The description of motion through position, velocity and acceleration, before specifying the forces responsible. For constant acceleration these are related by integrating time derivatives. Projectile motion under an approximately uniform gravitational acceleration is an elementary example.
The position and velocity of one body relative to another are the differences of their corresponding vectors in the same reference frame. Constant relative velocity gives an affine function of time, even when neither body is at rest.
If the relative position is with , completing the square givesThis proves the minimum distance over all real times. If time is restricted to an interval, project onto that interval; if , the distance is constant. Both particles are evaluated at the same time, unlike the two independent parameters in the distance between skew lines.
Motion is a change of position with time, described relative to a reference frame. A book resting on a train has constant position in the train's frame while its position changes in a frame attached to the track. Neither description is complete until the frame and the measured time have been specified.
A record of successive positions gives a trajectory. Comparing nearby measurements gives velocity, and changes of velocity give acceleration. A camera recording a rolling ball at known time intervals provides a simple measurement: unequal gaps between successive images reveal a changing speed. The description of this record is kinematics; explaining the change using forces belongs to particle dynamics.
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Kinematics is a branch of classical mechanics that deals with the motion of objects without considering the forces that cause the motion. It focuses on describing the positions, velocities, and accelerations of objects as functions of time. Kinematics involves analyzing the paths followed by moving bodies, the time it takes to move from one position to another, and other characteristics of motion.