Snap, Crackle and Pop

If you studied physics, like I did, you likely began your academic journey with a semester of classical mechanics. To analyze the motion of objects, we always followed the same steps: identifying all the forces at play, applying the well-known \(F=ma\) equation, breaking the system down into chosen coordinates, and deriving the equation of motion.

Throughout the semester, the exercises, though sometimes challenging, were always manageable, especially once you could compare your answers with the provided solutions. However, during exams, when faced with problems like a train moving along a circular track while simultaneously rotating, solving them required far more than just the back of an envelope. But that only made the victory all the sweeter.

I recently discovered something new about these dynamical quantities. You see, in the context I described earlier, we only considered position (denoted as \(x\)), velocity (\(v\)), and acceleration (\(a\)). From a physics point of view, the last two quantities represent the first and second derivatives of the first one, such that: \[\vec v \, =\frac {d \vec x} {dt^\phantom{2}} \, \text{ and }\, \vec a \, =\frac {d \vec v} {dt^\phantom{2}} = \frac {d^2 \vec x} {dt^2}\] But it turns out there are even more quantities beyond these. First, there is the jerk (\(j\)), also known as jolt, which is the third derivative of position with respect to time. It is commonly used in engineering and physics, particularly in motion control and vehicle dynamics, where sudden changes in acceleration can impact comfort and structural integrity.

The fourth, fifth, and sixth derivatives are less frequently encountered, so their names are not as widely standardized. The fourth derivative, known as snap (\(s\)) (or jounce), appears in fields like robotics and aerospace engineering, where precise motion planning requires higher-order derivatives. The fifth and sixth derivatives, crackle (\(c\)) and pop (\(p\)), are primarily used in specialized applications such as high-precision navigation and mechanical vibrations analysis. Their names were inspired by the Rice Krispies mascots Snap, Crackle, and Pop.

The relationship between all these quantities can be mathematically summarized as: \[\vec c \, = \frac {d \vec s \phantom{j}} {dt^\phantom{2}} = \frac {d^2 \vec \jmath} {dt^2} = \frac {d^3 \vec a} {dt^3} = \frac {d^4 \vec v} {dt^4}= \frac {d^5 \vec x} {dt^5}, \] or even visualized by the curves in the following plot:

Even though I never had the occasion to use these quantities, I appreciate learning something new, especially when it connects back to the fundamental concepts I studied at the very beginning of my academical journey.