После удара на стекле появилась трещина.

Because после meaning after normally requires the genitive case.

So:

• dictionary form: удар = blow, hit, impact
• genitive singular: удара

That gives:

• после удара = after the удар / after the impact

This is a very common pattern in Russian:

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

So here, удара is simply the genitive singular form required by после.

In this sentence, удар means something like a blow, a hit, or an impact.

Depending on context, После удара на стекле появилась трещина could mean:

• after someone hit the glass
• after the glass was struck by something
• after an impact to the glass

So удар is a fairly general word for a physical strike or impact.

This is a very natural question, because in English we often say a crack in the glass.

In Russian, трещина на стекле is very natural because the crack is viewed as something on the surface of the glass. Russian often uses на for marks, damage, or visible features located on a surface:

• царапина на столе = a scratch on the table
• пятно на рубашке = a stain on the shirt
• трещина на стекле = a crack on the glass

English and Russian do not always picture these things the same way. In English, in the glass is common; in Russian, на стекле is often the more idiomatic choice.

Стекле is the prepositional singular form of стекло.

Why? Because на can take different cases, and here it means location:

• на стекле = on the glass

For this meaning, на usually takes the prepositional case.

Forms of стекло here:

• nominative: стекло
• prepositional: на стекле

So the -е ending shows the prepositional singular.

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

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

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

Here the subject is:

• трещина = crack

So the verb must be:

• появилась = appeared

Even though стекле comes earlier in the sentence, it is not the subject. The subject is трещина.

Because появилась is perfective, and this sentence describes a single completed event/result:

• there was an impact
• as a result, a crack appeared

That is exactly the kind of situation where Russian prefers the perfective.

Compare:

• появилась = appeared, came into being, showed up as a completed event
• появлялась = was appearing / used to appear / appeared repeatedly, depending on context

Here we are talking about one specific crack coming into existence after one impact, so появилась is the natural choice.

Yes, basically. Появиться means to appear, to show up, to come into existence / become visible.

In this sentence, it works very naturally with трещина:

• появилась трещина = a crack appeared / a crack formed

In English, we might also translate it more naturally as:

• A crack formed in the glass after the impact.

So the Russian verb is literally appeared, but in context English may prefer formed.

Russian word order is flexible, and this order is very natural because it presents the information in a smooth way:

1. После удара = sets the time/context
2. на стекле = tells us where
3. появилась трещина = gives the new main event

This makes трещина the new piece of information at the end, which is very common in Russian.

Other orders are possible, for example:

• После удара трещина появилась на стекле.
• Трещина появилась на стекле после удара.

But the original sentence sounds especially natural if the speaker is introducing the crack as new information.

Because Russian has no articles like English a/an/the.

So:

• трещина can mean a crack or the crack
• the exact meaning depends on context

In this sentence, the most natural English translation is a crack appeared, because the crack is being introduced for the first time.

Russian relies on context and word order instead of articles.

It can mean either, depending on context.

Стекло can refer to:

• glass as a material
• a piece/pane of glass
• the glass part of a window, door, screen, etc.

In this sentence, на стекле most likely means on the glass surface or on the pane of glass. English might translate it as:

• on the glass
• in the glass
• on the windowpane

The exact choice depends on the situation.

Yes. A few natural alternatives are possible, with slightly different wording:

• После удара на стекле образовалась трещина.
  = After the impact, a crack formed on the glass.
  This sounds a bit more like formed than appeared.

• После удара стекло треснуло.
  = After the impact, the glass cracked.
  This focuses on the glass itself, not specifically on the crack.

• От удара на стекле появилась трещина.
  = From the impact, a crack appeared on the glass.
  От удара emphasizes cause a little more directly.

The original sentence is completely normal and idiomatic; these are just alternative ways to express a similar idea.