I've decided to try some old Metallica songs, but I'm having trouble on Master of Puppets. Specifically, the first part of the 2nd solo, where Kirk is playing that fast hammer on/pull off lick

Normally I bar the E and the B, but I can't hit the pull off from the A to the E fast or cleanly enough. It's kind of a big stretch for my finger. I can do it more easily if I take my finger off the B, but then I can't change strings fast enough. How do I go about learning this part?
Barring, imo, sticks your hand in a certain position which severely compromises mobility, I never barred it and gave it a sweep feel with the only notes being picked being the A(17) and C(13) as well as only picking the E when coming back up.

I split up the solo like this

E|[-[color="DarkRed"]17[/COLOR]p12----------[color="DarkRed"]12[/COLOR]- ][[color="DarkRed"]17[/COLOR]p12----------[color="DarkRed"]12[/COLOR]-] [[color="DarkRed"]17[/COLOR]p12----------[color="DarkRed"]12[/COLOR]-][[color="DarkRed"]15[/COLOR]p12----------[color="DarkRed"]12[/COLOR]-|]
B|[-------[color="DarkRed"]13[/COLOR]-12-13---- ][------[color="DarkRed"]13[/COLOR]-12-13----] [------[color="DarkRed"]13[/COLOR]-12-13-----][-----[color="DarkRed"]13[/COLOR]-12-13----|]

Where the red were the notes I picked and everything else was legato. It's probably not the 'correct' way to play the solo, but I really liked how fluid it made it sound.
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Your main issue, probably, is that it's 200+ Bpm, and you're trying to play it as such off the bat. Knock it down a lot, and focus on what you're having trouble with. If it's a tough stretch, which no matter how small your hands are, 17-12 isn't too much (I can hit that comfortably and functionally with my index and middle, and I don't have large hands, really, but because I put a lot of time into fixing my shitty posture, so don't blame it on small hands) work on your wrist angle, the actual form of your arm, and thumb placement. Just your whole posture in general.

I'd say it's best not to bar it, and I'm pretty sure most would agree. So, you'll have trouble getting from string to string, which means a.) you need to work on economy of motion, and b.) you need to work on making all of your movements fit rhythmically; You can't fall off the beat just because there's a string change.

All of these things are going to be impossible to work on at 210 (?) ish Bpm. Which leads us to the golden rule: slow down.

But it shouldn't be very difficult for you to get properly, if you do slow it down to the point that you can check each of the things I listed with each movement you make. So literally play it at like 60bpm, with each note being triplet eighth (or slower, if need be, each note a quarter, whatever you gotta do, it'll pay off big time in the long run if you fix this stuff early, particularly the posture,) And then it's just a matter of working it up to speed as you have to think about your form & technique less.

Good luck with it, and I hope this helps you. Also, back when I was learning that stuff, I did the same shit, figure out the notes, and just try to jump in at full speed, and I can tell you, it won't do much for your technique, but you'll have a much easier time working it back up to speed after you apply better techniques.
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treat it as an exercise lick. strict alternate picking and practice it slowly for weeks until you get it in your muscle memory CORRECT then build speed like always. It takes time and patience to get that at full tempo so it is deceptive. Cycle this pattern. D D U D U