A Rock Is Thrown From One Side Of A River To The Other And Follows The Path Modeled By The Function: H ( T ) = − 16 T 2 + 80 T + 30 H(t) = -16t^2 + 80t + 30 H ( T ) = − 16 T 2 + 80 T + 30 .1. When Does The Rock Reach Its Maximum Height? A. 1.5 Seconds B. 2.5 Seconds C. 3 Seconds 2. What Is

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Introduction

The study of projectile motion is a fundamental concept in physics, and it has numerous applications in various fields, including engineering, sports, and even everyday life. When a rock is thrown from one side of a river to the other, it follows a curved path, which can be modeled using a quadratic function. In this article, we will explore the path of the rock, determine when it reaches its maximum height, and analyze the factors that affect its trajectory.

The Path of the Rock

The path of the rock is modeled by the function:

h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30

where h(t)h(t) is the height of the rock at time tt.

Understanding the Function

The function h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30 is a quadratic function, which means it has a parabolic shape. The graph of this function will be a parabola that opens downward, since the coefficient of the t2t^2 term is negative.

Finding the Maximum Height

To find the maximum height of the rock, we need to find the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction, and it is the highest or lowest point on the graph.

The x-coordinate of the vertex of a parabola can be found using the formula:

x=−b2ax = -\frac{b}{2a}

where aa and bb are the coefficients of the t2t^2 and tt terms, respectively.

In this case, a=−16a = -16 and b=80b = 80, so:

x=−802(−16)=2.5x = -\frac{80}{2(-16)} = 2.5

This means that the rock reaches its maximum height at t=2.5t = 2.5 seconds.

Verifying the Answer

To verify our answer, we can plug t=2.5t = 2.5 into the original function:

h(2.5)=−16(2.5)2+80(2.5)+30h(2.5) = -16(2.5)^2 + 80(2.5) + 30

h(2.5)=−16(6.25)+200+30h(2.5) = -16(6.25) + 200 + 30

h(2.5)=−100+200+30h(2.5) = -100 + 200 + 30

h(2.5)=130h(2.5) = 130

This means that the rock reaches a height of 130 units at t=2.5t = 2.5 seconds.

Conclusion

In conclusion, the rock reaches its maximum height at t=2.5t = 2.5 seconds, which is option B. This is because the vertex of the parabola is at t=2.5t = 2.5 seconds, and this is the point where the rock reaches its highest height.

Factors Affecting the Trajectory

There are several factors that can affect the trajectory of the rock, including:

  • Gravity: The force of gravity pulls the rock downward, causing it to follow a curved path.
  • Air Resistance: Air resistance can slow down the rock and cause it to follow a more curved path.
  • Initial Velocity: The initial velocity of the rock affects its trajectory, with faster initial velocities resulting in more curved paths.
  • Angle of Projection: The angle at which the rock is projected affects its trajectory, with more shallow angles resulting in more curved paths.

Real-World Applications

The study of projectile motion has numerous real-world applications, including:

  • Sports: Understanding the trajectory of a projectile is essential for athletes, coaches, and sports analysts.
  • Engineering: The study of projectile motion is used in the design of various engineering systems, including bridges, buildings, and aircraft.
  • Everyday Life: Understanding the trajectory of a projectile is essential for everyday activities, such as throwing a ball or launching a projectile.

Conclusion

Introduction

In our previous article, we explored the path of a rock thrown from one side of a river to the other, modeled by the function h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30. We determined that the rock reaches its maximum height at t=2.5t = 2.5 seconds. In this article, we will answer some frequently asked questions about the path of the rock and the factors that affect its trajectory.

Q&A

Q: What is the initial velocity of the rock?

A: The initial velocity of the rock is not explicitly given in the function h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30. However, we can determine the initial velocity by finding the derivative of the function and evaluating it at t=0t = 0.

Q: How does air resistance affect the trajectory of the rock?

A: Air resistance can slow down the rock and cause it to follow a more curved path. However, in this case, we are assuming that air resistance is negligible, and the rock follows a parabolic path.

Q: What is the maximum height of the rock?

A: The maximum height of the rock is 130 units, which occurs at t=2.5t = 2.5 seconds.

Q: How does the angle of projection affect the trajectory of the rock?

A: The angle of projection affects the trajectory of the rock, with more shallow angles resulting in more curved paths. However, in this case, we are assuming that the rock is thrown at a 45-degree angle, which results in a parabolic path.

Q: Can the rock be thrown at a different angle to achieve a different trajectory?

A: Yes, the rock can be thrown at a different angle to achieve a different trajectory. However, the function h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30 assumes a 45-degree angle of projection.

Q: How does the initial velocity affect the trajectory of the rock?

A: The initial velocity affects the trajectory of the rock, with faster initial velocities resulting in more curved paths. However, in this case, we are assuming a constant initial velocity, which results in a parabolic path.

Q: Can the rock be thrown with a different initial velocity to achieve a different trajectory?

A: Yes, the rock can be thrown with a different initial velocity to achieve a different trajectory. However, the function h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30 assumes a constant initial velocity.

Q: What is the significance of the vertex of the parabola?

A: The vertex of the parabola represents the maximum height of the rock, which occurs at t=2.5t = 2.5 seconds.

Q: How does the force of gravity affect the trajectory of the rock?

A: The force of gravity pulls the rock downward, causing it to follow a curved path.

Q: Can the rock be thrown in a different direction to achieve a different trajectory?

A: Yes, the rock can be thrown in a different direction to achieve a different trajectory. However, the function h(t)=−16t2+80t+30h(t) = -16t^2 + 80t + 30 assumes a horizontal direction of projection.

Conclusion

In conclusion, the study of projectile motion is a fundamental concept in physics, and it has numerous applications in various fields. The path of a rock thrown from one side of a river to the other can be modeled using a quadratic function, and the maximum height of the rock can be found by finding the vertex of the parabola. The factors that affect the trajectory of the rock include gravity, air resistance, initial velocity, and angle of projection. We hope that this Q&A article has provided a better understanding of the path of the rock and the factors that affect its trajectory.