# Nonrecursive Movement Formulas

Since arithmetico-geometric sequences have explicit formulas, we can build non-recursive functions to calculate simple but useful results, such as the height of the player on any given tick, or the distance of a jump in terms of the initial speed and duration.

**Definitions:**

- is the player's initial speed (speed on , before jumping)
- is the number of ticks considered (ex: the jump duration on flat ground is t=12, see Tiers)
- is the "jump bonus" (0.3274 for sprintjump, 0.291924 for strafed sprintjump, 0.1 for 45° no-sprint jump...)
- is the movement multiplier after jumping (1.3 for 45° sprint, 1.274 for normal sprint, 1.0 for no-sprint 45°...)

## Vertical Movement (jump) [1.8]

Vertical speed after jumping ()

Relative height after jumping ()

For , see below.

## Vertical Movement (jump) [1.9+]

Vertical speed after jumping ()

Relative height after jumping ()

## Horizontal Movement (instant jump)

Assuming the player was airborne before jumping.

Horizontal speed after sprintjumping ()

Sprintjump distance ()

**Note:** These formulas are accurate for most values of , but some negative values can wind up activating the speed threshold and reset the player's speed at some point, thus rendering these formulas inaccurate.

## Horizontal Movement (delayed jump)

Assuming the player is on ground before jumping (at least 1 tick since landing).

Horizontal speed after sprintjumping ()

Sprintjump distance ()

## Advanced Formulas

Horizontal speed after consecutive sprintjumps on a momentum of period (, ).

If the first sprintjump is delayed, multiply by 0.6