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Which 22 bullets will drop a C dog but not exit or blow out half the side and is there a rule of thumb on velocity thresholds for this to happen reliably.
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Your're looking for the "Holy Grail" of bullets, and arguably there isn't such a bullet.
If you can control your hits well, then the light 40 grain plastic tipped bullets like the BlitzKing and V-Max are great... but hit a shoulder and they splash and make a mess.
Use heavier bullets on the shoulder and they work great, but hit soft tissue, and they go right through, and rip up the other side.
The folks that love to collect hides swear by the original 17 Remington.
... and ask them. That site is probably the best when you want to save pelts. But be warned, they are "very weird" over there.