A displacement is the shortest distance between an object's initial and final positions, represented by a displacement vector. A position vector indicates an object's distance and direction from a reference point. Displacement can be described as the difference between initial and final position vectors. Velocity is the rate of change of an object's displacement over time and indicates both speed and direction, while speed only refers to how fast an object is moving. Average speed is the total distance traveled divided by the total time, unlike instantaneous speed which refers to speed at an instant in time.

1. A 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:

2. 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 station
For motion over a given interval of time, the displacement divided by the length
of the time interval defines the average velocity.

3. Rigid body
In 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.

4. Distance
Distance 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".

6. 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).

7. 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.

8. 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.

10. 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

11. 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