Final answer:
The relationship between the speeds of the two satellites can be determined using the principle of conservation of angular momentum. The orbital period, T, of satellite B can be calculated using Kepler's third law.
Step-by-step explanation:
The Relationship Between the Speeds of the Two Satellites
The relationship between the speeds of the two satellites can be determined using the principle of conservation of angular momentum. Since both satellites are in the same circular orbit, their angular momentum is conserved. The angular momentum of a satellite in a circular orbit is given by L = mvr, where m is the mass of the satellite, v is its linear velocity, and r is the distance from the center of the orbit. Therefore, we have:
LA = mAvArA
LB = mBvBrB
Since the linear velocity is inversely proportional to the radius, we can write:
vB/vA = rA/rB = (R + h)/(R) = (1 + h/R),
where R is the radius of the Earth and h is the distance of the satellites from the Earth's surface.
Finding the Orbital Period of Satellite B
The orbital period, T, of a satellite is the time it takes to complete one orbit. It can be calculated using Kepler's third law:
T² = 4π²r³/(GM),
where r is the average radius of the orbit, G is the gravitational constant, and M is the mass of the Earth.
For satellite B, the average radius of the orbit is R + h, where R is the radius of the Earth and h is the distance of the satellite from the Earth's surface. Substituting the values into the equation, we have:
T² = 4π²(R + h)³/(GM),
Simplifying the equation and solving for T, we find:
T = √((4π²(R + h)³)/(GM)).