Respuesta :
Let's break down the problem step by step:
a. To calculate the potential energy of the satellite at the initial altitude, we can use the formula for gravitational potential energy:
\[ U = -\frac{G \cdot M \cdot m}{r} \]
Where:
- \( U \) is the potential energy
- \( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2\))
- \( M \) is the mass of Mars (\(6.42 \times 10^{23} \, \text{kg}\))
- \( m \) is the mass of the satellite (\(1337 \, \text{kg}\))
- \( r \) is the distance from the center of Mars to the satellite's initial altitude (\(45000 \, \text{km} + 3.39 \times 10^6 \, \text{m}\))
Substituting the values:
\[ U = -\frac{(6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2) \cdot (6.42 \times 10^{23} \, \text{kg}) \cdot (1337 \, \text{kg})}{(45000 \times 10^3 \, \text{m} + 3.39 \times 10^6 \, \text{m})} \]
Calculating this will give us the potential energy at the initial altitude.
b. The change in energy of the satellite during the descent will be the difference between the potential energy at the initial altitude and the potential energy at the final altitude.
c. To calculate the orbital speed of the satellite after descending to the altitude of 316 km, we can use the formula for orbital speed:
\[ v = \sqrt{\frac{G \cdot M}{r}} \]
Where:
- \( v \) is the orbital speed
- \( G \) is the gravitational constant
- \( M \) is the mass of Mars
- \( r \) is the distance from the center of Mars to the satellite's altitude (316 km + radius of Mars)
Substituting the values, we can calculate the orbital speed.
a. To calculate the potential energy of the satellite at the initial altitude, we can use the formula for gravitational potential energy:
\[ U = -\frac{G \cdot M \cdot m}{r} \]
Where:
- \( U \) is the potential energy
- \( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2\))
- \( M \) is the mass of Mars (\(6.42 \times 10^{23} \, \text{kg}\))
- \( m \) is the mass of the satellite (\(1337 \, \text{kg}\))
- \( r \) is the distance from the center of Mars to the satellite's initial altitude (\(45000 \, \text{km} + 3.39 \times 10^6 \, \text{m}\))
Substituting the values:
\[ U = -\frac{(6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2) \cdot (6.42 \times 10^{23} \, \text{kg}) \cdot (1337 \, \text{kg})}{(45000 \times 10^3 \, \text{m} + 3.39 \times 10^6 \, \text{m})} \]
Calculating this will give us the potential energy at the initial altitude.
b. The change in energy of the satellite during the descent will be the difference between the potential energy at the initial altitude and the potential energy at the final altitude.
c. To calculate the orbital speed of the satellite after descending to the altitude of 316 km, we can use the formula for orbital speed:
\[ v = \sqrt{\frac{G \cdot M}{r}} \]
Where:
- \( v \) is the orbital speed
- \( G \) is the gravitational constant
- \( M \) is the mass of Mars
- \( r \) is the distance from the center of Mars to the satellite's altitude (316 km + radius of Mars)
Substituting the values, we can calculate the orbital speed.
a. To calculate the potential energy of the satellite at the initial altitude, we use the formula for gravitational potential energy:
PE = (G * M * m) / r
Where:
- G is the gravitational constant (6.674 × 10^-11 Nm^2/kg^2)
- M is the mass of the planet Mars (6.42 × 10^23 kg)
- m is the mass of the satellite (1337 kg)
- r is the distance from the center of Mars to the satellite's initial altitude (in meters)
Given:
- r = altitude + radius of Mars = 45000 km + 3.39 × 10^6 m
b. To calculate the change in energy of the satellite during the descent, we subtract the potential energy at the initial altitude from the potential energy at the final altitude.
c. To calculate the orbital speed of the satellite after descending to the altitude of 316 km, we use the formula for orbital speed:
v = sqrt((G * M) / r)
Where:
- v is the orbital speed
- G is the gravitational constant
- M is the mass of the planet Mars
- r is the distance from the center of Mars to the satellite's altitude after descending (in meters)
Given the new altitude, we calculate r and then use the formula to find v. Let's calculate these values.
PE = (G * M * m) / r
Where:
- G is the gravitational constant (6.674 × 10^-11 Nm^2/kg^2)
- M is the mass of the planet Mars (6.42 × 10^23 kg)
- m is the mass of the satellite (1337 kg)
- r is the distance from the center of Mars to the satellite's initial altitude (in meters)
Given:
- r = altitude + radius of Mars = 45000 km + 3.39 × 10^6 m
b. To calculate the change in energy of the satellite during the descent, we subtract the potential energy at the initial altitude from the potential energy at the final altitude.
c. To calculate the orbital speed of the satellite after descending to the altitude of 316 km, we use the formula for orbital speed:
v = sqrt((G * M) / r)
Where:
- v is the orbital speed
- G is the gravitational constant
- M is the mass of the planet Mars
- r is the distance from the center of Mars to the satellite's altitude after descending (in meters)
Given the new altitude, we calculate r and then use the formula to find v. Let's calculate these values.
