Crackle (physics)

Crackle is the facetious name of a high-order derivative, and more specifically, the fifth derivative of the displacement. There is little consensus on what to call derivatives past the 4th derivative, jounce, due to there being few well-defined practical applications. The terms are, however, utilized within the fields of robotics & human motion.

Notation

Crackle is given by the notation:

${\vec {c}}={\frac {d^{5}{\vec {r}}}{dt^{5}}},$

meaning crackle is equal to the 5th-order derivative of position vector over time, equal to the vector s.

The following equations are used for constant crackle:

${\vec {s}}={\vec {s}}_{0}+{\vec {c}}\,t$

${\vec {\jmath }}={\vec {\jmath }}_{0}+{\vec {s}}_{0}\,t+{\frac {1}{2}}{\vec {c}}\,t^{2}$

${\vec {a}}={\vec {a}}_{0}+{\vec {\jmath }}_{0}\,t+{\frac {1}{2}}{\vec {s}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {c}}\,t^{3}$

${\vec {v}}={\vec {v}}_{0}+{\vec {a}}_{0}\,t+{\frac {1}{2}}{\vec {\jmath }}_{0}\,t^{2}+{\frac {1}{6}}{\vec {s}}_{0}\,t^{3}+{\frac {1}{24}}{\vec {c}}\,t^{4}$

${\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}\,t+{\frac {1}{2}}{\vec {a}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {\jmath }}_{0}\,t^{3}+{\frac {1}{24}}{\vec {s}}_{0}\,t^{4}+{\frac {1}{120}}{\vec {c}}\,t^{5}$

where

${\vec {c}}$ : constant crackle,

${\vec {s}}_{0}$ : initial jounce,

${\vec {s}}$ : final jounce,

${\vec {j}}_{0}$ : initial jerk,

${\vec {j}}$ : final jerk,

${\vec {a}}_{0}$ : initial acceleration,

${\vec {a}}$ : final acceleration,

${\vec {v}}_{0}$ : initial velocity,

${\vec {v}}$ : final velocity,

${\vec {r}}_{0}$ : initial position,

${\vec {r}}$ : final position,

$t$ : time between initial and final states.

See also

• Second derivative