Kinematics relates displacement, velocity, acceleration, and time. Most engineering problems in kinematics can be solved by assuming that the moving body is rigid and the motions occur in one plane.
Plane motion of a rigid body may be divided into four categories: rectilinear translation, in which all points of the rigid body move in straight lines; curvilinear translation, in which all points of the body move on congruent curves; rotation, in which all particles move in a circular path; and plane motion, a combination of translation and rotation in a plane.
Rectilinear translation is often of particular interest to designers. Let an arbitrary point P displace a distance s to P during time interval At. The average velocity of the point during this interval is As /At. The instantaneous velocity is obtained by letting At approach zero:

Let Av be the difference between the instantaneous velocities at points P and P’ during the time interval At. The average acceleration is Av/At. The instantaneous acceleration is

Suppose, for example, that the motion of a particle is described by the time-dependent displacement function s(t) = t^4 - 2t^2  1. By Eq. (3.19), the velocity of the particle would be

With the same relationships, the displacement function s(t) could be determined from a
given acceleration function a(t). This can be done by integrating the acceleration function twice with respect to time t. The first integration would yield the velocity function v(t)  a(t) dt, and the second would yield the displacement function s(t) =  a(t) dt dt.
These concepts can be extended to incorporate the relative motion of two points A and
B in a plane. In general, the displacement sA of A equals the vector sum of the displacement of sB of B and the displacement sAB of A relative to B:

These equations hold for any two points in a plane. They need not be points on a rigid body. (J. L. Meriam and L. G. Kraige, Mechanics, Part II: Dynamics, John Wiley & Son, Inc., New York; F. P. Beer and E. R. Johnston, Vector Mechanics for Engineers Statics and Dynamics, McGraw-Hill, Inc., New York.)

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