Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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![### Rocket Launch Problem
A rocketry club is holding a competition. There is cloud cover at 1000 feet. If a rocket is launched with an initial upward velocity of 315 ft/sec, use the function:
\[ h(t) = -16t^2 + vt + s \]
to determine how long the rocket is out of sight.
### Explanation
The function \( h(t) = -16t^2 + vt + s \) is a mathematical model used to calculate the height \( h \) of the rocket over time \( t \). In this equation:
- \( -16t^2 \) represents the effect of gravity (assuming \( t \) is in seconds and \( h \) in feet).
- \( v \) is the initial velocity of the rocket, given as 315 ft/sec.
- \( s \) is the initial position of the rocket, typically the launchpad height, assumed here to be 0 feet.
#### Objective
The goal is to find the duration for which the rocket remains above the cloud cover (1000 feet), making it out of sight.
### Calculations & Graphs
To solve the problem, graph the function to find the time interval during which the height \( h(t) \) is above 1000 feet. Initially, set \( h(t) = 1000 \) and solve for \( t \):
\[ 1000 = -16t^2 + 315t \]
By solving this quadratic equation, the values of \( t \) where the rocket crosses 1000 feet will provide the required time interval. Use these roots to find when the rocket passes above and then below 1000 feet, marking the "out of sight" duration.](https://content.bartleby.com/qna-images/question/2b9a38c0-77e3-442f-bc6a-7392be38ce1a/8c106fc5-c224-417e-9a4c-2d766698ff95/f8j3pvr_processed.jpeg)
Transcribed Image Text:### Rocket Launch Problem
A rocketry club is holding a competition. There is cloud cover at 1000 feet. If a rocket is launched with an initial upward velocity of 315 ft/sec, use the function:
\[ h(t) = -16t^2 + vt + s \]
to determine how long the rocket is out of sight.
### Explanation
The function \( h(t) = -16t^2 + vt + s \) is a mathematical model used to calculate the height \( h \) of the rocket over time \( t \). In this equation:
- \( -16t^2 \) represents the effect of gravity (assuming \( t \) is in seconds and \( h \) in feet).
- \( v \) is the initial velocity of the rocket, given as 315 ft/sec.
- \( s \) is the initial position of the rocket, typically the launchpad height, assumed here to be 0 feet.
#### Objective
The goal is to find the duration for which the rocket remains above the cloud cover (1000 feet), making it out of sight.
### Calculations & Graphs
To solve the problem, graph the function to find the time interval during which the height \( h(t) \) is above 1000 feet. Initially, set \( h(t) = 1000 \) and solve for \( t \):
\[ 1000 = -16t^2 + 315t \]
By solving this quadratic equation, the values of \( t \) where the rocket crosses 1000 feet will provide the required time interval. Use these roots to find when the rocket passes above and then below 1000 feet, marking the "out of sight" duration.
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