An artificial Earth satellite, of mass 3.00 × 10^3 kg, has an elliptical orbit, with a mean altitude of 500 km.
(a) What is its mean value of gravitational potential energy while in orbit?
(b) What is its mean value of orbital kinetic energy?
(c) What is its total energy while in orbit?
(d) If its perigee is 150 km, what is its orbital velocity at perigee?
http://www.pha.jhu.edu/~broholm/l24/node1.html
See the post just before this one.
To calculate the values, we will need to use some formulas related to gravitational potential energy, orbital kinetic energy, and total energy.
(a) The gravitational potential energy of an object in a gravitational field is given by the formula:
Potential energy = - (G * M * m) / r
Where G is the gravitational constant (approximately 6.674 × 10^-11 Nm^2/kg^2), M is the mass of the Earth (approximately 5.972 × 10^24 kg), m is the mass of the satellite (3.00 × 10^3 kg), and r is the distance from the center of the Earth to the satellite (mean altitude + radius of the Earth).
To calculate the mean value of gravitational potential energy, we need to find the potential energy at both the apogee and perigee of the satellite's elliptical orbit, and then take the average of the two values.
(b) The orbital kinetic energy of an object in motion around another object is given by the formula:
Kinetic energy = (1/2) * m * v^2
Where m is the mass of the satellite and v is its velocity. In this case, we need to find the orbital velocity at the mean altitude of 500 km.
(c) The total energy of the satellite in orbit is the sum of the gravitational potential energy and the orbital kinetic energy.
Total energy = Potential energy + Kinetic energy
(d) To calculate the orbital velocity at perigee, we can use the concept of specific orbital energy, which is a constant for a given orbit. The specific orbital energy is the sum of the gravitational potential energy and the orbital kinetic energy per unit mass of the satellite.
Specific orbital energy = Total energy / m = - (G * M) / (2 * a)
Where a is the semi-major axis of the elliptical orbit (mean altitude + radius of the Earth).
We can then use the specific orbital energy to calculate the orbital velocity at perigee using the formula:
Orbital velocity = sqrt(2 * (specific orbital energy + (G * M) / r))
Let's plug in the values and calculate the answers:
(a) Mean value of gravitational potential energy:
Gravitational potential energy at apogee = - (G * M * m) / (apogee)
Gravitational potential energy at perigee = - (G * M * m) / (perigee)
Mean value = (potential energy at apogee + potential energy at perigee) / 2
(b) Mean value of orbital kinetic energy:
Calculate the orbital velocity at mean altitude using the formula: Orbital velocity = sqrt((G * M) / r)
Orbital kinetic energy at mean altitude = (1/2) * m * (orbital velocity at mean altitude)^2
(c) Total energy = Mean value of gravitational potential energy + Mean value of orbital kinetic energy
(d) Orbital velocity at perigee:
Calculate the specific orbital energy using the formula: Specific orbital energy = - (G * M) / (2 * a)
Orbital velocity at perigee = sqrt(2 * (Specific orbital energy + (G * M) / perigee))
By applying these formulas and calculations, we can find the values for each question.