Після уроку я ховаю зошит у рюкзак.

Because після always takes the genitive case.

• урок = nominative singular
• уроку = genitive singular

So після уроку means after the lesson / after class.

Зошит is the direct object of the verb ховаю, so it is in the accusative case.

For masculine inanimate nouns in the singular, the accusative looks the same as the nominative:

• nominative: зошит
• accusative: зошит

So even though it is accusative here, the form does not visibly change.

This is about the difference between direction/destination and location.

• у/в + accusative often means into
• у/в + locative often means in / inside

So:

• у рюкзак = into the backpack
• у рюкзаку = in the backpack

In this sentence, the notebook is being put into the backpack, so рюкзак is in the accusative.

Compare:

• Я ховаю зошит у рюкзак. = I put/hide the notebook into the backpack.
• Зошит у рюкзаку. = The notebook is in the backpack.

Yes, ховати / ховаю basically means to hide, but in everyday speech it can also mean to put away, to tuck away, or to stow something out of sight.

So in this sentence it can sound like:

• I put my notebook away in my backpack
• I tuck my notebook into my backpack

If you want a more neutral put, Ukrainian might also use кладу:

• Після уроку я кладу зошит у рюкзак.

But ховаю is perfectly understandable and adds the idea of putting it away/out of sight.

Ховаю is present tense, 1st person singular: I hide / I am hiding / I put away.

It comes from the imperfective verb ховати.

Imperfective present in Ukrainian can describe:

• something happening now
• a regular/habitual action
• a general routine

So this sentence could mean:

• After class, I put my notebook in my backpack
• After class, I’m putting my notebook in my backpack
  depending on context.

Not necessarily.

Ukrainian verb endings usually already show the subject. Ховаю already means I hide / I put away.

So both are possible:

• Після уроку я ховаю зошит у рюкзак.
• Після уроку ховаю зошит у рюкзак.

Including я can make the subject more explicit or slightly more emphatic.

No, Ukrainian word order is fairly flexible.

The version here is a natural, neutral order:

• Після уроку я ховаю зошит у рюкзак.

But other orders are also possible, depending on emphasis:

• Я після уроку ховаю зошит у рюкзак.
  Emphasizes I
• Зошит я ховаю у рюкзак після уроку.
  More marked; emphasizes the notebook

So word order in Ukrainian often reflects focus and style, not just grammar.

Usually після уроку means after the lesson or after class — one class period.

If you want after classes / after school, Ukrainian more often says:

• після уроків

So there is a useful difference:

• після уроку = after one lesson
• після уроків = after classes / after school

У and в both mean in / into. Ukrainian often alternates between them for euphony — to make pronunciation smoother.

So both may be possible in many contexts, but one may sound more natural than the other depending on surrounding sounds.

Here у рюкзак sounds smooth and natural.

This alternation usually does not change the meaning.

Ukrainian has no articles like a or the.

So зошит can mean:

• a notebook
• the notebook
• my notebook
  if the context makes that clear

Ukrainian relies on context much more than English does for definiteness.

A helpful stress-marked version is:

Пі́сля уро́ку я хова́ю зо́шит у рюкза́к.

A rough English-style approximation:

PEES-lya oo-ROH-koo ya khoh-VA-yu ZOH-shyt oo ryuk-ZAK

A few notes:

• я = ya
• ю in рюкзак gives a ryu sound
• ш in зошит sounds like sh

It is the 1st person singular present-tense form of ховати.

The pattern is:

• я ховаю = I hide / I put away
• ти ховаєш = you hide
• він/вона ховає = he/she hides
• ми ховаємо = we hide
• ви ховаєте = you hide
• вони ховають = they hide

So the ending -ю helps tell you that the subject is I.