How does the acceleration due to gravity affect the motion of the rock?

A: The acceleration due to gravity affects the motion of the rock by causing it to accelerate downward. This means that the rock will fall faster and faster as it approaches the ground.

Q: What are some real-world applications of the equation h=−4.9t2+10h = -4.9t^2 + 10?

A: The equation h=−4.9t2+10h = -4.9t^2 + 10 has many real-world applications, including:

  • Modeling the motion of projectiles, such as rocks, balls, and bullets
  • Calculating the time it takes for an object to fall from a certain height
  • Determining the maximum height reached by an object in flight
  • Designing safety nets and other protective systems to catch falling objects

Q: What are some limitations of the equation h=−4.9t2+10h = -4.9t^2 + 10?

A: The equation h=−4.9t2+10h = -4.9t^2 + 10 has several limitations, including:

  • It assumes a constant acceleration due to gravity, which is not always the case
  • It does not take into account air resistance or other external forces that can affect the motion of the rock
  • It is only applicable to objects that are thrown or dropped from a height, and not to objects that are propelled horizontally or at an angle.

Q: How can I use the equation h=−4.9t2+10h = -4.9t^2 + 10 in my own projects or applications?

A: You can use the equation h=−4.9t2+10h = -4.9t^2 + 10 to model the motion of projectiles, calculate the time it takes for an object to fall from a certain height, or determine the maximum height reached by an object in flight. Simply plug in the values for the initial height and time, and solve for the height or time as needed.