To get this number I added: DC For high jumping 1 ft (see reasoning below) + DC for long jumping 2 ft = 4 + 2 = 6
I wasn't sure how "vaulting" should impact the DC, or whether an additional penalty (beyond adding DCs) should be introduced because the PC is trying to jump both vertically and horizontally, so you might fudge it a little one way or another, but I would be inclined to make the final DC lower, not higher (I personally think adding DCs is good enough, and think use of the hands should lower it to DC 4 or so).
Why not use High Jump rules for a height of 3 feet?
I think it's a mistake to treat this as a High jump of 3 feet.
The rules say that "A high jump is a vertical leap made to reach a ledge high above or to grasp something overhead."
Real life experience shows that "getting over something 3 feet tall" and "jumping 3 feet up" are (very) different difficulty tasks. After looking at some real world examples of jumps in this range, I think the high jump rules make sense for the latter, and the former should be handled by first calculating (or guestimating) how high the PC actually needs to raise their Center of Mass (COM) and then using the High Jump rules for that.
In real life
College students use hurdles that are 42 inches tall, and many (non hurdle specialist) members of said track team could clear a series of these hurdles with only a few minutes of instruction and practice (doing it quickly and efficiently is another matter). I struggled to find hard numbers on how high they jump, but this site claims that a top US women's hurdler only raises her COM by 11 inches when she clears a 32 inch hurdle.
For contrast the NBA draft has a "max vertical jump" component which allows an (admittedly small) run-up and is measured by how much higher an athlete can touch the tips of their fingers compared to when they are standing flat on the ground. The highest this year was 43 inches, and only 2 NBA draftees (which is a much more selective group than college track in general) cleared 42". See NBA site for more information
Clearly, these are two very different things (my personal experience in track and field with long jump, high jump, and hurdles supports this as well).
Mediocre high school athletes (like myself) who can guarantee clearing 39" hurdles are not on the same level as PCs who can guarantee passing a DC 12+ Acrobatics check in Pathfinder (and anyways, it just doesn't seem reasonable that jumping over a 1 ft obstacle would have any chance of failure for any reasonable person, even DC 4 seems excessive for that).
Back to the Game
Thus, I would rule that High jump applies to raising your COM by X feet.
You only have to raise your COM by ~1 ft (DC 4) to clear a 36 inch obstacle, two feet is barely a legitimate long jump, but we'll count it anyways (DC 2). I'm not sure whether adding the DCs is the best way to handle trying to jump both vertically and horizontally at once, but even if you add a further penalty I think being able to use your hands to assist in a vault should offset that.
Why add DCs for Jumps with a vertical and horizontal component?
Caveat: This is the part of my answer I'm least confident in, and would welcome any corrections or suggestions on how to combine DCs that's backed by at the table experience: the main thrust of my answer is to encourage DMs to treat "clearing X ft obstacle" differently from "Jump up X ft" for the vertical component of the jump, I only chose a method for combining DCs because it was necessary to arrive at a complete final answer. I think this is necessary even though other answers currently ignore the horizontal component because in the other answers the DC of the horizontal component (2) is very small compared the DC of the vertical (12) and can be somewhat safely ignored. In my answer the horizontal component (DC2) is half the DC of the vertical component (DC4), which feels wrong to fully ignore. My reasoning for adding them follows:
Reasoning: The rules don't say what to do. Looking at all the simple rulings we could make (to keep the game fast and not turn it into a physics simulator), the others have obvious flaws.
- Max of DCs Taking the higher of the two DCs and ignoring the lower ignores an important component of the difficulty (no extra difficulty at all for jumping up 4 ft if you long jump 20 ft?)
- Multiplying DCs Multiplying the DCs leads to very high DCs very quickly and doesn't pass basic sanity checks when applying it to real world situations (you naturally raise your COM on any jump. A long jump of 20 ft will definitely cause your COM to raise by 1 ft, but under multiplication rules a 20 ft long jump with a 1 ft vertical component would be DC 20 * DC 4 = DC 80 -- an impossible task).
- Separate Rolls Making two separate rolls (one for the vertical component to see if they get the height required and one for the horizontal component to see if they get distance required) is the other option I considered most heavily. I rejected it after considering the following case: if a PC has a bonus of 19 to his Acrobatics rolls, he's guaranteed to be able to high jump 5 feet, or long jump 20 feet, but only just barely. He would not be able to guarantee a long jump of 21 feet, but he would be able to guarantee a long jump of 20 feet while also jumping up 5 feet which seems fishy.
- Adding DCs Adding DCs is the simplest method of combining the DCs that makes reasonable DCs for realistic situations I have a good intuition for (in the range where the PC is high jumping 0-5 ft and long jumping 0-20ft). As noted in the multiplication section, in reality anyone long jumping 20ft (DC 20) will definitely raise their COM 1 ft as well (DC 4), so one could argue the final DC should remain 20, not 24, but it's A) less broken than the other simple options and B) a small enough margin of error that I wouldn't complicate the rules with exceptions or fancier math to try to eliminate it (play speed is important too).
One thing all of these methods have in common is that they produce a DC <=10 for your PC in this situation, even before considering the fact the PC used their hands to "Vault" the barrel:
- Max of DCs: Max(DC4, DC2) = DC4
- Multiply DCs: DC4 * DC2 = DC8
- Separate Rolls: DC4 (without bonuses) = 85% of success; DC2 (without bonuses) = 95% chance of success, overall chance of success ~80% = DC5
- Adding DCs: DC4 + DC2 = DC 6
Regardless of how you choose to combine the DCs, in this particular case the DC should not be greater than 8.