You are an astronomer investigating four astronomical sources of infrared radiation. You have identified the nature of each source, so you know the frequency of each when it is at rest relative to you. Your detector, which is at rest relative to the earth, measures the frequency f of the moving source. Your results are given in the table.

(a) Which source is moving at the highest speed relative to your detector? What is its speed? Is that source moving toward or away from the detector?

(b) Which source is moving at the lowest speed relative to your detector? What is its speed? Is that source moving toward or away from the detector? (c) For source B, what frequency would your detector measure if the source were moving at the same speed relative to the detector but toward it rather than away from it?

The source A has $f=7.1THz<f_0=9.2THz$, so it is moving away from the detector with a relative speed satisfies Eq. (37.26):$f=f_0\sqrt{\frac{c-u}{c+u}}$

Solving this equation for u and substituting the known values off and $f_0$, we find

$u=\frac{1-{\left(\frac{f}{f_0}\right)}^2}{1+{\left(\frac{f}{f_0}\right)}^2}c=\frac{1-{\left(\frac{7.1}{9.2}\right)}^2}{1+{\left(\frac{7.1}{9.2}\right)}^2}c=0.253c$

Similarly, the source B with$f=5.4THz<f_0=8.6THz$ is moving away from the detector with a relative speed of

$u=\frac{1-{\left(\frac{f}{f_0}\right)}^2}{1+{\left(\frac{f}{f_0}\right)}^2}c=\frac{1-{\left(\frac{5.4}{8.6}\right)}^2}{1+{\left(\frac{5.4}{8.6}\right)}^2}c=0.434c$

The source C with$f=6.1THz<f_0=7.9THz$ is moving away from the detector with a relative speed of

$u=\frac{1-{\left(\frac{f}{f_0}\right)}^2}{1+{\left(\frac{f}{f_0}\right)}^2}c=\frac{1-{\left(\frac{6.1}{7.9}\right)}^2}{1+{\left(\frac{6.1}{7.9}\right)}^2}c=0.253c$

The source D with$f=8.1THz<f_0=8.9THz$ is moving away from the detector with a relative speed of

$u=\frac{1-{\left(\frac{f}{f_0}\right)}^2}{1+{\left(\frac{f}{f_0}\right)}^2}c=\frac{1-{\left(\frac{8.1}{8.9}\right)}^2}{1+{\left(\frac{8.1}{8.9}\right)}^2}c=0.094c$

From the results, it is obvious that source B is moving at the highest relative speed, which is ; B is u=0.434c; moving away from the detector.

From the results of part (a), we notice that source D is the one moving at the lowest relative speed, which is u=0.094 ; D is also moving away from the detector.

From Eq. (37.25) the frequency f our detector would measure for source B is

$f=f_0\sqrt{\frac{c+u}{c-u}}$

Substituting$f_0=8.6THz$ and the same relative speed $u=0.434c$, we find

$f=8.6\sqrt{\frac{c+0.434c}{c-0.434c}}-=13.7THz$