User:CanvasK/Jump
In Smash 4, jumps were dictated by a character's height and gravity following the formula f=sqrt(2gh)+(g/2) where f is the jump force (i.e. the speed on frame 1), g is gravity, and h is jump height. In Ultimate, most jumps also follow this formula except for grounded full jumps. Grounded full jumps now have an extra value called "initial height" (or initial speed, neither are full accurate), which determines the height the character reaches in the first 4 frames. Once these 4 frames are over, the formula is used again from the height the character is currently at, making the formula f=sqrt(2g(h-i))+(g/2) where i is initial height. The formula to determine the force used to reach the initial height is not currently known.
Spirits can alter jumps with certain skills, traits, and styles, but this is done in an odd way. Most have values for jump_y, jump_initial_speed_y, mini_jump_speed_y, jump_aerial_speed_y, tread_jump_speed_y, and tread_mini_jump_speed_y; these modify ground full jump height, initial height speed, short hop speed, air jump speed, footstool speed, and mini footstool speed. Note how only grounded full jumps have the height modified whereas everything else has speed modified. The speed multipliers are applied after the force formula is used, whereas height is during the force formula. Since height is within a squareroot, height multipliers contribute significantly less than speed multipliers which are linear. This will result in jumps that have a speed multiplier (read, everything except grounded full jumps) being able to reach much greater heights than jumps with height multipliers (read, grounded full jumps).
For example, below is a graph of Steve's jump heights under different formulations with a spirit team with high jump multipliers. Red, blue, and green are grounded full jumps and purple, orange, and black are short hops. What is observed in game are the red and orange curves, with orange (a short hop) being much higher than the grounded full jump in red because short hops use speed multipliers and grounded full jumps use height multipliers. If short hops used height multipliers like grounded full hops, that gives the purple curve; lower than the red curve as would be expected out of a short hop. If grounded full jumps used speed multipliers like short hops, then the blue curve would seen; expectedly higher than the short hop curve, though perhaps too high in practical use. If both speed and height multipliers are used, then the green and black curves would appear which... is a little much (it is enough to go from the bottom of Palutena's Temple to the top in one jump).
To reiterate, this would be for Steve, the character with the worst jumps. Falco's grounded full jumps are the highest, but his short hop is much shorter than his grounded full jump, which is enough for his grounded full jump to still beat his short hop (by 14 units, 218 vs 204)—character's whose grounded full jump is less than 2.5× their short hop will have short hops greater than their grounded full jump, which is all but 4 characters (at least with this specific spirit team). This is not true for his air jump which has the same height but uses speed multipliers, making it go much higher (288). If speed and height multipliers would be used, his short hop would reach 863 units and his grounded full hop would reach 1525 units, or 3 Palutena's Temples.
Bunny Hoods can further exaggerate the gap between grounded full jumps and short hops when combined with spirits; on its own only air jumps are noticeably higher, but not by much. Softhops appear to be unaffected by spirits, though I haven't and don't really care to test, but Steve jumped about his normal height which is much less than the spirit height.
