Velocity Vectors

Velocity vectors is a complex term used in many studies in order to figure out and solve scientific problems. This term is created with the help of two words = Velocity + Vector. Both of these words have their own different meaning. Let us read about Velocity Vectors in detail. 

Velocity is simply the rate of change of movement. Whereas, on the other hand, Vector is the physical quantity of which has both magnitude and direction. The graphical representation of a vector would be assumed as a line with an arrowhead on it. 

Here, the length of the line drawn over the actual amount addresses the greatness of the vector, and the bolt shows the course of that vector. The velocity is the speed with heading and extent. Thus, it is a vector quantity. So, a velocity vector addresses the pace of progress of the position of the body. The size of a velocity vector shows the speed of an article while the vector bearing provides its guidance. 

What is Velocity Vector in terms of Definition?

For instance, consider a body is moving with a uniform velocity V beside a straight line OX

Let O be the point of position of measurement, and also then time at the point O is ‘t’.

Let the position of the object at points A and B at the instants of time ‘t₁’ and ‘t₂’.

In way that \[\overrightarrow{OA}\] = \[\overrightarrow{x_1}\] and \[\overrightarrow{OB}\] = \[\overrightarrow{x_2}\]. 

Then, the body is displaced in the time interval of -

(t₂ - t₁) = \[\overrightarrow{AB}\] - \[\overrightarrow{OA}\] = \[\overrightarrow{x_2}\] - \[\overrightarrow{x_1}\]

Hence, Velocity =  \[\frac{\text {Displacement}}{\text{Time interval}}\]

Then, the velocity vector, \[\overrightarrow{V}\] = \[\frac {(\overrightarrow{x_2} - \overrightarrow{x_1})}{t_2 - t_1}\]

What is the Instantaneous Velocity Vector?

As you read above when a total displacement of a body is divided by the total time taken then it is the average velocity of the body. 

The average velocity is represented as -

V_avg =   \[\frac {\Delta x}{\Delta t}\]

T is the instant time, Δx is the distance covered by the object while moving in a small interval of time that is Δt. 

For the calculation of instantaneous velocity at an instant time t, Δt will approach straight to zero, which is - Δt → 0 .

The formula for instantaneous velocity is -

v_{instantaneous} = Limit   \[\frac {\Delta x}{\Delta t}= \frac {\partial x}{\partial t}\]

Δt → 0

Hence, the limit of the average velocity of an object elapsed time to approach zero, or it can be x or t considering instantaneous velocity.

Also, when considering dimension length per unit time the instantaneous velocity vector is a vector. 

Relative Velocity Vector

Let’s understand what relative velocity is.

When two objects P and Q are moving with different velocities, then the velocity of object P with respect to the object Q is called the relative velocity.

Consider two objects moving with uniform velocities of v₁ and v₂, respectively, along with the parallel tracks in the same direction.

Let x_0a and x_0b be their displacements at an instant, t = 0. If at time t, x₁ and x₂  are the two displacements of the two objects regarding the origin of the position axis, then for the object P, we have

                                     x₁  = x_0a+ v₁t…(1)

For object Q, 

                                    x₂  = x_0b+ v₂t…(2)

Now, (2) - (1), we get,

                                    x₂  - x₁ = (x_0b - x_0a) + (v₂ - v₁)t…(3)

Here, (x_0b - x_0a) = x₀, the initial displacement of object Q with respect to object P at t =0, and (x₂ - x₁) = x,  the relative displacement of object Q with respect to object P at time t. This relation (3) can be re-written as

           x  = x₀ + (v₂ - v₁)t 

Or,      \[\frac {(x - x_0)}{t}\] = (v2 - v1)  

 So,     vQP = (v₂ - v₁)

                = velocity of object Q - velocity of object P

Breaking Vectors into Components   

We’ll understand the concept of breaking vectors into two components by adding velocity vectors and finding the resultant velocity vector.

Let’s use the triangular law of vector addition by considering a velocity vector example:

A swimmer is swimming across a river, she aims straight across the river, and the river pulls her downstream. We call the swimmer’s velocity as ‘u’ and the water’s velocity as ‘v’. Taking OA=u and OP= v. The resultant of a velocity vector can be determined by the adding velocity vectors u and v. 

Now, construct a vector to complete the third side of the triangle OAP.

This is how we can use the triangular law of addition to find the resultant velocity vector OP by adding two velocity vectors u and v.

This method is similar to the parallelogram law of vector addition. We can represent this by constructing a copy of v,tail-to-tail to u to obtain the parallelogram as shown below:

Using the position vector notation here, the triangular law of vector addition can be written as follows: 

For any three points, P, Q, and R.