Two satellites are in circular orbits around the earth. The orbit for satellite A is at a height of 418 km above the earth’s surface, while that for satellite B is at a height of 731 km. Find the orbital speed for (a) satellite A and (b) satellite B.
F = G M m/R^2 = m v^2/R
so v^2 = G M/R
G = 6.67 * 110^-11
M = 6 * 10^24
Re = 6.38*10^6 meters
Ra = 6.38*10^6 + .418*10^6
Rb = 6.38*10^6 + .731*10^6
Ra = 680700
Rb = 7253000
To find the orbital speed of a satellite, we can use the formula:
v = √(G * M / r)
Where:
- v is the orbital speed of the satellite
- G is the gravitational constant (approximately 6.67430 x 10^-11 N(m/kg)^2)
- M is the mass of the Earth (approximately 5.972 x 10^24 kg)
- r is the distance between the satellite and the center of the Earth
(a) To find the orbital speed of satellite A, we need to calculate the distance between the satellite and the center of the Earth. The orbit for satellite A is at a height of 418 km above the Earth's surface.
The distance between the satellite and the center of the Earth can be found by adding the radius of the Earth (approximately 6371 km) to the height of the satellite above the Earth's surface:
r = (6371 km + 418 km) = 6790 km
Now we can calculate the orbital speed:
v = √(G * M / r)
= √((6.67430 x 10^-11 N(m/kg)^2) * (5.972 x 10^24 kg) / (6790 km * 1000 m/km))
= √(40.2706 N(m/kg)^2 * 5.972 x 10^27 kg / 6790000 m)
= √(242.0562 x 10^(10+27-6))
= √(242.0562 x 10^31)
≈ 1.033 x 10^4 m/s
So, the orbital speed of satellite A is approximately 1.033 x 10^4 m/s.
(b) To find the orbital speed of satellite B, we follow the same steps as in part (a) but with a different distance between the satellite and the center of the Earth. The orbit for satellite B is at a height of 731 km above the Earth's surface:
r = (6371 km + 731 km) = 7102 km
Now we can calculate the orbital speed:
v = √(G * M / r)
= √((6.67430 x 10^-11 N(m/kg)^2) * (5.972 x 10^24 kg) / (7102 km * 1000 m/km))
= √(40.2706 N(m/kg)^2 * 5.972 x 10^27 kg / 7102000 m)
= √(286.726 x 10^(10+27-6))
= √(286.726 x 10^31)
≈ 1.074 x 10^4 m/s
So, the orbital speed of satellite B is approximately 1.074 x 10^4 m/s.