Pop

Integrals and derivatives of displacement, including pop, as well as integrals and derivatives of energy, including actergy.^[1]

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.^[2][3] Pop is defined by any of the following equivalent expressions:

[math]\displaystyle{ \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} }[/math]

The following equations are used for constant pop:

[math]\displaystyle{ \vec c = \vec c_0 + \vec p \,t }[/math]

[math]\displaystyle{ \vec s = \vec s_0 + \vec c_0 \,t + \frac{1}{2} \vec p \,t^2 }[/math]

[math]\displaystyle{ \vec \jmath = \vec \jmath_0 + \vec s_0 \,t + \frac{1}{2} \vec c_0 \,t^2 + \frac{1}{6} \vec p \,t^3 }[/math]

[math]\displaystyle{ \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 }[/math]

[math]\displaystyle{ \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 }[/math]

[math]\displaystyle{ \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 }[/math]

where

[math]\displaystyle{ \vec p }[/math] : constant pop,

[math]\displaystyle{ \vec c_0 }[/math] : initial crackle,

[math]\displaystyle{ \vec c }[/math] : final crackle,

[math]\displaystyle{ \vec s_0 }[/math] : initial snap,

[math]\displaystyle{ \vec s }[/math] : final snap,

[math]\displaystyle{ \vec \jmath_0 }[/math] : initial jerk,

[math]\displaystyle{ \vec \jmath }[/math] : final jerk,

[math]\displaystyle{ \vec a_0 }[/math] : initial acceleration,

[math]\displaystyle{ \vec a }[/math] : final acceleration,

[math]\displaystyle{ \vec v_0 }[/math] : initial velocity,

[math]\displaystyle{ \vec v }[/math] : final velocity,

[math]\displaystyle{ \vec r_0 }[/math] : initial position,

[math]\displaystyle{ \vec r }[/math] : final position,

[math]\displaystyle{ t }[/math] : 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.^[3]

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.

References

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