Pop (physics)

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

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

The following equations are used for constant pop:

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

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

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

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

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

${\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}\,t+{\frac {1}{2}}{\vec {a}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {\jmath }}_{0}\,t^{3}+{\frac {1}{24}}{\vec {s}}_{0}\,t^{4}+{\frac {1}{120}}{\vec {c}}_{0}\,t^{5}+{\frac {1}{720}}{\vec {p}}\,t^{6}$

where

${\vec {p}}$ : constant pop,

${\vec {c}}_{0}$ : initial crackle,

${\vec {c}}$ : final 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 name "pop", along with "snap" (also referred to as jounce) and "crackle" are somewhat facetious terms for the fourth, fifth, and sixth derivatives of position, being a reference to Snap, Crackle, and Pop.

Unit and dimension

The dimensions of pop are LT⁻⁶. In SI units, this is m/s⁶, and in CGS units, 100 Gal per quartic second. This pattern continues for higher order derivatives.