Calculate the speed and altitude of a satellite in a circular orbit around Earth, given Earth's mass and radius. Verify that the satellite follows Kepler's third law.
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To calculate the speed and altitude of a satellite in a circular orbit around Earth, we can use the following equations:
1. Speed of satellite (v) = √(GM/r)
where G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2), M is Earth's mass (5.97219 x 10^24 kg), and r is the radius of the orbit.
2. Altitude of satellite (h) = r - R
where R is Earth's radius (6,371 km).
To verify Kepler's third law, we need to compare the satellite's orbital period (T) with the average distance from the center of the Earth (a). According to Kepler's third law:
T^2 = (4π^2 / GM) * a^3
Let's use these equations to find the speed, altitude, and verify Kepler's third law.
Calculations:
1. Speed of satellite:
Earth's mass (M) = 5.97219 x 10^24 kg
Earth's radius (R) = 6,371 km = 6,371,000 m
Radius of circular orbit (r) = R + h (where h = altitude of satellite)
Plug in the values:
v = √(GM/r)
= √((6.67430 x 10^-11 m^3 kg^-1 s^-2) * (5.97219 x 10^24 kg) / (R + h))
2. Altitude of satellite:
h = r - R
3. Kepler's Third Law:
Calculate orbital period (T):
Use the time for one revolution around Earth, which is equal to 24 hours or 86,400 seconds.
T^2 = (4π^2 / GM) * a^3
Solve for a:
a = (T^2 * GM / (4π^2)) ^(1/3)
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