Question

Two identical satellites orbit the earth in stable orbits. one satellite orbits with a speed v at a distance r from the center of the earth. the second satellite travels at a speed that is less than v. at what distance from the center of the earth does the second satellite orbit? view available hint(s)

Answer 1

The available options are: (found the complete text on internet)
A- at a distance less than r
B- at a distance equal to r
C- at a distance greater than r

Solution:
The correct answer is C) at a distance greater than r.

In fact, the gravitational attraction between the satellite and the Earth provides the centripetal force that keeps the satellite in circular orbit, so we can write
$G \frac{Mm}{r^2}=m \frac{v^2}{r}$
where the term on the left is the gravitational force, while the term on the right is the centripetal force, and where
G is the gravitational constant
M is the Earth mass
m is the satellite mass
r is the distance of the satellite from the Earth's center
v is the satellite speed

Re-arranging the equation, we get
$r= \frac{GM}{v^2}$
and we see from this formula that, if the second satellite has a speed less than the speed v of the first satellite, it means that the denominator of the fraction is smaller, and so r is larger for the second satellite.