The National weather Service operates network of 159 Doppler weather radars that

Question

The National weather Service operates network of 159 Doppler weather radars that are used to monitor Weather across the nation. These radars constantly scan the sky with electromagnetic (EM) waves that travel at the speed of light (symbolized with c = 3 times 10^3 m/s).typically with a wavelength of 10 cm and a frequency of 3 GHz (1 GHz = 10^3 HZ). The EM waves reflect off precipitation (i.e. clouds, rain, snow, etc.) in the air, and are collected by the radar that originally emitted the waves. The intensity of the reflected waves corresponds to the speed. is amount of precipitation in the air, and the Doppler frequency shift of the reflected waves depends on how fast the precipitation is moving towards or away from the radar, which allows for the remote detection of wind speed. A Doppler weather radar is scanning a distant storm and detects reflected EM waves with a frequency that was 400 Hz higher than the 3 GHz waves it had emitted. What speed and direction must this storm be moving in?

Explanation / Answer

From relativistic Doppler effect

                  fD = fo (sqrroot(1-v2/c2 ) /1-(v/c)cos(theta) )

as v<<c,

so

                  fD = fo (1+ (v/c) cos(theta) )      …(1)

                  fD - fo = fo (v/c)cos(theta)      

                  fD - fo = 400 Hz

                  fo = 3 GHz = 3*10^9 Hz

so           v cos(theta) = 400*3*10^8 / 3*10^9   = 40 m/s

                as v cos(theta)= radial velocity of storm

so storm has velocity 40 m/s in radial direction.

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