The first U.S. satellite, the 14-kg Explorer 1, launched in March 1958, waS placed in an orbit for which the distances from the center of the earth ot perigee and apogee where rp= G6650 km and rA= 9920 km. (A.) Find the mechanical energy of the satellite (B.) Find the period
The first U.S. satellite, the 14-kg Explorer 1, launched in March 1958, waS placed in an orbit for which the distances from the center of the earth ot perigee and apogee where rp= G6650 km and rA= 9920 km. (A.) Find the mechanical energy of the satellite (B.) Find the period
Related questions
Question
![**Title: Orbital Mechanics of the First U.S. Satellite, Explorer I**
**Introduction:**
The first U.S. satellite, the 14-kg Explorer I, launched in March 1958, was placed in an orbit for which the distances from the center of the Earth at perigee and apogee were \( r_p = 6650 \) km and \( r_a = 9920 \) km respectively.
**Problem Statement:**
**(A) Calculation of the Mechanical Energy of the Satellite**
To determine the mechanical energy of the satellite in its orbit, you need to calculate both the kinetic energy and potential energy at a specific point in the orbit and then sum them up.
**(B) Calculation of the Orbital Period**
To find the period of the satellite, apply Kepler's Third Law or use the formula for the orbital period of an elliptical orbit based on given distances or orbital parameters.
**Explanation of Notations:**
- \( r_p \): Distance from the center of the Earth at perigee (the closest point to the Earth) = 6650 km.
- \( r_a \): Distance from the center of the Earth at apogee (the farthest point from the Earth) = 9920 km.
**Discussion:**
**(A) Mechanical Energy Calculation:**
The mechanical energy per unit mass (\( E \)) of a satellite in orbit is given by:
\[ E = - \frac{GM}{2a} \]
where:
- \( G \) is the gravitational constant.
- \( M \) is the mass of the Earth.
- \( a \) is the semi-major axis of the orbit, which can be calculated as:
\[ a = \frac{r_p + r_a}{2} \]
**(B) Orbital Period Calculation:**
The period \( T \) of the orbit can be calculated using:
\[ T = 2\pi \sqrt{\frac{a^3}{GM}} \]
**Graphical Representations (Hypothetical):**
This section could include graphical depictions like:
1. **Diagram of the Elliptical Orbit**:
- Showing Earth at one focus of the ellipse.
- Marking distances for perigee \( r_p \) and apogee \( r_a \).
2. **Graphical Analysis of Orbital Parameters:**
- A plot relating potential energy, kinetic energy](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c95aabf-9eb4-4d63-86ef-475a7c27077f%2F23a556d3-c3f0-4ae0-a4af-eaf0e4b59818%2Fn1hve8k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Orbital Mechanics of the First U.S. Satellite, Explorer I**
**Introduction:**
The first U.S. satellite, the 14-kg Explorer I, launched in March 1958, was placed in an orbit for which the distances from the center of the Earth at perigee and apogee were \( r_p = 6650 \) km and \( r_a = 9920 \) km respectively.
**Problem Statement:**
**(A) Calculation of the Mechanical Energy of the Satellite**
To determine the mechanical energy of the satellite in its orbit, you need to calculate both the kinetic energy and potential energy at a specific point in the orbit and then sum them up.
**(B) Calculation of the Orbital Period**
To find the period of the satellite, apply Kepler's Third Law or use the formula for the orbital period of an elliptical orbit based on given distances or orbital parameters.
**Explanation of Notations:**
- \( r_p \): Distance from the center of the Earth at perigee (the closest point to the Earth) = 6650 km.
- \( r_a \): Distance from the center of the Earth at apogee (the farthest point from the Earth) = 9920 km.
**Discussion:**
**(A) Mechanical Energy Calculation:**
The mechanical energy per unit mass (\( E \)) of a satellite in orbit is given by:
\[ E = - \frac{GM}{2a} \]
where:
- \( G \) is the gravitational constant.
- \( M \) is the mass of the Earth.
- \( a \) is the semi-major axis of the orbit, which can be calculated as:
\[ a = \frac{r_p + r_a}{2} \]
**(B) Orbital Period Calculation:**
The period \( T \) of the orbit can be calculated using:
\[ T = 2\pi \sqrt{\frac{a^3}{GM}} \]
**Graphical Representations (Hypothetical):**
This section could include graphical depictions like:
1. **Diagram of the Elliptical Orbit**:
- Showing Earth at one focus of the ellipse.
- Marking distances for perigee \( r_p \) and apogee \( r_a \).
2. **Graphical Analysis of Orbital Parameters:**
- A plot relating potential energy, kinetic energy
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
