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; in other words, the pop is 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 jounce,

${\vec {s}}$ : final jounce,

${\vec {\jmath }}_{0}$ : 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. Currently, there are no well-accepted designations for the derivatives of pop. Higher-order derivatives of position are not commonly useful. Thus, there has been no consensus among physicists on the proper names for derivatives above pop.

Unit and dimension

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