- Introduction
- |
- Average velocity
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- Instantaneous velocity
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- Average and instantaneous acceleration
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- Motion with constant acceleration
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- Projectile Motion
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- Uniform circular motion
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- Motion in three dimensions

- In previous chapter we have learned about the motion of any particle along a straight line

- Straight line motion or rectilinear motion is motion in one dimension.Now in this chapter ,we will consider both motion in two dimension and three dimension.

- In two dimensional motion path of the particle is constrained to lie in a fixed plane.Example of such motion motion are projectile shot from a gun ,motion of moon around the earth,circular motion and many more.

- To solve problems of motion in a plane,we need to generalize kinematic language of previous chapter to a more general using vector notations in two and three dimensions.

(1) Scalars are physical quantities that only have magnitude for example mass, length, time, temperature ,the distance between two points, mass of an object, and the time at which a certain event happened. The rules for combining scalars are the rules of ordinary algebra. Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers

(2)Vectors are physical quantities having both magnitude and direction for example velocity, force, electric field, torque etc. It can be represented by an arrow in space.It obeys the triangle law of addition or equivalently the parallelogram law of addition.

- A quantity that has magnitude as well as direction is called vector.From a geometric point of view, a vector can be defined as a line segment having a specific direction and a specific length
- It is denoted by the letter bold letter a or it can be denoted as a
- Magnitude of a vector a is denoted by |a| or a.It is a positive quantity
- It obeys the triangle law of addition or equivalently the parallelogram law of addition
- Example velocity, force, electric field, torque,acceleration etc

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(iX)

(1) Vector addition is commutative i.e.

(2) Vector addition is associative i.e.

(3)

4)

Now

So we will first reverse the direction of vector

From the terminal point of

Another method to find substraction of vectors would be

Let draw vector

Scalar multiplication of vectors is distributive i.e.,

n(

(i) |k a| = |k| |a|

(ii) k O = O

(iii) m (-a) = – ma = – (m a)

(iv) (-m) (-a) = m a

(v) m (n a) = mn a = n(m a)

(vi) (m + n)a = m a+ n a

(vii) m (a+b) = m a + m b

But since

Therefore,

We say that

Similarly We can represent any vector in rectangular components form. Let us assume an xyz coordinate plane and unit vector

Then we can represent any vector in the components forms like

r= x

- x,y and z are scalar components of vector r
- x
**i**,y**j**,z**k**are called the vector components - x,y,z are termed as rectagular components
- Length of vector or magnitude of the vector is defined as
- x,y,z are called the direction ratio of vector r
- In case it is given l,m,n are direction cosines of a vector then

In component form addition of two vectors is

Where,

Thus in component form resultant vector C becomes,

Cx = Ax+ Bx : Cy = Ay+ By : Cz = Az+ Bz

In component form substraction of two vectors is

D = (Ax- Bx)

where, A = (Ax, Ay, Az) and B = (Bx, By, Bz)

Thus in component form resultant vector D becomes,

Dx = Ax - Bx : Dy = Ay- By : Dz = Az- Bz

Equality of vector in components form

Axi +Ayj+Azk= Bxi+Byj+Bzk

Ax=Bx

Ay=By

Az=Bz

Multiplication of scalar by vector in components form

=K(Bx

=(kBx)

In the coordinate system, the line joining the origin O to the point P in the system is called the position vector of Point P

Let Point P and Q are there

Postion Vector of Point P

Postion Vector of Point Q

=(x

- Consider a particle moving along a curved path in x-y plane shown below in the figue

- Suppose at any time,particle is at the point P and after some time 't' is at point Q where points P and Q represents the position of particle at two different points.

- Position of particle at point P is described by the Position vector
**r**from origin O to P given by

**r**=x**i**+y**j**

where x and y are components of**r**along x and y axis

- As particle moves from P to Q,its displacement would be would be Δ
**r**which is equal to the difference in position vectors**r**and**r'**.Thus

Δ**r**=**r'**-**r**= (x'**i**+y'**j**)-(x**i**+y**j**) = (x'-x)**i**+(y'-y)**j**= Δx**i**+Δy**j**(1)

where Δx=(x'-x) and Δy=(y'-y)

- If Δt is the time interval during which the particle moves from point P to Q along the curved path then average velocity(
**v**_{avg}) of particle is the ratio of displacement and corresponding time interval

since**v**_{avg}=Δ**r**/Δt , the direction of average velocity is same as that of Δ**r**

- Magnitude of Δ
**r**is always the straight line distance from P to Q regardless of any shape of actual path taken by the particle.

- Hence average velocity of particle from point P to Q in time interval Δt would be same for any path taken by the particle.

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