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The Global Positioning System (GPS) is based on a rather simple principle. Satellites on 12-hour orbits emit perfectly synchronized signals then detected by a receiver. The receiver cannot carry perfect clock and therefore can detect only differences of distances to the satellites (i.e. it cannot measure the distance, but only the difference between two distances). Four satellites are enough to calculate the position.

Explain, why is the accuracy of GPS significantly better in the horizontal direction than in the vertical direction.

Measure the coefficient of reflexivity of aluminium foil in the visible light. Suggest an appropriate method. Do not forget to describe the side of the foil you measure.

In this problem we analyse and interpret measurements made in 1994 on radio wave emition from a source consisting multiple bodies within our galaxy. The distance to the central celestial body from Earth is estimated to be $R = 3,86.10^{20}$ m. The angular velocities of two objects ejected from the centre in opposite directions were measuredyo be: $\omega _{1} = 9,73.10^{-13} rad.s^{-1}$ and $\omega _{2} = 4,42.10^{-13} rad.s^{-1}$. We calculate the transverse velocities: $v_{1} = R\omega _{1} =3,76.10^{8} m.s^{-1}$ and $v_{2} = R\omega _{2} = 1,71.10^{8} m.s^{-1}$. The first object is faster than light! How is it possible?

Let's consider an object moving with velocity $v$. The angle between the velocity vector and the direction to the observer is $\varphi$. The distance to the observer is denoted $R$. Calculate the angular velocity as seen by the observer. Can $Rω$ be greater than the speed of light? Using your results calculate the real velocities of the two objects. Assume that the velocities are equal.