A Rocket Is Launched From The Top Of An 80-foot Cliff With An Initial Velocity Of 100 Feet Per Second. The Height, { H $}$, Of The Rocket After { T $}$ Seconds Is Given By The Equation:${ H = -16t^2 + 100t + 80 }$How
Introduction
Projectile motion is a fundamental concept in physics that describes the motion of an object under the influence of gravity. In this article, we will explore the trajectory of a rocket launched from the top of an 80-foot cliff with an initial velocity of 100 feet per second. The height of the rocket after t seconds is given by the equation h = -16t^2 + 100t + 80. We will analyze this equation, understand its components, and use it to determine the maximum height reached by the rocket, the time it takes to reach the ground, and the velocity of the rocket at various points in its trajectory.
The Equation of Motion
The equation h = -16t^2 + 100t + 80 describes the height of the rocket as a function of time. This equation is a quadratic equation, which means it can be written in the form h = at^2 + bt + c, where a, b, and c are constants. In this case, a = -16, b = 100, and c = 80.
Understanding the Components of the Equation
Let's break down the components of the equation and understand their significance.
- a = -16: This is the coefficient of the t^2 term, which represents the acceleration due to gravity. The negative sign indicates that the acceleration is downward, which means the rocket is under the influence of gravity.
- b = 100: This is the coefficient of the t term, which represents the initial velocity of the rocket. The positive sign indicates that the initial velocity is upward, which means the rocket is launched with an initial velocity of 100 feet per second.
- c = 80: This is the constant term, which represents the initial height of the rocket. The positive sign indicates that the initial height is 80 feet above the ground.
Determining the Maximum Height
To determine the maximum height reached by the rocket, we need to find the vertex of the parabola described by the equation. The vertex of a parabola is the point where the parabola changes direction, and it is given by the equation t = -b / 2a.
In this case, t = -100 / (2 * -16) = 3.125 seconds. Plugging this value of t into the equation, we get h = -16(3.125)^2 + 100(3.125) + 80 = 262.5 feet.
Therefore, the maximum height reached by the rocket is 262.5 feet.
Determining the Time to Reach the Ground
To determine the time it takes for the rocket to reach the ground, we need to find the value of t when h = 0. Plugging h = 0 into the equation, we get:
0 = -16t^2 + 100t + 80
Rearranging the equation, we get:
16t^2 - 100t - 80 = 0
Using the quadratic formula, we get:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 16, b = -100, and c = -80. Plugging these values into the formula, we get:
t = (100 ± √((-100)^2 - 4(16)(-80))) / (2 * 16)
Simplifying the expression, we get:
t = (100 ± √(10000 + 5120)) / 32
t = (100 ± √15220) / 32
t = (100 ± 123.5) / 32
Therefore, the two possible values of t are:
t = (100 + 123.5) / 32 = 5.5 seconds
t = (100 - 123.5) / 32 = -0.5 seconds
Since time cannot be negative, we discard the second solution. Therefore, the time it takes for the rocket to reach the ground is 5.5 seconds.
Determining the Velocity of the Rocket
To determine the velocity of the rocket at various points in its trajectory, we need to find the derivative of the equation with respect to time. The derivative of the equation is given by:
dh/dt = -32t + 100
This equation represents the velocity of the rocket as a function of time.
To find the velocity at a particular point in time, we need to plug the value of t into the equation. For example, to find the velocity at t = 3.125 seconds, we get:
dh/dt = -32(3.125) + 100 = 25 feet per second
Therefore, the velocity of the rocket at t = 3.125 seconds is 25 feet per second.
Conclusion
In this article, we analyzed the trajectory of a rocket launched from the top of an 80-foot cliff with an initial velocity of 100 feet per second. We used the equation h = -16t^2 + 100t + 80 to determine the maximum height reached by the rocket, the time it takes to reach the ground, and the velocity of the rocket at various points in its trajectory. We found that the maximum height reached by the rocket is 262.5 feet, the time it takes to reach the ground is 5.5 seconds, and the velocity of the rocket at t = 3.125 seconds is 25 feet per second.
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.
Glossary
- Projectile motion: The motion of an object under the influence of gravity.
- Quadratic equation: An equation of the form h = at^2 + bt + c, where a, b, and c are constants.
- Vertex: The point where a parabola changes direction.
- Derivative: The rate of change of a function with respect to a variable.
A Rocket's Descent: Understanding the Trajectory of a Projectile - Q&A ====================================================================
Introduction
In our previous article, we explored the trajectory of a rocket launched from the top of an 80-foot cliff with an initial velocity of 100 feet per second. We used the equation h = -16t^2 + 100t + 80 to determine the maximum height reached by the rocket, the time it takes to reach the ground, and the velocity of the rocket at various points in its trajectory. In this article, we will answer some of the most frequently asked questions related to the topic.
Q: What is the significance of the coefficient a in the equation h = -16t^2 + 100t + 80?
A: The coefficient a in the equation represents the acceleration due to gravity. In this case, a = -16, which means the acceleration due to gravity is 16 feet per second squared.
Q: Why is the initial velocity of the rocket represented by the coefficient b in the equation?
A: The initial velocity of the rocket is represented by the coefficient b in the equation because it is the rate of change of the height with respect to time at the initial point. In this case, b = 100, which means the initial velocity of the rocket is 100 feet per second.
Q: How do you determine the maximum height reached by the rocket?
A: To determine the maximum height reached by the rocket, we need to find the vertex of the parabola described by the equation. The vertex of a parabola is the point where the parabola changes direction, and it is given by the equation t = -b / 2a. In this case, t = -100 / (2 * -16) = 3.125 seconds. Plugging this value of t into the equation, we get h = -16(3.125)^2 + 100(3.125) + 80 = 262.5 feet.
Q: How do you determine the time it takes for the rocket to reach the ground?
A: To determine the time it takes for the rocket to reach the ground, we need to find the value of t when h = 0. Plugging h = 0 into the equation, we get:
0 = -16t^2 + 100t + 80
Rearranging the equation, we get:
16t^2 - 100t - 80 = 0
Using the quadratic formula, we get:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 16, b = -100, and c = -80. Plugging these values into the formula, we get:
t = (100 ± √((-100)^2 - 4(16)(-80))) / (2 * 16)
Simplifying the expression, we get:
t = (100 ± √(10000 + 5120)) / 32
t = (100 ± √15220) / 32
t = (100 ± 123.5) / 32
Therefore, the two possible values of t are:
t = (100 + 123.5) / 32 = 5.5 seconds
t = (100 - 123.5) / 32 = -0.5 seconds
Since time cannot be negative, we discard the second solution. Therefore, the time it takes for the rocket to reach the ground is 5.5 seconds.
Q: How do you determine the velocity of the rocket at various points in its trajectory?
A: To determine the velocity of the rocket at various points in its trajectory, we need to find the derivative of the equation with respect to time. The derivative of the equation is given by:
dh/dt = -32t + 100
This equation represents the velocity of the rocket as a function of time.
Q: What is the significance of the derivative of the equation?
A: The derivative of the equation represents the rate of change of the height with respect to time. In other words, it represents the velocity of the rocket at a particular point in time.
Q: How do you use the equation h = -16t^2 + 100t + 80 to solve real-world problems?
A: The equation h = -16t^2 + 100t + 80 can be used to solve a variety of real-world problems, such as:
- Designing rocket trajectories: The equation can be used to design the trajectory of a rocket, taking into account the initial velocity, acceleration due to gravity, and other factors.
- Predicting the motion of projectiles: The equation can be used to predict the motion of projectiles, such as baseballs, basketballs, and other objects that are thrown or launched.
- Analyzing the motion of objects under gravity: The equation can be used to analyze the motion of objects under gravity, such as the motion of a ball thrown upwards or the motion of a object dropped from a height.
Conclusion
In this article, we answered some of the most frequently asked questions related to the trajectory of a rocket launched from the top of an 80-foot cliff with an initial velocity of 100 feet per second. We used the equation h = -16t^2 + 100t + 80 to determine the maximum height reached by the rocket, the time it takes to reach the ground, and the velocity of the rocket at various points in its trajectory. We also discussed the significance of the derivative of the equation and how it can be used to solve real-world problems.
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.
Glossary
- Projectile motion: The motion of an object under the influence of gravity.
- Quadratic equation: An equation of the form h = at^2 + bt + c, where a, b, and c are constants.
- Vertex: The point where a parabola changes direction.
- Derivative: The rate of change of a function with respect to a variable.