Just after launch from the earth, the space-shuttle orbiter is in the 36 x 162-mi orbit shown. At the apogee point A, its speed is 17211 mi/hr. If nothing were done to modify the orbit, what would its speed be at the perigee P? Neglect aerodynamic drag. (Note that the normal practice is to add speed at A, which raises the perigee altitude to a value that is well above the bulk of the atmosphere.) The radius of the earth is 3959 mi. 17211 mi/hr 36 mi 162 mi-
Just after launch from the earth, the space-shuttle orbiter is in the 36 x 162-mi orbit shown. At the apogee point A, its speed is 17211 mi/hr. If nothing were done to modify the orbit, what would its speed be at the perigee P? Neglect aerodynamic drag. (Note that the normal practice is to add speed at A, which raises the perigee altitude to a value that is well above the bulk of the atmosphere.) The radius of the earth is 3959 mi. 17211 mi/hr 36 mi 162 mi-
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![### Orbital Speed Calculations for Space-Shuttle Orbiter
**Context:**
Just after launch from Earth, the space-shuttle orbiter is in a 36 x 162-mile orbit as shown below. At the apogee point A, its speed is 17,211 mi/hr. Our objective is to determine the speed at the perigee P if nothing is done to modify the orbit. For this calculation, we neglect aerodynamic drag. Note that typically, speed is added at point A to raise the perigee altitude above the bulk of the atmosphere.
The radius of the Earth is 3,959 miles.
**Diagram Explanation:**
Below is a diagram illustrating the space-shuttle orbiter's elliptical orbit around the Earth:
- **Earth is depicted with a radius of 3,959 miles.**
- **Point P (perigee) is the closest point of the orbit to Earth, at a distance of 36 miles from the surface.**
- **Point A (apogee) is the furthest point from Earth at a distance of 162 miles from the surface.**
- **The speed at the apogee (A) is labeled as 17,211 mi/hr.**
- **The objective is to find the speed at the perigee (P).**
**Calculation:** (not shown in diagram)
To find the speed at perigee P, we use the principle of conservation of angular momentum and the vis-viva equation.
1. **Angular Momentum Conservation:**
Angular momentum \( L \) = \( m \cdot v \cdot r \) (where \( m \) is mass, \( v \) is velocity, and \( r \) is radius).
At apogee (A):
\[
L_A = m \cdot v_A \cdot r_A
\]
At perigee (P):
\[
L_P = m \cdot v_P \cdot r_P
\]
Since \( L_A = L_P \):
\[
v_A \cdot r_A = v_P \cdot r_P
\]
2. **Solve for \( v_P \):**
\[
v_P = \frac{v_A \cdot r_A}{r_P}
\]
3. **Determine Radii:**
- \( r_A \) (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b7dd8f1-dbab-4607-8b33-bde9974f18c9%2F4eaa8232-2a67-4818-bba6-750c3991310d%2Fobv8g58_processed.png&w=3840&q=75)
Transcribed Image Text:### Orbital Speed Calculations for Space-Shuttle Orbiter
**Context:**
Just after launch from Earth, the space-shuttle orbiter is in a 36 x 162-mile orbit as shown below. At the apogee point A, its speed is 17,211 mi/hr. Our objective is to determine the speed at the perigee P if nothing is done to modify the orbit. For this calculation, we neglect aerodynamic drag. Note that typically, speed is added at point A to raise the perigee altitude above the bulk of the atmosphere.
The radius of the Earth is 3,959 miles.
**Diagram Explanation:**
Below is a diagram illustrating the space-shuttle orbiter's elliptical orbit around the Earth:
- **Earth is depicted with a radius of 3,959 miles.**
- **Point P (perigee) is the closest point of the orbit to Earth, at a distance of 36 miles from the surface.**
- **Point A (apogee) is the furthest point from Earth at a distance of 162 miles from the surface.**
- **The speed at the apogee (A) is labeled as 17,211 mi/hr.**
- **The objective is to find the speed at the perigee (P).**
**Calculation:** (not shown in diagram)
To find the speed at perigee P, we use the principle of conservation of angular momentum and the vis-viva equation.
1. **Angular Momentum Conservation:**
Angular momentum \( L \) = \( m \cdot v \cdot r \) (where \( m \) is mass, \( v \) is velocity, and \( r \) is radius).
At apogee (A):
\[
L_A = m \cdot v_A \cdot r_A
\]
At perigee (P):
\[
L_P = m \cdot v_P \cdot r_P
\]
Since \( L_A = L_P \):
\[
v_A \cdot r_A = v_P \cdot r_P
\]
2. **Solve for \( v_P \):**
\[
v_P = \frac{v_A \cdot r_A}{r_P}
\]
3. **Determine Radii:**
- \( r_A \) (
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