As a general rule, you wanna aim for somewhere between 4 and 5 hours of sleep. The reason for this is that you want to wake up near the end or at the end of a REM cycle. Typically, one REM cycle lasts about 90 minutes, so 4 1/2 hours is 3 cycles. Usually, by the fourth cycle, your cycles consist of more REM sleep than non-REM. Of course, your REM cycle could be shorter or longer (mine's about 85 minutes, last time I checked), hence the room for error.
It sounds like you're getting 5 hours of sleep, which is probably interrupting you right in the middle of your 4th sleep cycle. If you go to bed at 2 or 3 a.m., maybe set your alarm for 6 or 7 a.m. instead.
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