Question

A rocket is used to place a synchronous satellite in orbit about the earth. What is the speed of the satellite in orbit?

Answer 1

Answer: 3071.79 m/s

Explanation:

Approaching satellite's orbit around the Earth to a circular orbit, we can use the equation of velocity in the case of uniform circular motion:  

$V=\sqrt{G\frac{M}{r}}$ (1)  

Where:

$V$ is the velocity of the satellite

$G=6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}$ is the Gravitational Constant  

$M=5.972(10)^{24} kg$ is the mass of the Earth

$r$ is the radius of the orbit

Now, if we want the satellite to move in a geosynchronous orbit, it needs to travel at a specific orbiting radius and period to fulfill this condition. This means the satellite's orbital period $T$ must match Earth's rotation on its axis, which takes one sidereal day ( approximately 24 h).

So, if the satellite travels one complete circle $C=2\pi r$ in a period $T$, its velocity is also expressed as:

$V=\frac{2\pi r}{T}$ (2)  

Where $T=24 h \frac{3600 s}{1 h}=86400 s$

Combining (1) and (2):

$\sqrt{G\frac{M}{r}}=\frac{2\pi r}{T}$ (3)  

Isolating $r$:

$r=\sqrt[3]{x\frac{GMT^{2}}{4 \pi^{2}}}$ (4)  

$r=42,240,234.3 m$ (5)  

Substituting (5) in (1):

$V=\sqrt{6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}\frac{5.972(10)^{24} kg}{42,240,234.3 m}}$ (6)  

Finally:

$V=3071.79 m/s$