## Tuesday, October 03, 2017

Chapter – 4
Motion in plain (vectors)

Scalar quantity: The physical quantities which are completely specified by their magnitude or size alone are called scalar quantities. E.g. – length, mass, volume, density, etc.
Vector quantity: The physical quantities which have both magnitude and direction are called vector quantities. E.g. – Velocity, force, angular momentum, electric field, etc.
Or
The physical quantities which have both magnitude and direction and which follows vectors laws of addition, subtraction, multiplication are called vector quantities.
Difference between scalar and vector quantity
 Scalar quantity Vector quantity 1. Scalar quantities are specified by magnitude only. 2. Scalar quantities change with magnitude only. 3. Scalar quantities with same unit can be added or subtracted according to the ordinary rules of algebra. 1. Vector quantities are specified by both magnitudes and direction. 2. Vector quantities change either with the change in magnitude or with the change in direction or with the change of both magnitude and direction. 3. Vectors quantities cannot be added or subtracted algebraically.

Representation of a vector: A vector quantity has two things. (i) Magnitude; (ii) Direction

Types of vector:
(i) Polar vector: A vector quantity whose direction is along the direction of the motion of a body or particle is known as polar vector. E.g. – Displacement, velocity, linear momentum and force, etc.
(ii) Axial vector: A vector quantity whose direction is along the axis of rotation of the body or particle is called an axial vector. E.g. – angular velocity, angular acceleration, torque, angular momentum,
Position vector: A vector drawn from the origin to the position of a particle at any instant is called position vector.

Unit vector: A vector of unit magnitude and whose direction is same as that of the given vector is called unit vector. A unit vector represents the direction of the given vector.
Consider a vectorthen vector can be written as vector = magnitude of a vector x direction.
Or
is the unit vector drawn in the direction of
Note: (1) Magnitude of unit vector is 1.
(2) Unit vector always gives the direction of the given vector.
(3) It has no unit and has no dimension.
Addition and Subtraction of vectors:
Addition of two vectors pointing in different direction
1. Triangular law of vector addition:
Statement: If two vectors are represented both in magnitude and direction by the two side of a triangle taken in the same order, than the resultant of this vectors is represented both in magnitude and direction by the third side of the triangle taken in the opposite order.
Analytical method to determine the resultant vector using triangle of vector addition: Consider two vectorandat an angle Q. According to triangle law of vector addition, the resultant of these vectors A and B is given by.

Draw which meetsof when produced forward to point.
In angled triangle

In
Whenis the magnitude of.

Also
Alsoand.
Substituting the values (ii) (iii) and (iv) in equation we have,
Which is the magnitude of resultant?
Direction of
Letmake an anglewith the direction of , then from right angled triangle.
Q. A old man walk 10 m due east from his house and then turn to this left at an angle of 600 with the east. He than walk 10 m in that direction and sits on a bench His grandson after seating him runs straight to him from the position on old man started his journey. How much distance his grandson travels to reach him in and in what direction to ran?
Ans.
Parallelogram law of vector addition
Statement: According to this law if two vector are represented both in magnitude and direction by the adjacent sides of a gm drawn from a point, than the resultant vector is represented both in magnitude and direction. By this diagonals of the gm passing through at same point
Proof: Let adjacent side OA and OB of a gm representsandrespectively. Letbe the angle betweenand.
Draw which meetsat

From right angle

Special Case:

Q. Two para of each of 4N acts on a body at anoffind the magnitude and direction of the resultant force acting on the body.
Ans. Given,

Collinear Vectors: Vectors having equal or unequal magnitude but drawn in the same magnitude are called collinear vectors.
Note: Angle between two collinear vectors is 00.
Co initial vectors: Vectors, ,andshown in the figure below are said to be co initial vector.

Scalar product or dot product of two vectors: The dot product of two vectors is defined as the product of the magnitude of the first vector and the component of second vector in direction first vector.
Dot product of two vectors gives the scalar physical quantity.

Properties of dot product:
1. Commutative:

1. Distributive:

1. Dot product of two parallel vectors:
I.e. between two vectors is 0.

1. Dot product of two equal vectors:
The angle between two equal vectors is 0.

1. Dot product of perpendicular vectors:
The angle between two perpendicular vectors is 900
Q. Show thatandareto each other.
Ans.
Homework
E.g. 4.4:
Cross product or Vector product: The cross product of two vectors is a single vector whose magnitude is equal to the product of the magnitude of two vectors multiplied by the sine of the angle between the two vectors.
The direction of the vector given by the cross product of the two vectors is perpendicular to the plane containing the two vectors i.e.
Whenis the unit vector which gives the direction of

The direction is given by Right hand Screw rule.
Properties of Cross product:
1. Anti commutative i.e.
2. Distributive
3. Cross product of two parallel vectors or equal vectors i.e. angle between two vectors is
For unit vector:
Cross product of two perpendicular vector i.e. =90
For unit vector:

4. Find vector product ofand vector

5. Cross product in terms of their component
Solution:

6. Magnitude of Cross Product of two vectorandrepresents the area of the kgm: Consider a kgm OPRQ whose adjacent side OP and OQ are represented both in magnitude and direction by the two vectorsand.
Now

Since, the kgm has two.andof area,
Resolution of a vector: The process of splitting a vector into two or more vectors is called resolution of a vector. The vectors so obtained are known as the component of the vector.

Projectile: An object thrown with initial velocity and which is then allowed to move under the action of gravity alone is called projectile.
The path followed by projectile during its flight is called trajectory.
E.g. (i) A bomb released from an aeroplane.
(ii) A bullet shoots from a gun.
(iii) A javelin thrown.
Projectile fired at an angle with the horizontal: Let a projectile be thrown with initial velocityat an anglewith the horizontal direction. Thus, and (horizontal component of the initial velocity) and(vertical component of the initial velocity).

Let at any instant of time t the projectile be at point P. Letandbe the horizontal and vertical distance travelled by projectile in time t.
The distance travelled by the projectile in the horizontal direction in time t is given by
Since, because projectile is not accelerated as horizontally as no force acts on it in horizontal direction.
Distance travelled by the projectile in the vertical direction in time t is given by
Since
Putting the value of t from equation (ii) and equation (iii) we have
Which is the equation of a parabola of second order symmetric? Codes. Hence the path followed by a projectile is parabolic.
Maximum height attained by a projectile: The maximum vertical distance travelled by the projectile during its flight or journey is called the maximum height attained by the projectile. It is denoted by H.
For vertical upward motion
Thus maximum height depends upon
1. The initial velocity of the projectile and
2. The angle of projection with the horizontal.
Time of flight of a projectile: The total time taken by the projectile from the point of projection till it heats the horizontal plain having point of projection is called time of flight.
We know
Let,
Here,
Since,
Thus time of flight depends upon
1. The initial velocity of the projectile.
2. The angle of projection with the horizontal.

Horizontal range of a projectile: The maximum horizontal distance between the point of projection and the point on the horizontal plan when the projectile hits are called horizontal range.
Or
The maximum horizontal distance travelled by projectile is called horizontal range.
For horizontal motion
Here,

Maximum range of the projectile: The range of the projectile will be maximum if(i.e. (i)). So for maximum rangeor.
Thus the projectile has maximum range if it is projected at an angle of 450 with the horizontal.
Velocity of a projectile at any instance: Consider a projectile projected from O with initial velocityat an angle Q with the horizontal direction.
Resolvinginto two component

Let the projectile be a point P after time T
For horizontal motion,
Here
For vertical motion
Sinceand are perpendicular to each other, therefore resultant ofandis given by
which is the velocity of the projectile at timeduring its flight
Letmakes an anglewith the horizontal direction.

Q. A cricket ball is thrown at a speed of 28m/s in a direction of 300 above the horizontal. Calculate the maximum height B time taken by the ball to return the same level C the distance from the thrown to the point where the ball returns to the same level.
Ans.
Q. Show that
Ans.