“к” vs “ко” preposition

Question

Is there any common rule of when to append the “о” vowel to “к” preposition in front of words starting with “в”?

Nowadays almost nobody says e.g. “ко врачу,” it sounds quite archaic. I hear a slight semantic difference in intentions gravity, though:

• Этот отчет должен быть готов ко вторнику.
• Вечер понедельника удался, и к вторнику я оказался не готов.

“не готов ко вторнику” here was nearly “have not completed some task until Tuesday,” while “не готов к вторнику” states for “did not consider that Tuesday comes so suddenly.”

Are these the alien voices in my head or the meaning really differs?

Answer 1

This is a preposition that governs the Egressive semantical case, meaning "by the time of" (while the morphological case is Dative).

Egressive is not morphologically independent case in Russian, its forms are always the same as of Dative.

There are other examples:

я готовился к встрече
ко встрече я уже был осведомлен о новостях

In many respects the preposition ко is similar to the preposition во, which is used with Locative. If the word is monosyllabic and has a flying vowel in root, then in Locative case the preposition becomes во, while in Prepositional case it remains в:

во сне я летал
в сне много пользы

лежать во льну 
в льне много пользы

бежать во ржи
в ржи много белка

болячка во рту 

под насыпью во рву

Some other words also require the во for Locative:

Я разочаровался во главе правительства 
(= I became disappointed while being the head of government)

Я разочаровался в главе правительства 
(= I became disappointed with the head of government)