In physics, the fourth, fifth and sixth derivatives of position are defined as derivatives of the position vector with respect to time – with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively. The higher-order derivatives are less common than the first three;^[1][2] thus their names are not as standardized, though the concept of a minimum snap trajectory has been used in robotics and is implemented in MATLAB.^[3]

The fourth derivative is often referred to as snap or jounce. The name "snap" for the fourth derivative led to crackle and pop for the fifth and sixth derivatives respectively,^[4] inspired by the Rice Krispies mascots Snap, Crackle, and Pop.^[5] These terms are occasionally used, though "sometimes somewhat facetiously".^[5]

Fourth derivative (snap/jounce)

Snap,^[6] or jounce,^[2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time.^[5] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions:

$${\displaystyle {\vec {s))={\frac {d\,{\vec {\jmath ))}{dt))={\frac {d^{2}{\vec {a))}{dt^{2))}={\frac {d^{3}{\vec {v))}{dt^{3))}={\frac {d^{4}{\vec {r))}{dt^{4))}.}$$

The following equations are used for constant snap:

$${\displaystyle {\begin{aligned}{\vec {\jmath ))&={\vec {\jmath ))_{0}+{\vec {s))t,\\{\vec {a))&={\vec {a))_{0}+{\vec {\jmath ))_{0}t+{\tfrac {1}{2)){\vec {s))t^{2},\\{\vec {v))&={\vec {v))_{0}+{\vec {a))_{0}t+{\tfrac {1}{2)){\vec {\jmath ))_{0}t^{2}+{\tfrac {1}{6)){\vec {s))t^{3},\\{\vec {r))&={\vec {r))_{0}+{\vec {v))_{0}t+{\tfrac {1}{2)){\vec {a))_{0}t^{2}+{\tfrac {1}{6)){\vec {\jmath ))_{0}t^{3}+{\tfrac {1}{24)){\vec {s))t^{4},\end{aligned))}$$

where

The notation ${\displaystyle {\vec {s))}$ (used by Visser^[5]) is not to be confused with the displacement vector commonly denoted similarly.

The dimensions of snap are distance per fourth power of time (LT⁻⁴). The corresponding SI unit is metre per second to the fourth power, m/s⁴, m⋅s⁻⁴.

Fifth derivative

The fifth derivative of the position vector with respect to time is sometimes referred to as crackle.^[4] It is the rate of change of snap with respect to time.^[4][5] Crackle is defined by any of the following equivalent expressions:

$${\displaystyle {\vec {c))={\frac {d{\vec {s))}{dt))={\frac {d^{2}{\vec {\jmath ))}{dt^{2))}={\frac {d^{3}{\vec {a))}{dt^{3))}={\frac {d^{4}{\vec {v))}{dt^{4))}={\frac {d^{5}{\vec {r))}{dt^{5))))$$

The following equations are used for constant crackle:

$${\displaystyle {\begin{aligned}{\vec {s))&={\vec {s))_{0}+{\vec {c))\,t\\[1ex]{\vec {\jmath ))&={\vec {\jmath ))_{0}+{\vec {s))_{0}\,t+{\tfrac {1}{2)){\vec {c))\,t^{2}\\[1ex]{\vec {a))&={\vec {a))_{0}+{\vec {\jmath ))_{0}\,t+{\tfrac {1}{2)){\vec {s))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {c))\,t^{3}\\[1ex]{\vec {v))&={\vec {v))_{0}+{\vec {a))_{0}\,t+{\tfrac {1}{2)){\vec {\jmath ))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {s))_{0}\,t^{3}+{\tfrac {1}{24)){\vec {c))\,t^{4}\\[1ex]{\vec {r))&={\vec {r))_{0}+{\vec {v))_{0}\,t+{\tfrac {1}{2)){\vec {a))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {\jmath ))_{0}\,t^{3}+{\tfrac {1}{24)){\vec {s))_{0}\,t^{4}+{\tfrac {1}{120)){\vec {c))\,t^{5}\end{aligned))}$$

where

The dimensions of crackle are LT⁻⁵. The corresponding SI unit is m/s⁵.

Sixth derivative

The sixth derivative of the position vector with respect to time is sometimes referred to as pop.^[4] It is the rate of change of crackle with respect to time.^[4][5] Pop is defined by any of the following equivalent expressions:

$${\displaystyle {\vec {p))={\frac {d{\vec {c))}{dt))={\frac {d^{2}{\vec {s))}{dt^{2))}={\frac {d^{3}{\vec {\jmath ))}{dt^{3))}={\frac {d^{4}{\vec {a))}{dt^{4))}={\frac {d^{5}{\vec {v))}{dt^{5))}={\frac {d^{6}{\vec {r))}{dt^{6))))$$

The following equations are used for constant pop:

$${\displaystyle {\begin{aligned}{\vec {c))&={\vec {c))_{0}+{\vec {p))\,t\\{\vec {s))&={\vec {s))_{0}+{\vec {c))_{0}\,t+{\tfrac {1}{2)){\vec {p))\,t^{2}\\{\vec {\jmath ))&={\vec {\jmath ))_{0}+{\vec {s))_{0}\,t+{\tfrac {1}{2)){\vec {c))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {p))\,t^{3}\\{\vec {a))&={\vec {a))_{0}+{\vec {\jmath ))_{0}\,t+{\tfrac {1}{2)){\vec {s))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {c))_{0}\,t^{3}+{\tfrac {1}{24)){\vec {p))\,t^{4}\\{\vec {v))&={\vec {v))_{0}+{\vec {a))_{0}\,t+{\tfrac {1}{2)){\vec {\jmath ))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {s))_{0}\,t^{3}+{\tfrac {1}{24)){\vec {c))_{0}\,t^{4}+{\tfrac {1}{120)){\vec {p))\,t^{5}\\{\vec {r))&={\vec {r))_{0}+{\vec {v))_{0}\,t+{\tfrac {1}{2)){\vec {a))_{0}\,t^{2}+{\tfrac {1}{6)){\vec {\jmath ))_{0}\,t^{3}+{\tfrac {1}{24)){\vec {s))_{0}\,t^{4}+{\tfrac {1}{120)){\vec {c))_{0}\,t^{5}+{\tfrac {1}{720)){\vec {p))\,t^{6}\end{aligned))}$$

where

The dimensions of pop are LT⁻⁶. The corresponding SI unit is m/s⁶.