Problem Set 11
(15.6) GPS Satellites and Relativity
(a) How fast do GPS satellites travel?
🛰️ GPS satellites stay in orbit because gravity pulls them inward while they move sideways fast enough to fall around the Earth. We use the orbital velocity formula:
Set centripetal force equal to gravitational force:
\[ \frac{mv^2}{r} = \frac{GMm}{r^2} \quad \Rightarrow \quad v = \sqrt{\frac{GM}{r}} \]
\[ v = \sqrt{\frac{GM}{r}} \]
- Gravitational constant: \( G = 6.67 \times 10^{-11} \, \text{m}^3/\text{kg}/\text{s}^2 \)
- Earth’s mass: \( M = 5.97 \times 10^{24} \, \text{kg} \)
- Earth’s radius: \( R_E = 6.37 \times 10^6 \, \text{m} \)
- Altitude of satellite: \( h = 2.018 \times 10^7 \, \text{m} \)
- Total orbital radius: \( r = R_E + h = 2.6558 \times 10^7 \, \text{m} \)
Substitute into the equation: \[ v = \sqrt{\frac{6.67 \times 10^{-11} \cdot 5.97 \times 10^{24}}{2.6558 \times 10^7}} \approx 3.87 \times 10^3 \, \text{m/s} \]
Conclusion: GPS satellites travel at about 3.87 km/s.
(b) What is their orbital period?
Imagine a GPS satellite zooming around the Earth way up in space.
To find how long one trip around the Earth takes, we use a simple idea:
If you know how far something has to go, and how fast it’s moving,
you can figure out how long it takes!
The Formula:
We use this equation:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
The distance is one full circle around Earth, so we use:
\[ \text{Distance} = 2\pi r \]
- \( r = 26,\!558,\!000 \, \text{m} \) (the satellite's distance from Earth's center)
- \( v = 3,\!870 \, \text{m/s} \) (the satellite's speed)
Plug in the numbers:
\[ T = \frac{2\pi \cdot 26,\!558,\!000}{3,\!870} \approx 43,\!100 \, \text{seconds} \]
How long is that in hours?
\[ \frac{43,\!100}{3600} \approx 11.96 \, \text{hours} \]
Final Answer:
A GPS satellite takes about 12 hours to go around the Earth once.
That means it completes two full orbits every day!
(c) Special-relativistic correction: which clock goes slower?
Einstein told us something really surprising:
Time goes slower for things that move really fast!
This is called special relativity.
GPS satellites move very fast — over 3,800 meters every second!
From our point of view on Earth, their clocks tick just a little bit slower than ours.
We use a formula to figure out how much slower:
\[ \Delta t = \frac{1}{2} \left( \frac{v}{c} \right)^2 \cdot T \]
- \( v = 3,\!870 \, \text{m/s} \) → the satellite’s speed
- \( c = 300,\!000,\!000 \, \text{m/s} \) → the speed of light
- \( T = 43,\!100 \, \text{seconds} \) → time for one full orbit
Plug in the numbers:
\[ \Delta t = \frac{1}{2} \left( \frac{3,\!870}{300,\!000,\!000} \right)^2 \cdot 43,\!100 \approx 3.59 \, \mu\text{s} \]
This means the clock on the GPS satellite ticks 3.59 microseconds slower than clocks on Earth during each orbit.
Final Answer:
The satellite clock loses about 3.6 millionths of a second every orbit just because it’s moving fast. That’s super tiny — but for GPS, even this matters a lot!
(d) General-relativistic correction: which clock goes slower?
Einstein had another amazing idea:
Gravity also slows down time!This is called general relativity.
The closer you are to Earth, the slower time goes.
So:
- Clocks on Earth tick slower
- Clocks in orbit tick faster
Why? Because the satellite is farther from Earth's gravity, so time runs a little faster for it.
The formula to calculate the difference is:
\[ \Delta t = \left( \frac{GM}{R_E c^2} - \frac{GM}{r c^2} \right) \cdot T \]
- \( GM = 3.99 \times 10^{14} \, \text{m}^3/\text{s}^2 \)
- \( R_E = 6.37 \times 10^6 \, \text{m} \)
- \( r = 2.6558 \times 10^7 \, \text{m} \)
- \( c = 3 \times 10^8 \, \text{m/s} \)
- \( T = 43,\!100 \, \text{s} \)
Plug in the numbers:
\[ \Delta t \approx 2.28 \times 10^{-5} \, \text{seconds} = 22.8 \, \mu\text{s} \]
Final Answer:
Because Earth’s gravity is stronger at the surface, clocks on Earth tick slower.
The satellite clock ticks about 22.8 microseconds faster per orbit than a clock on Earth!