Revision Notes on Kinematics

Inertial frame of reference:- Reference frame in which Newtonian mechanics holds are called inertial reference frames or inertial frames. Reference frame in which Newtonian mechanics does not hold are called non-inertial reference frames or non-inertial frames.
The average speed v_av and average velocity of a body during a time interval ?t is defined as,

v_av= average speed

= ?s/?t

Instantaneous speed and velocity are defined at a particular instant and are given by

Note:

(a) A change in either speed or direction of motion results in a change in velocity

(b) A particle which completes one revolution, along a circular path, with uniform speed is said to possess zero velocity and non-zero speed.

Average acceleration is defined as the change in velocity over a time interval ?t.

The instantaneous acceleration of a particle is the rate at which its velocity is changing at that instant.

The three equations of motion for an object with constant acceleration are given below.

(a) v= u+at

(b) s= ut+1/2 at²

Here u is the initial velocity, v is the final velocity, a is the acceleration , s is the displacement travelled by the body and t is the time.

Note: Take ‘+ve’ sign for a when the body accelerates and takes ‘–ve’ sign when the body decelerates.

The displacement by the body in n^thsecond is given by,

s_n= u + a/2 (2n-1)

Position-time (x vs t), velocity-time (v vs t) and acceleration-time (a vs t) graph for motion in one-dimension:

(i) Variation of displacement (x), velocity (v) and acceleration (a) with respect to time for different types of motion.

	Displacement(x)	Velocity(v)	Acceleration (a)
(a) At rest
(b) Motion with constant velocity
(c) Motion with constant acceleration
(d) Motion with constant deceleration

Scalar Quantities:- Scalar quantities are those quantities which require only magnitude for their complete specification.(e.g-mass, length, volume, density)
Vector Quantities:- Vector quantities are those quantities which require magnitude as well as direction for their complete specification. (e.g-displacement, velocity, acceleration, force)
Null Vector (Zero Vectors):- It is a vector having zero magnitude and an arbitrary direction.

When a null vector is added or subtracted from a given vector the resultant vector is same as the given vector.

Dot product of a null vector with any arbitrary is always zero. Cross product of a null vector with any other vector is also a null vector.

Collinear vector:- Vectors having a common line of action are called collinear vector. There are two types.

Parallel vector (θ=0°):- Two vectors acting along same direction are called parallel vectors.

Anti parallel vector (θ=180°):-Two vectors which are directed in opposite directions are called anti-parallel vectors.

Co-planar vectors- Vectors situated in one plane, irrespective of their directions, are known as co-planar vectors.
Vector addition:-

Vector addition is commutative-

Vector addition is associative-

Vector addition is distributive-

Triangles Law of Vector addition:- If two vectors are represented by two sides of a triangle, taken in the same order, then their resultant in represented by the third side of the triangle taken in opposite order.

Triangle Law of Vector Addition

Magnitude of resultant vector :-

R=√(A²+B²+2ABcosθ)

Here θ is the angle between and .

If β is the angle between and ,

then,

If three vectors acting simultaneously on a particle can be represented by the three sides of a triangle taken in the same order, then the particle will remain in equilibrium.

So,

Parallelogram law of vector addition:-

R=√(A²+B²+2ABcosθ),

Cases 1:- When, θ=0°, then,

R= A+B (maximum), β=0°

Cases 2:- When, θ=180°, then,

R= A-B (minimum), β=0°

Cases 3:- When, θ=90°, then,

R=√(A²+B²), β = tan^-1 (B/A)

The process of subtracting one vector from another is equivalent to adding, vectorially, the negative of the vector to be subtracted.

So,

Resolution of vector in a plane:-

Resolution of Vector

Product of two vectors:-

(a) Dot product or scalar product:-

Here A is the magnitude of , B is the magnitude of and θ is the angle between and .

(i) Perpendicular vector:-

(ii) Collinear vector:-

When, Parallel vector (θ=0°),

When, Anti parallel vector (θ=180°),

(b) Cross product or Vector product:-

Or,

Here A is the magnitude of , B is the magnitude of ,θ is the angle between and and is the unit vector in a direction perpendicular to the plane containing and .