Find the speed of a satellite in a circular orbit around the Earth with a radius 3.65 times the mean radius of the Earth; RE= 6.37·103 km ME= 5.98·1024 kg
I must admit I am not clear on what you mean by "mean radius" of the earth; average, arithmetic, geometric, harmonic, etc. I am making my own interpretation to be the radius that divides the spherical earth into two equal volumes, ignoring the equatorial bulge.
Using an earth radius of 3963 miles, yields a volume of 3.8376x10^22 cub.ft.From the half volume of 1.9188x10^22 cub.ft., I derive a half volume radius of 3145 miles or 16,607,898 ft. The radius of the orbit you defined is therefore 3145 x 3.65 = 11,795.25 miles or 60,610,440 ft.
The orbital circular velocity of a satellite at any altitude derives from Vc = sqrt(µ/Ro) where Vc = the orbital velocity in feet per second, µ = the earth's gravitational constant = Re^2g, Ro = the orbital radius and Re = the earth radius. With µ = 3963(5280)32.19 ~= 1.407974x10^16, the satellite's orbital velocity becomes
Vc = sqrt[{20,924,640)^2 x 32.19}/60,610,440] = 15,249fps = 10,397mph.
If this is not exactly the result you desired, please post your specific wants.
If you should find any inconsistencies in my math-magic, please don't hesitate to let me know.
To find the speed of a satellite in a circular orbit around the Earth, we can use the formula for the orbital speed:
v = sqrt(G * M / r)
where:
- v is the speed of the satellite
- G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the Earth
- r is the radius of the circular orbit
To solve the problem, we need to convert the values given into SI units (meters and kilograms) and then substitute them into the formula.
The radius of the circular orbit is given as 3.65 times the mean radius of the Earth, which can be calculated as:
r = 3.65 * RE
where RE is the mean radius of the Earth.
The mean radius of the Earth is given as 6.37 x 10^3 km, which can be converted to meters by multiplying by 1000:
RE = 6.37 x 10^3 km * 1000 = 6.37 x 10^6 meters
Substituting the values into the formula, we get:
v = sqrt(G * M / r)
v = sqrt(6.67430 x 10^-11 m^3 kg^-1 s^-2 * 5.98 x 10^24 kg / (3.65 * 6.37 x 10^6 meters))
Now we can calculate the value using a scientific calculator or an online calculator.