Crackle (physics)

In physics, crackle is the fifth derivative of the position vector with respect to time, with the first, second, third, and fourth derivatives being velocity, acceleration, jerk, and snap, respectively; crackle is thus the rate of change of the snap with respect to time. Crackle is defined by any of the following equivalent expressions:

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

The following equations are used for constant crackle:

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

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

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

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

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

where

${\vec {p}}$ : constant crackle,

${\vec {s}}_{0}$ : initial snap,

${\vec {s}}$ : final snap,

${\vec {\jmath }}_{0}$ : initial jerk,

${\vec {\jmath }}$ : 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.

The terms snap (also referred to as jounce), crackle, and pop‍—‌for the fourth, fifth, and sixth derivatives of position‍—‌were inspired by the advertising mascots Snap, Crackle, and Pop.

Unit and dimension

The dimensions of crackle are LT⁻⁵. In SI units, this is m/s⁵, and in CGS units, 100 gal per cubed second.

References

This article "Crackle (physics)" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Crackle (physics). Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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