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Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
Integrating jerk over time across the Dirac delta yields the jump-discontinuity. For example, consider a path along an arc of radius r, which 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 tangential acceleration is
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Consider the ratio formed by dividing the difference of two positions of a particle (displacement) by the time interval. This ratio is called the average velocity over that time interval and is defined as ¯ = = ^ + ^ + ^ = ¯ ^ + ¯ ^ + ¯ ^ where is the displacement vector during the time interval .
Another method to describe the motion of a Brownian particle was described by Langevin, now known for its namesake as the Langevin equation.) (,) = (,), given the initial condition (, =) = (); where () is the position of the particle at some given time, is the tagged particle's initial position, and is the diffusion constant with the S.I. units ...
The expression in brackets is a total or material derivative as mentioned above, [74] in which the first term indicates how the function being differentiated changes over time at a fixed location, and the second term captures how a moving particle will see different values of that function as it travels from place to place: [+ ()] = [+] =.
Relation between Cartesian coordinates (x,y) and polar coordinates (r,θ). For example, consider a particle moving in a circular path. Its position is given by the displacement vector = ^ + ȷ ^, related to the angle, θ, and radial distance, r, as defined in the figure:
The branch of physics describing the motion of objects without reference to their cause is called kinematics, while the branch studying forces and their effect on motion is called dynamics. If an object is not in motion relative to a given frame of reference, it is said to be at rest , motionless , immobile , stationary , or to have a constant ...