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The player's movement can be accurately calculated with sequences.

The following formulas come from analyzing the game's source code.

Note that these formulas are not exact, due to how floats are computed. When used for calculations, only the first 4-6 decimals should be considered accurate. For a completely accurate simulation, you would need to replicate the source code.

In this article, we only consider standard movement, and ignore mechanics specific to certain blocks.

You will find further documentation of movement physics in these articles:

Note: Minecraft's coordinate system is oriented differently: 0° points towards "positive Z", and 90° points towards "negative X". We choose to work in the standard coordinate system to make calculations more intuitive. If need be, we can simply invert the X axis to match Minecraft's coordinate system.

Vertical Movement

Jump Formula

$V\displaystyle _{Y,1}=0.42$
$V\displaystyle _{Y,t}=\left(V_{Y,t-1}-{\underset {gravity}{0.08}}\right)\times {\underset {drag}{0.98}}$

If

{\textstyle \left|V\displaystyle _{Y,t}\right|<0.005}

,

{\textstyle V\displaystyle _{Y,t}}

is set to 0 instead (the player's height doesn't change for that tick)

In 1.9+, it's compared to 0.003 instead.

Notes

${\textstyle V\displaystyle _{Y,0}}$ isn't assigned a value because it has no importance. By convention, the 0^th tick corresponds to the player's initial velocity before jumping.
${\textstyle V\displaystyle _{Y,1}}$ corresponds to the initial jump motion. It is increased by 0.1 per level of Jump Boost
Terminal velocity is -3.92 m/t
When the player collides vertically with a block, vertical momentum is cancelled and only the acceleration is left.

Vertical Position

To get the position on a given tick, you simply need to sum

{\textstyle V\displaystyle _{Y}}

{\textstyle Y(n)=\sum _{t=1}^{n}V_{Y,t}}

Jump duration

The duration of a jump is the number of ticks between jumping and landing.

It also corresponds to the period of that jump's cycle when performed repeatedly.

This notion is linked to the notion of Tiers.

Description	Duration
Flat Jump	12 t
3bc Jump	11 t
+0.5 Jump	10 t
+1 Jump	9 t
2.5bc Jump	6 t
2bc Jump	3 t
1.8125bc Jump	2 t

Source code

from EntityLivingBase

/* Code unrelated to vertical movement is cut out */

protected float getJumpUpwardsMotion(){
    return 0.42F;
}


protected void jump()
{
    this.motionY = this.getJumpUpwardsMotion();
    if (this.isPotionActive(Potion.jump))
    {
        this.motionY += (this.getActivePotionEffect(Potion.jump).getAmplifier() + 1) * 0.1F;
    }
    this.isAirBorne = true;
}


public void moveEntityWithHeading(float strafe, float forward)
{
    ... /* also moves the player horizontally */

    this.motionY -= 0.08;
    this.motionY *= 0.98;
}


public void onLivingUpdate()
{
    if (this.jumpTicks > 0)
        --this.jumpTicks;

    if (Math.abs(this.motionY) < 0.005D)
        this.motionY = 0.0D;


    if (this.isJumping)
    {
        ... /* different if in water or lava */

        if (this.onGround && this.jumpTicks == 0)
        {
            this.jump();
            this.jumpTicks = 10; //activate autojump cooldown (0.5s)
        }
    }
        
    else
    {
        this.jumpTicks = 0; //reset autojump cooldown
    }

    ...
    
    this.moveEntityWithHeading(this.moveStrafing, this.moveForward);
}

Horizontal Movement

Horizontal Movement is a bit more complex than Vertical Movement, as it relies on many more factors: player actions, direction, and ground slipperiness.

On every tick, the game does these three steps:

Acceleration is added to the player's velocity.
The player is moved (new position = position + velocity).
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)

Failed to parse (unknown function "\begin{Bmatrix}"): {\displaystyle 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)

Failed to parse (syntax error): {\displaystyle 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).
${\textstyle V_{H,t}}$ is the player's speed on tick ${\textstyle t}$ .

Ground Speed:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (syntax error): {\displaystyle 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:

Failed to parse (syntax error): {\displaystyle 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:

${\textstyle D_{t}}$ , The player's Direction in degrees (defined by their inputs and rotation)
${\textstyle F_{t}}$ , 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:

${\textstyle V_{X,0}}$ and ${\textstyle V_{Z,0}}$ correspond to the player's initial velocity.
${\textstyle V_{X,t}}$ and ${\textstyle V_{Z,t}}$ correspond to the player's velocity on tick ${\textstyle t}$
Ground Velocity:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (syntax error): {\displaystyle 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}) }

Failed to parse (syntax error): {\displaystyle 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:

Failed to parse (syntax error): {\displaystyle 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}) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

{\textstyle V\displaystyle _{X,t}}

is set to 0 and the player is placed against the wall.

If the player hits a Z-facing wall, then

{\textstyle V\displaystyle _{Z,t}}

is set to 0 and the player is placed against the wall.

Negligible Speed Threshold:

If

{\textstyle \left|V\displaystyle _{X,t}\times S_{t}\times 0.91\right|<0.005}

, momentum is cancelled and only the acceleration is left.

If

{\textstyle \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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