Concept of Velocity

 The term speed was used to measure several physical things. As it is insufficient only speed does not give full details about any motion.

By the term speed we can get information about the fastness of the approaching of any object but it doesn’t tell any information about the direction. Without direction, it cannot be understood completely the motion at each and every moment in space.

From the means of knowledge about the speed of an enemy army, we cannot decide the way by which the attack is going to happen. Suppose if the king travels with his army in the direction completely opposite to the direction of enemies, then it is completely a nuisance.

We know that, motion is “the change in orientation of the point in space with respect to time”.

The orientation of any moving object can only be pointed out, only if it was known about the distance and the direction of that object from a fixed reference point.

It was able to get the information of distance from the term “speed” though it is not adequate.

Besides to get a clear understanding of motion, it was started to designate new terms and measurements to define direction of any object.

           

The Direction of any motion or anything is completely indicated with

“The Cartesian Co-ordinate system”. 

As Direction is the one of the fundamental concept for the understanding of physics and mathematics, it was separately studied and analyzed in a detailed manner in the chapter of “Vectors”.

And that is how it was created two new kinds of quantities in physics so-called Vector and Scalar quantities.

Vector quantities are those which have both magnitude and direction. It gives information about both the number value and the direction of any physical quantity.  

Subsequently, Scalar quantities are those which don’t have any direction but only the magnitude.

E.g. Speed is a scalar quantity, which has only magnitude. Similarly distance and time is also a scalar quantity.

From the definition of Vectors, we can conclude Motion is a vector thing as it changes with direction. Hence to define Motion, we need vector quantities.

            Already the most successful term that defines is motion is Speed. If Speed could be defined in a vector form then it might be possible to get information about Motion in the simplest form. 

            Henceforth Speed defined in a vector manner by giving “direction to the distance” travelled by the object.

Since “time” has no direction in mechanics, “direction to the distance” gives “direction to the speed” by way of the formula,

                        Speed (s) = Distance (d) / time (t)

In addition, it was named new terms for the quantities that have direction. And the new names are,

Displacement (r)          -           distance that has direction

Velocity (v)                  -           speed that has direction

To point out the difference between vector and scalar quantities,

            Vector quantities are written by putting a small arrow mark above the top as

But for the sake of simplicity in our text format, it will be often written all vectors by its name and not by the sign.

e.g. 

Displacement vector    –          vector r
Velocity vector             –          vector v

The introduction of vectors made slight changes in the measurements of distance from the previous measurements as it was done in speed.

When coming to directions, it needs to be careful to note that “direction is represented only by the straight lines of rays and not by the curved ones”.

Though the motion happens in a curved path, displacement is measured using only straight lines of rays.

In any motion straight line of rays are used to represent the direction of any point and that ray is called the position vector of the moving point at that instant of time.

Since Motion is the change in orientation [change in position], Displacement is calculated by the change in position vectors of the initial and the final points.  

It was not considered the whole path of the motion but the initial and the final.    

The “change in position vector” is called the displacement vector and it is illustrated by drawing a ray from the initial to the final point of the motion.  

•          The numerical size of the ray is measured to be the magnitude of the displacement vector.
•          The direction of that ray with respect to a fixed reference direction is called the direction of the displacement vector.

In most of the physical problems, horizontal direction is taken as the fixed ones and all other directions are measured from this horizontal. Where it can be seen in Cartesian Co-ordinate system, the fixed horizontal direction is known as positive X - axis.

Thus it was able to measure both the magnitude and direction of any displacement vector using positional vectors.  

             If you take the motion as from the origin O to point A, then the displacement vector is denoted simply as vector r and it is directed θ angles from the horizontal.  

            See Figure: Displacement vector of a motion that has initial point at origin O

Being the initial point is at origin, position vector of initial point is zero. And so, it was given only one positional vector which is “the positional vector of the final point – vector r”.

That is why it is not obvious to see the change in position vectors in the picture of motion that is given below figure. 

But if you take the same motion in some other reference frame that is situated somewhere else in the space, then it need to be drawn two rays from a new fixed point O` to the initial and the final point of the motion.

See Figure: Position vectors of a motion

The two rays are the position vectors of initial and final position O and A.  

And the direction of the position vectors of these two points is measured by the angles θ1 [theta 1] and θ2 [theta 2].

           

But here, the displacement vector is measured using the positional vectors.   

The ray joining the initial and the final position tells the direction of the motion by which the point object moves. And the magnitude of the joining vector is defined as the magnitude of the displacement and its direction is defined as the direction of the displaced motion.

Thus displacement vector is the joining vector line that connects the position vectors r1 and r2.

From the vector sum rule it is known that

           vector O`O   +   vector OA                    =    vector O`A

          vector r1       +   displacement vector r   =    vector r2

                      displacement vector r   =    vector r2 – vector r1

Hence the displacement vector is written mathematically 

as and it makes “theta - θ” angles from the horizontal.

The angle theta made by the displacement vector with the horizontal position can be deduced by the angles of the initial and the final positions.

            Depending on the motion, position vectors changes its directions and magnitude with respect to time.

           Thus speed was further redefined with direction using the newly created term – displacement.

And it is called the velocity which is used often in physics.

Velocity is defined in a same way how the speed was defined. Instead of the distance term, it was replaced by the displacement term.

     Velocity vector (v)      =     displacement vector (r) / time (t)      

(or)

Displacement Vector ( r ) = Change in position vectors, therefore,

           

Velocity vector (v)  =  Change in position 

                                   vectors  / time (t)

It should be noted a relation between speed and the velocity. It was known that, displacement is the line drawn between the initial and the final points of the motion.

From the chapter of geometry in mathematics, it was known that the shortest distance between any two points in the space is the length of the line that connects the two points.

Henceforth displacement is the shortest distance between any two points of the motion that is considered.

            All other distances [lengths] that measured between those two points should be equal or greater than the length of the displacement vector.

            So, with correct mathematical words it is stated as,    

Magnitude of displacement vector is always less than or equal to magnitude of the distance of the path in any motion

Magnitude of displacement is the numerical value of the length of the displacement vector.

To represent the magnitude of any quantity it will be regularly used the symbol | |   [drawing two lines on both sides of the physical term].

The statement can be written as,

      |displacement vector|          ≤          |distance of path|

  

Dividing a scalar quantity will not affect the above relation and so divide by time on both sides

→   |displacement vector / time|   ≤      |distance of the path / time|

 →     |velocity|                             ≤      |speed|

Thus magnitude of velocity is always less than or equal to speed.

Note:

Writing the same statement as,

           displacement vector       ≤          distance of the path

is wrong because,  

    “Displacement vector” is a vector that has both magnitude and direction but

    “Distance of the path” has only magnitude.

      

      Comparison can be done only between same kinds of properties.

An apple cannot be compared with a mango in mathematical calculations. That is why it was compared only the magnitudes that is common for both these physical quantities.