A GPS satellite orbits at an altitude of 2.0 x 10^7 m and a speed of 3.9 x 10^3 m/s, Earth’s radius is 6.4 x 10^6 m, and a point at its equator has tangential speed of 460 m/s, By approximately what fraction must the time be adjusted to account for both regular/speed-dependent and gravitational time dilation?
t(Earth) = t(satellite) / (square root) 1 - (v/c)^2
= t(satellite) /
(square root) 1 - [ (3.9 x 10^3) / (3x 10^8)^2 ]
= (1 - 1.7 x 10^-10)^ - 1/2 [ t(satellite) ]
t(Earth) = (1 + 8.5 x 10^-11) [ t(satellite) ]
GMEarth = (6.67 x 10^-11 N x m^2/kg^2)
(5.98 x 10^24 kg)
= 4.0 x 10^14 m^3/s^2
But how did you get (5.98 x 10^24 kg)??
To plug in
t(Earth) = t(satellite) [1 - (1/c^2) [ ( GM/r, Earth )
- ( GM/r, satellite )
To calculate the time dilation due to both regular/speed-dependent and gravitational effects, you can use the formula:
Δt' = Δt √(1 - (v^2/c^2)) / √(1 - (r_s/r)^2)
Where:
Δt' is the adjusted time interval
Δt is the original time interval
v is the speed of the GPS satellite
c is the speed of light (approximately 3 x 10^8 m/s)
r_s is the Schwarzschild radius of the Earth
r is the radius of the Earth plus the orbital altitude of the GPS satellite.
First, let's calculate the value of r_s using the formula:
r_s = 2GM/c^2
Where G is the gravitational constant (approximately 6.67 x 10^-11 Nm^2/kg^2), and M is the mass of the Earth (approximately 5.97 x 10^24 kg). Substituting the values, we have:
r_s = (2 * 6.67 x 10^-11 Nm^2/kg^2 * 5.97 x 10^24 kg) / (3 x 10^8 m/s)^2
Calculating this expression will give us the value of r_s, which is approximately 8.87 mm.
Now, let's calculate the adjusted time interval Δt'. We can use the given values:
Δt = 1 second (since we're considering a time interval of 1 second)
v = 3.9 x 10^3 m/s
c = 3 x 10^8 m/s
r = 6.4 x 10^6 m + 2.0 x 10^7 m
Plugging these values into the formula, we get:
Δt' = 1 √(1 - (3.9 x 10^3 m/s)^2 / (3 x 10^8 m/s)^2) / √(1 - (8.87 x 10^-3 m) / (6.4 x 10^6 m + 2.0 x 10^7 m)^2)
Calculating this expression will give us the adjusted time interval Δt', which represents the combined effect of regular/speed-dependent and gravitational time dilation.
Once you have the value of Δt', you can calculate the fraction by dividing Δt' by Δt:
Fraction = Δt' / Δt