Question

A satellite orbits the Earth such that it remains above the same point on the equator at all times. Another satellite orbits the Earth at a distance (from the Earth’s center) that is 4 times as far as the previous satellite. How long does it take the second satellite to orbit the Earth?

Answer 1

this is a matter of identifying the relationship between period and the radius

Answer 2

\[\frac{ T ^{2} }{ R ^{3} }=constant\]

Answer 3

let us suppose the period for the first planet is T1 radius R1 and for the second T2 and R2

Answer 4

\[\frac{ T _{1}^{2} }{ R _{1}^{3} }=\frac{ T _{2}^{2} }{ R _{2}^{3} }\]

Answer 5

express R2 in terms of R1

Answer 6

\[R _{2}=4R _{1} => R _{2}^{3}=64R _{1}^{3}\]

Answer 7

\[\frac{ T _{1}^{2} }{ R _{1}^{3} }=\frac{ T _{2} ^{2}}{ 64R _{1}^{3}}=>T _{2}^{2}=64T _{1}^{2}\]

Answer 8

\[T _{2}=8T_{1}\]