The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using "triangulation" and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational "fixes." The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical mile = 1.852 km = 6076 ft].
(a) Determine the speed of each satellite.
(b) Determine the period of each satellite.
Never mind I figured it out! You use the equation v^2=(GM)/r
Remember to add the radius of the earth (6.371 million m0 to r
and for part b you use v=(pi2r)/T
To determine the speed of each satellite, we can use the formula for the orbital speed of a satellite. The orbital speed of a satellite can be calculated using the following formula:
v = √(G * M / r)
Where:
- v is the orbital speed of the satellite
- G is the gravitational constant (6.67430 x 10^(-11) m^3 kg^(-1) s^(-2))
- M is the mass of the Earth (5.97219 x 10^24 kg)
- r is the distance from the center of the Earth to the satellite (11,000 nautical miles + radius of the Earth)
First, let's convert the altitude of the satellite to meters:
11,000 nautical miles = 11,000 * 1.852 km * 1000 m = 20,372,000 m
Since the altitude is given as distance from the center of the Earth, we need to add the radius of the Earth to the altitude. The radius of the Earth is approximately 6,376 ft, which is approximately 1,950 m (1 ft = 0.3048 m).
So, the total distance from the center of the Earth to the satellite is:
r = 20,372,000 m + 1,950 m = 20,373,950 m
Now, we can calculate the speed of each satellite:
v = √((6.67430 x 10^(-11) m^3 kg^(-1) s^(-2)) * (5.97219 x 10^24 kg) / (20,373,950 m))
Calculating the value, we get:
v ≈ 3,872 m/s
Therefore, the speed of each satellite in the GPS system is approximately 3,872 m/s.
To determine the period of each satellite, we can use Kepler's Third Law, which states that the square of an object's orbital period is directly proportional to the cube of its semi-major axis.
The semi-major axis of the satellite's orbit is equal to the distance from the center of the Earth to the satellite-altitude + radius of the Earth. From the previous part, we determined that this value is approximately 20,373,950 m.
The period (T) of each satellite can be calculated using the following formula:
T = 2π * √(a^3 / (G * M))
Where:
- T is the period of the satellite
- a is the semi-major axis of the orbit
- G is the gravitational constant (6.67430 x 10^(-11) m^3 kg^(-1) s^(-2))
- M is the mass of the Earth (5.97219 x 10^24 kg)
Plugging in the values, we get:
T = 2π * √((20,373,950 m)^3 / ((6.67430 x 10^(-11) m^3 kg^(-1) s^(-2)) * (5.97219 x 10^24 kg)))
Calculating the value, we get:
T ≈ 43,000 seconds
Therefore, the period of each satellite in the GPS system is approximately 43,000 seconds.
To determine the speed of each satellite in the Navstar GPS system, we can start with the basic equation for orbital motion:
v = sqrt(G * M / r)
where:
v = speed of the satellite
G = gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
M = mass of the Earth (5.9722 x 10^24 kg)
r = radius of the satellite's orbit (converted to meters)
To convert the altitude of 11,000 nautical miles to meters, we can use the conversion factor given:
1 nautical mile = 1852 meters
Therefore, the radius of the satellite's orbit would be:
r = (11,000 nautical miles) * (1852 meters / 1 nautical mile)
r = 20,372,000 meters
Plugging these values into the equation, we can find the speed of each satellite:
v = sqrt((6.67430 x 10^-11 m^3 kg^-1 s^-2) * (5.9722 x 10^24 kg) / (20,372,000 meters))
Calculating this equation gives us:
v ≈ 3,873.7 m/s
Thus, the speed of each satellite in the Navstar GPS system is approximately 3,873.7 meters per second.
To determine the period of each satellite, we can use the formula:
T = 2π * sqrt(r^3 / (G * M))
Using the same values for r, G, and M as before, we can calculate the period:
T = 2π * sqrt((20,372,000 meters)^3 / ((6.67430 x 10^-11 m^3 kg^-1 s^-2) * (5.9722 x 10^24 kg)))
Simplifying this equation yields:
T ≈ 24.01 hours
Therefore, the period of each satellite in the Navstar GPS system is approximately 24.01 hours.