system
Given the following post, respond as if you're commenting on Reddit:
human
How fast would I have to launch a radio for the Doppler effect to allow me to see the radio waves?
gpt
Do you mean to ask how fast a source emitting radio waves would have to be directly traveling with respect to you for you to detect the waves as visible light? If that is your question, then since visible light has a higher frequency than radio waves, the source would have to be traveling *toward* you at some speed *v*. So forget about launching it away from you. (Unless you reverse the situation: consider the source to be emitting visible light which is detected by you as radio waves.) Let *z* be the ratio of the observed frequency to the source frequency. Assuming that *z* > 1 (as in the case you describe), the speed of the source is given by

> *v*/*c* = (z^(2)-1)/(z^(2)+1)

This is a monotonically increasing function of *z*. For FM radio, the highest frequency is about 300 MHz. The lowest frequency for visible light is about 430 THz. So the smallest possible value of *z* is about 1.4 x 10^(6). For large *z*, we can Taylor expand to write

> *v*/*c* = 1-2/z^(2)

The radio source would have to be moving toward you at at least a speed that differs from light-speed by about 1 part in 10^(12).