The Relativistic Signal Mirror · AI Generator #GEN-980054B8

Physics

Problem

A deep space probe is equipped with a special mirror system designed to reflect communication signals back to Earth. The probe travels away from Earth at a constant velocity v=0.6cv = 0.6c along a straight line. At time t=0t = 0 (in Earth's frame), when the probe is at distance d=2.0×1012  md = 2.0 \times 10^{12}\;\text{m} from Earth, ground control sends a radio signal of frequency f0=1.5  GHzf_0 = 1.5\;\text{GHz} toward the probe. The signal reaches the probe's mirror and is immediately reflected back toward Earth. Due to the relativistic motion, the signal undergoes Doppler shifts both when traveling to the probe and when reflected back. (a) What is the frequency of the signal as measured by instruments on the probe when it arrives? (b) What is the frequency of the reflected signal as measured by Earth when it returns? (c) What is the total time elapsed on Earth from when the original signal was sent until the reflected signal is received back?

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