Revision Notes on Kinematics
- Inertial & Non 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}= 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.
(c) It is not possible for a particle to possess zero speed with a non-zero velocity.
- Average acceleration is defined as the change in velocity over a time interval delta 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 at^{2}
(c) v^{2}=u^{2}+2as
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.
- 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.
- 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.
Magnitude of resultant vector :-
R=√(A^{2}+B^{2}+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=√(A^{2}+B^{2}+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=√(A^{2}+B^{2}), 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.
- Resolution of vector in a plane:-
- 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 .
(i) Perpendicular vector (θ=90°):-
(ii) Collinear vector:-
When, Parallel vector (θ=0°),(null vector)
When, θ=180°,(null vector)
- Unit Vector:- Unit vector of any vector is a vector having a unit magnitude, drawn in the direction of the given vector.
In three dimension,
- Area:-
Area of triangle:-
Area of parallelogram:-
Volume of parallelopiped:-
- Equation of Motion in an Inclined Plane:
(i) Perpendicular vector :- At the top of the inclined plane (t = 0, u = 0 and a = g sinq ), the equation of motion will be,
(a) v= (g sinθ)t
(b) s = ½ (g sinθ) t^{2}
(c) v^{2 }= 2(g sinθ)s
(ii) If time taken by the body to reach the bottom is t, then s = ½ (gsinθ) t^{2}
t = √(2s/g sinθ)
But sinθ =h/s or s= h/sinθ
So, t =(1/sinθ) √(2h/g)
(iii) The velocity of the body at the bottom
v=g(sinθ)t
=√2gh
- The relative velocity of object A with respect to object B is given by
V_{AB}=V_{A}–V_{B}
Here, V_{B }is called reference object velocity.
- Variation of mass:- In accordance to Einstein’s mass-variation formula, the relativistic mass of body is defined as,
m= m_{0}/√(1-v^{2}/c^{2})
Here, m_{0 }is the rest mass of the body, v is the speed of the body and c is the speed of light.
- Projectile motion in a plane:- If a particle having initial speed u is projected at an angle θ (angle of projection) with x-axis, then,
Time of Flight, T = (2u sinα)/g
Horizontal Range, R = u^{2}sin2α/g
Maximum Height, H = u^{2}sin^{2}α/2g
Equation of trajectory, y = xtanα-(gx^{2}/2u^{2}cos^{2}α)
- Motion of a ball:-
(a) When dropped:- Time period, t=√(2h/g) and speed, v=√(2gh
(b) When thrown up:- Time period, t=u/g and height, h = u^{2}/2g
- Condition of equilibrium:-
(a)
(b) |F_{1}+F_{2}|≥|F_{3}|≥| F_{1}-F_{2}|