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Rom4ik [11]
3 years ago
8

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

Physics
1 answer:
worty [1.4K]3 years ago
7 0

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 <u>geosynchronous orbit</u>, 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

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