What rule governs the grammatical cases in these examples?

Question

I don't understand why the sentences Mы достигли своей цели and Пушка пробивала лобовую броню are correct. Why not Мы достигли свою цель? What's the rule for the grammatical cases used here? Why are they different? And which is correct: Пушка не пробивала лобовую броню or Пушка не пробивала лобовой брони?

Answer 1

The verb достигнуть requires you to put the thing achieved in the genitive. This is a function of the picture the verb paints. Your achievement is measured with the goal as a reference point. Your efforts move you toward the goal, they have no effect on the goal itself. Since it is not the direct target of your actions, the goal is not in the accusative case. The genitive is appropriate because it conveys the idea of "with reference to the goal". A close English parallel to the use of the genitive here is "Achievement of the goal".

In answer to your second question, either is grammatically correct. In the second variant the speaker exercises the option of putting the direct object of a negated verb in the genitive. This is still fairly common when the direct object is the name of an abstract concept or idea. But here the noun refers to a physical object and using the genitive in such cases is old-fashioned.