# Two satellites, A and B, are in different circular orbits about the earth. The orbital speed of...

## Question:

Two satellites, A and B, are in different circular orbits about the earth. The orbital speed of satellite A is sixty-six times that of satellite B. Find the ratio of the periods of the satellites.

## Satellites in Circular Orbits:

When a satellite is circling around the Earth, the gravitational force provides the source of the centripetal force to maintain the circular orbit. The centripetal acceleration is defined as the square of the orbital speed divided by the orbital radius. Also, the orbital period squared is proportional to the orbital radius cubed.

When considering a circular orbit of a satellite around the Earth, the orbital speed v is inversely proportional to the square root of the orbital radius r. That is, {eq}v^2\propto r^{-1} {/eq}

Let the orbital speed of satellite A be {eq}V_A {/eq} and that of satellite B be {eq}V_B {/eq}. The relationship between them is given as

{eq}V_A=66V_B {/eq}

Let the orbital radius of satellite A be {eq}R_A {/eq} and that of satellite B be {eq}R_B {/eq}

Using the above relation, we have

{eq}\dfrac{R_A}{R_B}=\left (\dfrac{V_B}{V_A}\right )^2 {/eq}

Let the orbital period of satellite A be {eq}T_A {/eq} and that of satellite B be {eq}T_B {/eq}

The circumference of the orbit is the orbital speed multiplied by the orbital period. So, the ratio between them is

{eq}\begin{align} \dfrac{T_A}{T_B}&=\dfrac{2\pi R_A/V_A}{2\pi R_B/V_B}\\\\ &=\dfrac{R_A}{R_B}\dfrac{V_B}{V_A}\\\\ &=\left( \dfrac{V_B}{V_A}\right )^3\\\\ &=\left(\dfrac{1}{66}\right )^3\\\\ &=\boxed{3.48\times 10^{-6}\, s} \end{align} {/eq} 