Horizontal Movement Formulas

On every tick, the game does these three steps:


 * 1) Acceleration is added to the player's velocity.
 * 2) The player is moved (new position = position + velocity).
 * 3) The player's velocity is reduced to simulate drag.

We'll start by introducing Multipliers in an effort to make formulas more readable.

Multipliers

 * Movement Multiplier (See 45° Strafe)


 * $$M_{t} = \begin{Bmatrix}1.3 & \textrm{Sprinting} \\ 1.0 & \textrm{Walking}\\ 0.3 & \textrm{Sneaking}\\ 0.0 & \textrm{Stopping} \end{Bmatrix} \times \begin{Bmatrix}0.98 & \textrm{Default}\\ 1.0 & \textrm{45° Strafe} \\ 0.98 \sqrt{2} & \textrm{45° Sneak} \end{Bmatrix}$$


 * Effects Multiplier (See Status Effects)


 * $$E_{t} = (\underset{Decreases \; by \; 15\% \; per \; level \; of \; Slowness}{\underset{Increases \; by \; 20\% \; per \; level \; of \; Speed}{\underbrace{\left ( 1 + 0.2\times Speed \right ) \: \times\: \left ( 1 - 0.15\times Slowness \right )}}} \geq 0$$


 * Slipperiness Multiplier (See Slipperiness)


 * $$S_{t} = \begin{Bmatrix}0.6 & \textrm{Default}\\ 0.8 & \textrm{Slime}\\ 0.98& \textrm{Ice} \\ 1.0 & \textrm{Airborne} \end{Bmatrix}$$

Linear Formulas
These simplified formulas only apply to linear movement (no change in direction). While this condition might seem very restrictive, these formulas are very useful to analyze conventional jumps and momentum We'll later expand on these formulas by including angles. Definition:
 * $$V_{H,0}$$ is the player's initial speed (default = 0).
 * $V_{H,t}$ is the player's speed on tick $t$.


 * Ground Speed:


 * $$V_{H,t} = \underset{Momentum}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}}  \right )^{3}}} $$


 * Jump Speed:


 * $$V_{H,t} = \underset{Momentum}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}}  \right )^{3}}} + \underset{Sprintjump \; Boost}{\underbrace{\begin{Bmatrix}0.2 & \textrm{Sprinting}\\ 0.0   & \textrm{Else}\end{Bmatrix} }}$$


 * Air Speed:


 * $$V_{H,t} = \underset{Momentum}{\underbrace{\underset{ }{V_{H,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{\underset{ }{0.02 \times M_{t}}}} $$

Complete Formulas
Let's introduce two more variables:
 * $ D_{t} $, The player's Direction in degrees (defined by their inputs and rotation)
 * $ F_{t} $, The player's Facing in degrees (defined by their rotation only)

In reality, angles aren't as simple as that, as there are a limited number of significant angles (see Facing and Angles).

For the purpose of simplicity, we'll ignore this fact.


 * Definition:
 * $V_{X,0}$ and $V_{Z,0}$  correspond to the player's initial velocity.
 * $V_{X,t}$ and $V_{Z,t}$  correspond to the player's velocity on tick $t$


 * Ground Velocity:


 * $$V\displaystyle _{X,t} = \underset{ }{V\displaystyle _{X,t-1} \times S_{t-1} \times 0.91 } \: + \: 0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}}  \right )^{3} \times \sin (D_{t}) $$
 * $$V\displaystyle _{Z,t} = \underset{Momentum}{\underbrace{\underset{ }{V\displaystyle _{Z,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}}  \right )^{3}}} \times \cos (D_{t}) $$


 * Jump Velocity:


 * $$V\displaystyle _{X,t} = \underset{ }{V\displaystyle _{X,t-1} \times S_{t-1} \times 0.91 } \: + \: 0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}}  \right )^{3} \times \sin (D_{t}) + \begin{Bmatrix} 0.2 & \textrm{Sprinting}\\ 0.0 & \textrm{Else}\end{Bmatrix} \times \sin (F_{t}) $$
 * $$V\displaystyle _{Z,t} = \underset{Momentum}{\underbrace{\underset{ }{V\displaystyle _{Z,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{0.1 \times M_{t} \times E_{t} \times \left (\frac{0.6}{S_{t}}  \right )^{3}}} \times \cos (D_{t}) + \underset{Sprintjump \; Boost }{\underbrace{\begin{Bmatrix} 0.2 & \textrm{Sprinting}\\ 0.0 & \textrm{Else}\end{Bmatrix} \times \cos (F_{t})}} $$


 * Air Velocity:


 * $$V\displaystyle _{X,t} = \underset{ }{V\displaystyle _{X,t-1} \times S_{t-1} \times 0.91 } \: + \: 0.02 \times M_{t} \times \sin (D_{t}) $$
 * $$V\displaystyle _{Z,t} = \underset{Momentum}{\underbrace{\underset{ }{V\displaystyle _{Z,t-1} \times S_{t-1} \times 0.91 }}} \: + \: \underset{Acceleration}{\underbrace{\underset{}{0.02 \times M_{t}}}} \times \cos (D_{t}) $$

Stopping Conditions
Horizontal speed is set to 0 if the player hits a wall, or if the speed is considered to be negligible.


 * Wall Collision:


 * If the player hits a X-facing wall, then momentum is cancelled and $V\displaystyle _{X,t}$ only includes acceleration.
 * If the player hits a Z-facing wall, then momentum is cancelled and $V\displaystyle _{Z,t}$ only includes acceleration.
 * In either of these case, the player stops sprinting.


 * Negligible Speed Threshold:


 * If $ \left | V\displaystyle _{X,t} \times S_{t} \times 0.91 \right | < 0.005 $, momentum is cancelled and only the acceleration is left.
 * If $ \left | V\displaystyle _{Z,t} \times S_{t} \times 0.91 \right | < 0.005 $, momentum is cancelled and only the acceleration is left.


 * In 1.9+, they are compared to 0.003 instead.

Source Code
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