Resolution

Class XI

Chapter 4 

Motion in a Plane

Resolution

          Let a and b be two non-zero vectors in a plane with different direction and let A be another vector in the same plane. A can be expressed as a sum of two vectors - one obtained by multiplying a by a real number and the other obtained by multiplying b by another real number. To see this, let O and P be the tail and head of the vector A. Then, through O, draw a straight line parallel to a, and through P, a straight line parallel to b. Let them intersect at Q. 

Therefore,

A = OP = OQ + QP

But since OQ is parallel to a, and QP is parallel to a, and QP is parallel to b , we can write:

OQ = ℷa and  𝜇b

Where  ℷ and  𝜇 are real numbers

Therefore,

A =  ℷa + 𝜇b

A has been resolved into two component vectors ℷa and  𝜇b along a and b respectively.

                   

           Resolve a vector A in terms of component vectors that lie along unit vectors i and j.

Consider a vector A that lies in x-y plane as shown in Figure. Draw a lines from the head of A perpendicular to the coordinate axes and get vectors A₁ and A₂ such that 

A₁ + A₂ = A
Since A₁ is parallel to i and A₂ is parallel to j 
A₁ = Aхⅰ
A₂ = Ayj
Therefore
Aхⅰ + Ayj = A

Aх and Ay are called the x and y components of  the vector A and the angle ፀ it makes with the x axis.
Aх = ACosፀ 
Ay = A Sinፀ 

If A and  ፀ are given                                                                 

Aх² + Ay² = A²Cos² ፀ   + A²Sin² ፀ  = A²

Tan ፀ  = Ay/Aх

Same procedure can be used to resolve vector A into three components along x, y and z axes in three dimension. If α, β and γ are the angles between x, y and z axes respectively.

Aх = ACosα
Ay = ACosβ 
Az = ACosγ

Therefore,
A = Aхi + Ayj + Azk

The magnitude of vector A
A = ✓Aх² + Ay² + Az²
r = xi + yj + zk 
Here x, y, and z are the components of r along x, y and z axes respectively.

Relative Velocity

Class XI Chapter 3

Relative Velocity

"The consideration of velocity of an object with respect to another object is said to be Relative Velocity."

Consider about two objects A and B moving in straight line with average velocities Va and Vb respectively.

Let the positions of the objects are at time t = 0 are Xa(0) and Xb(0) respectively.

After time T positions of the objects will be

Xa(t) = Xa(0) + Va.T

Xb(t) = Xb(0) + Vb.T

We know that 

Distance = Velocity x Time

Displacement from object A to object B 

= Xb(t) - Xa(t) = Xb(0) + Vb.T - Xa(t) - Va.T

or, Displacement from A to B = [Xa(0) - Xb(0)] + [Vb - Va].T

 The displacement A to B changes steadily by the amount Va - Vb.

This (Vb - Va) is also known as velocity of B relative to velocity of A.

Therefore, Relative velocity of B to A

Vba = Vb - Va.

Position - time graph of two objects with equal velocities.

When velocities are equal

Va = Va

Displacement from A to B = [Xa(0) - Xb(0)] 

The relative velocity will be zero and stay at constant distance.

Position - Time graph of two objects with unequal velocities.

When Va greater than Vb

The relative velocity of B to A will be Negative.

One graph is steeper than the other and they meet each other at a common point.

When Va and Vb have opposite sign.

 The magnitude of Vba will be greater than Va or Vb.

These three cases arises in Relative velocity.