In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to a reference frame.
The velocity of a particle is a vector quantity that describes the magnitude as well as direction of motion of the particle. More mathematically, the rate of change of the position vector of a point, with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle by the time interval. This ratio is called the average velocity over that time interval and is defined as
The speed of an object is the magnitude of its velocity. It is a scalar quantity:
The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio.
The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative,
which is the difference between the components of their position vectors.
The velocity of one point relative to another is simply the difference between their velocities
Additional relations between displacement, velocity, acceleration, and time can be derived. Since the acceleration is constant,
A relationship between velocity, position and acceleration without explicit time dependence can be had by solving the average acceleration for time and substituting and simplifying
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the Z axis:
The notation for angular velocity and angular acceleration is often defined as
Thus, for bodies in pure translation, the velocity and acceleration of every point P in the body are given by:
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:
A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases:
Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a cam joint. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.
For larger chains and their linkage topologies, see R. P. Sunkari and L. C. Schmidt, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", Mechanism and Machine Theory #41, pp. 1021–1030 (2006).