DisplacementA displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P. A 'displacement vector' represents the length and direction of that imaginary straight path.

A position vector expresses the position of a point P in space in terms of a displacement from an arbitrary reference point O (typically the origin of a coordinate system). Namely, it indicates both the distance and direction of an imaginary motion along a straight line from the reference position to the actual position of the point.A displacement may be also described as a 'relative position': the final position of a point relative to its initial position and a displacement vector can be mathematically defined as the difference between the final and initial position vectors:

In considering motions of objects over time the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The velocity then is distinct from the instantaneous speed which is the time rate of change of the distance traveled along a specific path. The velocity may be equivalently defined as the time rate of change of the position vector. If one considers a moving initial position, or equivalently a moving origin (e.g. an initial position or origin which is fixed to a train wagon, which in turn moves with respect to its rail track), the velocity of P (e.g. a point representing the position of a passenger walking on the train) may be referred to as a relative velocity, as opposed to an absolute velocity, which is computed with respect to a point which is considered to be 'fixed in space' (such as, for instance, a point fixed on the floor of the train stationFor motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity.

Rigid bodyIn dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement(displacement along a line), while the rotation of the body is called angular displacement.

For a position vector s that is a function of time t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, and other sciences and engineering disciplines.

These common names correspond to terminology used in basic kinematics. By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics.

DistanceDistance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and is a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other. In most cases, "distance from A to B" is interchangeable with "distance between B and A".

Distance is one of basic physical quantities hence it is "a property that can be quantified". It's basic SI unit is meter. Perhaps it's not much of an answer, but it's as far you can go. Geometrically speaking, distance between two places is a function of their coordinates or simply - just a number. Formula for the function that is used to calculate distance is called metric and can be as simple as Pythagoras theorem in plane or very complicated incorporating also time... (smart enough formula can describe how things fall down, it just needs to give smaller number (distance to the ground) for bigger time). 

Velocity velocity is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion. Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path (the object's path does not curve). Thus, a constant velocity means motion in a straight line at a constant speed. If there is a change in speed, direction, or both, then the object is said to have a changing velocity and is undergoing an acceleration. For example, a car moving at a constant 20 kilometers per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.

Velocity is a vector physical quantity; both magnitude and direction are required to define it. The scalar absolute value (magnitude) of velocity is called "speed", a quantity that is measured in meters per second (m/s or m⋅s−1) when using the SI (metric) system. For example, "5 meters per second" is a scalar (not a vector), whereas "5 meters per second east" is a vector. The rate of change of velocity (in m/s) as a function of time (in s) is "acceleration" (in m/s2 – stated "meters per second per second"), which describes how an object's speed and direction of travel change at each point in time. In science a "deceleration" is called a "negative acceleration", for example: −2 m/s2.

Speed is a scalar quantity in which direction of motion is unimportant (unlike the vector quantity velocity, in which both magnitude and direction must be taken into consideration). Movement can be described by using motion graphs. Plotting distance against time in a distance–time graph allows the total distance covered to be worked out. See also speed–time graph. 

Speed

Different from instantaneous speed, average speed is defined as the total distance covered over the time interval. For example, if a distance of 80 kilometers is driven in 1 hour, the average speed is 80 kilometers per hour. Likewise, if 320 kilometers are travelled in 4 hours, the average speed is also 80 kilometers per hour. When a distance in kilometers (km) is divided by a time in hours (h), the result is in kilometers per hour (km/h). Average speed does not describe the speed variations that may have taken place during shorter time intervals (as it is the entire distance covered divided by the total time of travel), and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to