Velocity Vectors

Bookmark added to your notes.
View Notes
×

Velocity vectors are a combination of two terms = velocity + vectors.

Let’s understand these two terms:

We interpret velocity as the rate of change of displacement.

Vectors - Those physical quantities having both direction and magnitude. 

Graphically we represent vectors by a line with an arrow-head on it.

Here, the length of the line drawn over the physical quantity represents the magnitude of the vector, and the arrow shows the direction of that vector.

The velocity is the speed with direction and magnitude. Therefore, it is a vector quantity.

So, a velocity vector represents the rate of change of position of the body. The magnitude of a velocity vector shows the speed of an object while the vector direction gives its direction.


Velocity Vector Definition

Let us consider a body moving with uniform velocity  V\[^{→}\] along with the straight line OX.

Let the position of measurement be at point O, and the time at point O is t0.

Let points A and B be the positions of the object at the instants of time t1 and t2, respectively, such that OA\[^{→}\] = x1\[^{→}\] and OB\[^{→}\] = x2\[^{→}\].

[Image will be Uploaded Soon]

Therefore, the displacement of the body in the time interval:

(t\[_{2}\] - t\[_{1}\]) = AB\[^{→}\] - OA\[^{→}\] = x2\[^{→}\] - x1\[^{→}\]

As velocity = displacement/ time interval

So,  velocity vector, V\[^{→}\]  = (x2\[^{→}\] - x1\[^{→}\])/(t\[_{2}\] - t\[_{1}\])


Instantaneous Velocity Vector

We know that the average velocity of an object is equal to its total displacement, divided by the total time taken. 

[Image will be Uploaded Soon]

We represent it as:

                     Vavg =  Δx/Δt

Let at instant time t, the object while moving covers a distance Δx in a small interval of time Δt around time t. 

If we wish to calculate the instantaneous velocity at an instant time t, then Δt approaches to zero, i.e., Δt → 0, then

Instantaneous velocity, vInstantaneous = Limit   Δx/Δt = ∂x/∂t

                                                       Δt → 0

So,  vInstantaneous of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x to t.

Therefore, the instantaneous velocity vector is a vector with a dimension of length per unit time.


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 x0a and x0b 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  = x0a+ v1t…(1)

For object Q, 

                                    x2  = x0b+ v2t…(2)

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

                                    x2  - x1 = (x0b - x0a) + (v2 - v1)t…(3)

Here, (x0b - x0a) = 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,      (x  -  x0)/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\[^{→}\].

[Image will be Uploaded Soon]

The resultant of a velocity vector can be determined by the adding velocity vectors u\[^{→}\] and v\[^{→}\]. 

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

 

The vector u\[^{→}\] + v\[^{→}\] is defined to be the vector OP\[^{→}\]

 

[Image will be Uploaded Soon]

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:

[Image will be Uploaded Soon]

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.                                  

PR\[^{→}\] = PQ\[^{→}\] + QY\[^{→}\]

FAQ (Frequently Asked Questions)

Q1: A Cyclist is Riding Her Bicycle on a Level Road at a Speed of 4 km/h. Raindrops Fall Vertically with a Speed of 6 km/h. Find the Velocity of Raindrops with Respect to the Woman. In Which Direction Should the Cyclist Hold His Umbrella to Protect Herself from the Rain?

Ans: Here, the velocity of a cyclist, vP = 4 km/h represented by OA, and

The velocity of the rain, vrP = 6 km/h is represented by OB 

To find the relative velocity of rain w.r.t. to the velocity of a cyclist (vrm), bring her to rest by imposing a velocity, - vp on her and apply it on rain as well.

Now, the resultant velocity of  vrP (OB) and -vP(=OA1) is represented by the diagonal OC of the rectangle OABC.

[Image will be Uploaded Soon]

Therefore, vrm  = √vr² + vm² = √4² + 6² = 10km/h.

So, tan Ө = OA/OB = 4/6 = 0.66 = tan 33.4

A cyclist should ride Ө = 33.4 in the vertical direction.

Q2: What is the Difference Between Vector and Velocity?

Ans: A vector is a directed quantity, having both magnitude and direction, while velocity is a speed with a directional component.

Q3: Can Velocity be Negative?

Ans: Yes. It can be negative.

Q4: Why is Speed Not a Vector?

Ans: Speed shows the distance traveled by a body, which is always positive that’s why it’s not a vector quantity.