## Third and Fourth derivative of displacement

Why is the third derivative of displacement called jerk?

The term jerk comes from mechanics where if you treat x as displacement and x' as its derivative with respect to time, then

x' = velocity

x'' = acceleration

and

x''' = jerk

x''' = Snap

x'''' = crackle

x'''''' = pop

To understand the physical interpretation of this third derivative, think of yourself as a driver of a car. Lets assume that throttle pedal directly controls acceleration of the car (in the real world, it controls this only indirectly but that's a separate matter) and hence the more you push on the pedal, the higher your car's acceleration. Now if you were to keep the pedal position fixed, your car would have constant acceleration and hence zero 'jerk' or jolting effect if you will. However, if you pump the accelerator pedal leading to quick changes in its position, the acceleration would vary with time and your car would be in jerky motion. Same thing holds while braking. If you brake with the brake pedal at a steady position, your car will slow down smoothly but if you really brake hard in a short span of time, your car would slow down or stop in a jerk.

Hope this helps relate the mathematical term to a physical interpretation!