BACKGROUND The Global Positioning System (GPS) is a network of at least 24 satellites that orbit Earth. Each satellite has a clock that is synchronized to the other satellites' clocks and to clocks on Earth. A GPS receiver on or near Earth's surface can quickly, accurately, and reliably determine its own position by communicating with this network of satellites. The GPS satellites are distributed uniformly along six different orbits (four satellites per orbit) at an average altitude of 20 200 km. This arrangement allows at least four satellites to be visible to any receiver on or near Earth's surface at any time. Each satellite orbits Earth twice per day, at an average speed of 14 000 km/h (0.00001296c). HOW DOES A RECEIVER DETERMINE ITS POSITION? Definitions of subscripts/symbols used subscript "Rreceiver's clock (on Earth) subscript "GPS" satellite's clock subscript "1" sent signal subscript "2"arriving signal Examples: Rtime on receiver's clock when signal arrives GPS time on satellite's clock when signal sent i. While orbiting, each satellite continuously broadcasts its orbital position (latitude, longitude, and altitude) via radio- i. Each position signal is "timestamped," meaning the satellite records the time at which it sends the signal (EaPs,) in ii. When a GPS receiver detects one of these signals, it records the time at which the signal arrives (t2) and compares it frequency signals. the signal itself to the timestamp indicating when it was sent (tars,) to calculate the signal's travel time (ttravet): (NOTE: This assumes both clocks are synchronizedtors) iv. Finally, knowing the signal's travel time (Ceravel, and that it travels at the speed of light (c), the receiver calculates its distance from the satellite's position when the signal was sent (i.e., how far the signal travelled): d vtctgravet This distance represents the radius of a spherical surface, centred at the sat which the receiver must be located. ellite's location when it sent the signal, on v. Distances from at least four satellites must be known in order to reduce the receiver's possible positions from a spherical surface to a single point in three-dimensional space (latitude, longitude, and altitude). SIMPLIFICATIONS FOR THIS ASSIGNMENT For this assignment, we will only consider the calculation of a receiver's distance from a single satellite. As well, we will assume that only time dilation, as predicted by special relativity, will affect the synchronicity of the clocks Assume that both clocks are synchronized initially (ttoes initially) . once they are synchronized, they remain synchronized (t tars at all times) at the moment the signal is sent, the time on the receiver's clock will be the same as that on the satellite's clock (t, = ters!), therefore, we can use the formula below to calculate ttrom Your GPS receiver receives a signal from a GPS satellite. The signal is timestamped with tars,-37.02376641 s and it arrives at your receiver at 37.09110074 s 1. a) Determine the signal's travel time. (Answer in seconds to eight desimal plases) b) How far are you from the satellite's position when it sent the signal? (Answer to the nearest metre.) Assume that: both clocks are synchronized initially (tn tars initialily) subsequently, they "drift" increasingly out of sync due to time dilation (teCons at all other times) at the moment the signal is sent, the time on the receiver's clock will NOT be the same as that on the satellite's clock (tR3 # tGPS, ), therefore, you CANNOT calculate ttre e, the same way you did in Part A. 2. Does the satellite's clock run slower or faster than the receiver's clock? 3. I 1 day has passed on Earth since the clocks were synchronized, by how much time will the clocks be HINTS i) Co l) Find tarure at the moment the signal was sent, where taurt raster cleck -ter clock If this time dilation is left uncorrected, the calculations of the signal's travel time and the receiver's distance from the satellite will also be incorrect! Complete the questions below to co rrect the calculations in Part A for time dilation 4. Determine the time on the receiver's clock when the signal is sent (Answer in seconds to eight decimal places ) 5. Determine the signal's true travel time. 6. Determine your true distance from the satellite's position when it sent the signal. If the receiver did NOT make corrections for time dilation, how far off would its calculation of your location be? Answer to the nearest metre) 7