Three ways to turn a redshift into a speed

Adapted from IOAA 2013 (Volos), theory problem 15. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The redshifts of three distant radio sources are 0.536 for 3C 279, 1.029 for 3C 245 and 3.8 for 4C 41.17.

(a) Find the apparent recession speed of each, using (1) the classical rule $v=cz$, (2) the approximate cosmological rule $v=c\ln (1+z)$, and (3) the relativistic Doppler formula. [5 points]

(b) For each rule, give the speeds as percentages of the speed of light. [3 points]

(c) Which of the three is most suitable for cosmological distances? [2 points]

Answer(a, b) Classical 54%, 103% and 380%. Logarithmic 43%, 71% and 157%. Relativistic 41%, 61% and 92%. (c) The approximate cosmological rule is the most suitable of the three, and the true speeds in our Universe, 47%, 79% and 164%, lie closest to it.