(II) Two Earth satellites, A and B, each of mass m = 950 kg , are launched into circular orbits around the Earth’s center. Satellite A orbits at an altitude of 4800 km, and satellite B orbits at an altitude of 12,600 km.
(b) What are the kinetic energies of the two satellites?

Verified step by step guidance

1. Step 1: Calculate the radius of each satellite's orbit by adding the Earth's radius to the altitude of each satellite. The radius of the Earth is approximately 6371 km. For satellite A, the total radius is 6371 km + 4800 km. For satellite B, the total radius is 6371 km + 12600 km.
2. Step 2: Use the formula for gravitational force to find the velocity of each satellite. The gravitational force acting on a satellite is given by F = \frac{G \cdot M \cdot m}{r^2}, where G is the gravitational constant, M is the mass of the Earth, and r is the radius of the orbit. The centripetal force needed to keep the satellite in orbit is provided by this gravitational force, and it is also given by F = \frac{m \cdot v^2}{r}.
3. Step 3: Equate the expressions for gravitational force and centripetal force to solve for the velocity v of each satellite. Set \frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r} and solve for v. Simplify to get v = \sqrt{\frac{G \cdot M}{r}}.
4. Step 4: Calculate the kinetic energy for each satellite using the formula KE = \frac{1}{2} m v^2. Substitute the mass of the satellite and the velocity expression from Step 3 into this formula.
5. Step 5: Simplify the kinetic energy expression by substituting v from Step 3 into the kinetic energy formula to get KE = \frac{1}{2} m \left(\sqrt{\frac{G \cdot M}{r}}\right)^2. This simplifies to KE = \frac{1}{2} m \frac{G \cdot M}{r}.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses due to its position in a gravitational field. For satellites, it is calculated using the formula U = -G(Mm/r), where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance from the center of the Earth to the satellite. This concept is essential for understanding the energy dynamics of satellites in orbit.

Kinetic Energy in Circular Motion

The kinetic energy (KE) of an object in circular motion is given by the formula KE = (1/2)mv², where m is the mass and v is the orbital speed. For satellites, the orbital speed can be derived from the gravitational force acting as the centripetal force, leading to the relationship v = √(GM/r). This concept is crucial for calculating the kinetic energies of the satellites in their respective orbits.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. It encompasses principles such as Kepler's laws of planetary motion and the equations governing circular orbits. Understanding these principles is vital for determining the parameters of satellite orbits, including altitude, speed, and energy, which are necessary for solving the problem at hand.