Two satellites S1 and S2 orbit around a planet P in circular orbits of radii r1 = 5.25*10^6 m,...

Question:

Two satellites S1 and S2 orbit around a planet P in circular orbits of radii r1 = 5.25*10{eq}^6 {/eq} m, and r2 = 8.60*10{eq}^6 {/eq} m respectively. If the speed of the first satellite S1 is 1.65*10{eq}^4 {/eq} m/s, what is the speed of the second satellite S2? m/s

The Orbital speed of a satellite about Earth:

A satellite that is orbiting the Earth at its orbit has an orbital speed.

The orbital speed of the satellite is given by;

{eq}v = \sqrt{\dfrac{GM}{r}} {/eq}

where,

  • {eq}M {/eq} is the mass of the Earth.
  • {eq}r {/eq} is orbiting radius of the satellite.
  • {eq}G = 6.67 \times 10^{-11} \ Nm^2/kg^2 {/eq} is the gravitational constant.

Answer and Explanation:

Given:

  • The orbiting radius of first satellite (S1) around the planet is {eq}r_1 = 5.25 \times 10^6\ m {/eq}
  • The orbitting radius of second satellite (S2) around the planet is {eq}r_2 = 8.60 \times 10^6\ m {/eq}
  • Speed of the first satellite is {eq}v_1 = 1.65\times 10^4\ m/s {/eq}

Let

  • Speed of the second satellite is {eq}v_2 {/eq}

As the orbital speed of the satellite is related to the radius is;

{eq}\begin{align} v &=\sqrt{\dfrac{GM}{r}}\\ \end{align} {/eq}

So, the speed of the first satellite is given by;

{eq}\begin{align} v_1 &=\sqrt{\dfrac{GM}{r_1}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Eqn. 1}\\ \end{align} {/eq}

Similarly, the speed of the second satellite is given by;

{eq}\begin{align} v_2 &=\sqrt{\dfrac{GM}{r_2}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Eqn. 2}\\ \end{align} {/eq}

Dividing equation (1) with (2), we get;

{eq}\begin{align} \dfrac{v_1}{v_2} &=\sqrt{\dfrac{r_2}{r_1}}\\ \implies v_2 &= v_1\sqrt{\dfrac{r_1}{r_2}}\\ &= (1.65\times 10^4\ m /s)\times\sqrt{\dfrac{5.5\times 10^6\ m}{8.60\times 10^6\ m}}\\ &=1.32\times 10^4\ m/s\\ \end{align} {/eq}

Therefore, the speed of the second satellite is {eq}v_2 =1.32\times 10^4\ m/s {/eq}


Learn more about this topic:

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How Orbits Are Influenced by Gravity & Energy

from Basics of Astronomy

Chapter 25 / Lesson 6
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