Editing Jerk (physics) (section)

==In an idealized setting==
Discontinuities in acceleration do not occur in real-world environments because of [[Deformation (engineering)|deformation]], [[quantum mechanics]] effects, and other causes. However, a jump-discontinuity in acceleration and, accordingly, unbounded jerk are feasible in an idealized setting, such as an idealized [[point mass]] moving along a [[piecewise]] [[Smooth function|smooth]], whole continuous path. The jump-discontinuity occurs at points where the path is not smooth. Extrapolating from these idealized settings, one can qualitatively describe, explain and predict the effects of jerk in real situations.

Jump-discontinuity in acceleration can be modeled using a [[Dirac delta|Dirac delta function]] in jerk, scaled to the height of the jump. Integrating jerk over time across the Dirac delta yields the jump-discontinuity.

For example, consider a path along an arc of radius {{mvar|r}}, which [[tangent#Tangent line to a curve|tangentially]] connects to a straight line. The whole path is continuous, and its pieces are smooth. Now assume a point particle moves with constant speed along this path, so its [[acceleration#Tangential and centripetal acceleration|tangential acceleration]] is zero. The [[acceleration#Tangential and centripetal acceleration|centripetal acceleration]] given by {{math|{{sfrac|''v''<sup>2</sup>|''r''}}}} is normal to the arc and inward. When the particle passes the connection of pieces, it experiences a jump-discontinuity in acceleration given by {{math|{{sfrac|''v''<sup>2</sup>|''r''}}}}, and it undergoes a jerk that can be modeled by a Dirac delta, scaled to the jump-discontinuity.

For a more tangible example of discontinuous acceleration, consider an ideal [[spring–mass system]] with the mass oscillating on an idealized surface with friction. The force on the mass is equal to the [[vector sum]] of the spring force and the [[friction|kinetic frictional force]]. When the velocity changes sign (at the maximum and minimum [[Displacement (vector)|displacements]]), the magnitude of the force on the mass changes by twice the magnitude of the frictional force, because the spring force is continuous and the frictional force reverses direction with velocity. The jump in acceleration equals the force on the mass divided by the mass. That is, each time the mass passes through a minimum or maximum displacement, the mass experiences a discontinuous acceleration, and the jerk contains a Dirac delta until the mass stops. The static friction force adapts to the residual spring force, establishing equilibrium with zero net force and zero velocity.

Consider the example of a braking and decelerating car. The [[brake pad]]s generate kinetic [[frictional force]]s and constant braking [[torque]]s on the [[disk brake|disks]] (or [[drum brake|drums]]) of the wheels. Rotational velocity decreases linearly to zero with constant angular deceleration. The frictional force, torque, and car deceleration suddenly reach zero, which indicates a Dirac delta in physical jerk. The Dirac delta is smoothed down by the real environment, the cumulative effects of which are analogous to damping of the physiologically perceived jerk. This example neglects the effects of tire sliding, suspension dipping, real deflection of all ideally rigid mechanisms, etc.

Another example of significant jerk, analogous to the first example, is the cutting of a rope with a particle on its end. Assume the particle is oscillating in a circular path with non-zero centripetal acceleration. When the rope is cut, the particle's path changes abruptly to a straight path, and the force in the inward direction changes suddenly to zero. Imagine a monomolecular fiber cut by a laser; the particle would experience very high rates of jerk because of the extremely short cutting time.