Relativity correction to GPS
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
GPS satellites move at about kilometerpersecond. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to one part in numpre.
Solution:
Let's calculate the magnitude of the time dilation effect: Delta t fracsqrt-fracv^c^ Delta t_ fracsqrt-numpr. Delta t_ Since the accuracy of most pocket calculators is not within ^- the calculator will say Delta t Delta t_. We need binomial expansion or taylor series to get results. The binomial expansion of pm x^n approx mp nx for xll . With n-frac we get -x^-frac -frac x i.e.: Delta t &approx -frac numpr. Delta t_ The time error divided by the time erval is: fracDelta t-Delta t_Delta t_ approx numpr Time dilation if not accounted for would roduce an error of about part in numpre which is numpr times greater than the precision of atomic clocks. Not correcting for time dilation means a receiver could give a much poorer position accuracy. GPS devices must make other corrections as well including effects associated with General Relativity. This effect is even bigger.
GPS satellites move at about kilometerpersecond. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to one part in numpre.
Solution:
Let's calculate the magnitude of the time dilation effect: Delta t fracsqrt-fracv^c^ Delta t_ fracsqrt-numpr. Delta t_ Since the accuracy of most pocket calculators is not within ^- the calculator will say Delta t Delta t_. We need binomial expansion or taylor series to get results. The binomial expansion of pm x^n approx mp nx for xll . With n-frac we get -x^-frac -frac x i.e.: Delta t &approx -frac numpr. Delta t_ The time error divided by the time erval is: fracDelta t-Delta t_Delta t_ approx numpr Time dilation if not accounted for would roduce an error of about part in numpre which is numpr times greater than the precision of atomic clocks. Not correcting for time dilation means a receiver could give a much poorer position accuracy. GPS devices must make other corrections as well including effects associated with General Relativity. This effect is even bigger.