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

Step-by-step 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(M)/(r)}` (1)

Where:

`V` is the velocity of the satellite

`G=6.674(10)^(-11)(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=(2\pi r)/(T)` (2)

Where
`T=24 h (3600 s)/(1 h)=86400 s`

Combining (1) and (2):

`\sqrt{G(M)/(r)}=(2\pi r)/(T)` (3)

Isolating
`r`:

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

`r=42,240,234.3 m` (5)

Substituting (5) in (1):

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

Finally:

`V=3071.79 m/s`