A satellite of mass 2500kg is orbiting the Earth in an elliptical orbit.At the farthest point from the Earth,its altitude is 3600km,while at the nearest point,it is 1100km.calculate the energy and angular momentm of the satellite and its speed at the aphelion and perihelion.
To calculate the energy and angular momentum of the satellite, as well as its speed at the aphelion and perihelion, we can use some basic principles of orbital mechanics.
First, let's calculate the energy of the satellite using the formula for specific orbital energy:
E = -GMm / 2a
Where:
E is the specific orbital energy of the satellite,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
M is the mass of the Earth (approximately 5.972 × 10^24 kg),
m is the mass of the satellite (given as 2500 kg), and
a is the semi-major axis of the elliptical orbit (half of the sum of the farthest and nearest distances from the Earth).
Given that the farthest point from the Earth is 3600 km and the nearest point is 1100 km, we can convert these distances to meters by multiplying by 1000:
a = (3600 km + 1100 km) / 2 = 4700 km / 2 = 2350 km = 2,350,000 meters
Substituting these values into the energy formula:
E = - (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.972 × 10^24 kg) * (2500 kg) / (2 * 2,350,000 m)
Calculating this expression will give you the value of the specific orbital energy.
To calculate the angular momentum of the satellite, we can use the formula:
L = mvr
Where:
L is the angular momentum of the satellite,
m is the mass of the satellite (given as 2500 kg),
v is the velocity of the satellite, and
r is the distance from the satellite to the center of the Earth.
At the farthest point from the Earth (aphelion), the distance from the satellite to the center of the Earth is 3600 km = 3,600,000 meters. At the nearest point (perihelion), it is 1100 km = 1,100,000 meters.
Using the formula, we can calculate the angular momentum of the satellite at both points.
Finally, to determine the speed of the satellite at the aphelion and perihelion, we can use the formula:
v = sqrt(GM (2 / r - 1 / a))
Where:
v is the velocity of the satellite,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
M is the mass of the Earth (approximately 5.972 × 10^24 kg),
r is the distance from the satellite to the center of the Earth, and
a is the semi-major axis of the elliptical orbit (as calculated before).
Using this formula, we can calculate the velocity of the satellite at both the aphelion and perihelion distances.
By following these calculations, you should be able to determine the energy and angular momentum of the satellite, as well as its speed at the aphelion and perihelion distances.