How to differentiate between adverb and adjective?

[Questor]

How do we differentiate whether 'up' is an adverb or an adjective in these two sentences?:

1) Your time is up.

2) The sun is up.

[Tdol]

They have different meanings- the second describes the position in the sky, the first is a prahsal verb, 'be up', meaning that the time has finished.

[Questor]

The sentences were taken from Merriam Webster online, which defines the first 'up' as an adverb and the second as an adjective. Why, I wonder.

Oxford Advanced Learners' dictionary defines it similarly.

I've always had the impression that 'up' is an adjective, especially in a 'noun is adjective' construction. Checking the dictionaries really had me quite confused.

[Tdol]

In the first the verb has two words: (be + up). They create a meaning together, so 'up' is modifying the verb, just as it is here:
They had a row but have since made up.
As the word is modifying the verb, to create a new meaning it is an adverb, though some also use the term particle.

[Questor]

I'm still having problem seeing 'up' in the first as an adverb... My mind tends to read 'up' as an adjective, as something equivalent to 'finished'. Looking up a similar word 'off' in the dictionary made me even more confused...

[Tdol]

Do you see that it is a phrasal verb (two-word verb)?

[Questor]

I can't see (be + up) as a phrasal verb in the first. I see up as more attached to the noun 'time' than modifying the verb 'is' . In fact, I didn't know that (be + up) is a phrasal verb as I usually associate a phrasal verb with (non-be verb + up) .

[Tdol]

There are phrasal verbs with 'be'- take many of the phrasal verbs with 'get' (get up/get over) and you'll see that 'be' has equivalents for after the process of getting our of bed, over an illness, etc.

[Questor]

In this case, how could we analyze the sentence to see that up is actually modifying is instead of time in 'Your time is up'?