После удара на экране телефона появилась заметная трещина.

Because после always takes the genitive.

• после = after
• удар = impact / blow / hit
• genitive singular of удар = удара

So после удара literally means after the impact or after the hit.

This is a fixed grammar pattern:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

In this sentence, удар means some kind of blow, hit, or impact.

With a phone, it usually suggests that the phone was:

• dropped,
• knocked against something,
• or otherwise hit hard enough to damage the screen.

So после удара is a natural way to say after the impact or after it was hit.

Because a crack appears on the surface of a screen, not inside it.

• на экране = on the screen
• в экране would sound unnatural here

Russian often uses на for surfaces:

• на столе = on the table
• на стене = on the wall
• на экране = on the screen

A crack is seen on the screen’s surface, so на экране is the natural choice.

Because на requires the prepositional case here.

• dictionary form: экран
• prepositional singular: экране

So:

• на экране = on the screen

Then телефона is in the genitive because it means of the phone:

• экран телефона = the phone’s screen / screen of the phone
• на экране телефона = on the phone’s screen

So the phrase is built like this:

• на
  • prepositional → на экране
• телефона = genitive dependent noun meaning possession or relation

Because the subject is трещина, and трещина is a feminine singular noun.

In the past tense, Russian verbs agree with the subject in:

• gender (in the singular)
• number

So:

• появился = masculine
• появилась = feminine
• появилось = neuter
• появились = plural

Here the subject is заметная трещина, so the verb must be появилась.

Because трещина is the subject of the sentence.

Even though it comes after the verb, it is still the thing that appeared:

• появилась трещина = a crack appeared

Russian word order is flexible, so the subject does not have to come first.

Here:

• появилась = appeared
• заметная трещина = a noticeable crack

So трещина stays in the nominative because it is the subject.

Because adjectives must agree with the noun they describe.

трещина is:

• feminine
• singular
• nominative

So the adjective must match:

• заметная трещина = a noticeable crack

Compare:

• заметный дефект = noticeable defect (masculine)
• заметное пятно = noticeable spot (neuter)
• заметные царапины = noticeable scratches (plural)

Yes, Russian could say it differently.

The sentence is:

После удара на экране телефона появилась заметная трещина.

This word order sounds natural and presents the information smoothly:

1. first the circumstance: after the impact
2. then the location: on the phone screen
3. then the new result: a noticeable crack appeared

Other word orders are possible, for example:

• После удара заметная трещина появилась на экране телефона.
• На экране телефона после удара появилась заметная трещина.

These are grammatical, but they shift the emphasis a bit. The original version sounds very natural and neutral.

Because Russian has no articles like a or the.

So заметная трещина can mean:

• a noticeable crack
• the noticeable crack

The exact meaning depends on context.

In this sentence, English would usually say a noticeable crack appeared, because the crack is being introduced as new information.

Появилась is perfective.

The infinitive is появиться = to appear (as a completed event).

That fits this sentence because it describes a single completed result:

• first there was no visible crack,
• then after the impact, the crack appeared.

The imperfective partner is появляться, which would be used for repeated, habitual, or ongoing situations, not for this one completed event.

Yes. That is a very close literal translation.

Piece by piece:

• После удара = after the blow / after the impact
• на экране телефона = on the phone screen
• появилась = appeared
• заметная трещина = a noticeable crack

A slightly more natural English translation might be:

• After the impact, a noticeable crack appeared on the phone screen.
• After it was hit, a noticeable crack appeared on the phone screen.

But the literal structure matches the Russian very well.

Yes, those are possible, but they have slightly different shades of meaning.

• появилась трещина = a crack appeared
  A neutral, very natural way to describe the result.

• возникла трещина = a crack arose / occurred
  A bit more formal or technical.

• образовалась трещина = a crack formed
  Often used when focusing on the process/result of damage.

For everyday description, появилась трещина is simple and natural.

No, there are several possibilities.

For example:

• заметная трещина = a noticeable crack
• видимая трещина = a visible crack
• большая трещина = a big crack
• трещина была хорошо видна = the crack was clearly visible

заметная suggests that the crack is clearly noticeable, not tiny or easy to miss. It is a good choice in this sentence.