Pop (physics)

Integrals and derivatives of displacement, including pop, as well as integrals and derivatives of energy, including actergy.

In physics, pop, also known as pounce, 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, 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 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 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.