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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.
(c) It is not possible for a particle to possess zero speed with a non-zero velocity.
The instantaneous acceleration of a particle is the rate at which its velocity is changing at that instant.
(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.
s_{n}= u + a/2 (2n-1)
(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.
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-
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,
So,
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)
Resolution of vector in a plane:-
(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°):-
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 of triangle:-
Area of parallelogram:-
Volume of parallelepiped:-
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 (g sinθ) 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
V_{AB}=V_{A}-V_{B}
Here, V_{B }is called reference object velocity.
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.
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)
(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
(a)
(b) |F_{1}+F_{2}|≥|F_{3}|≥| F_{1}-F_{2}|
You Might Like to Refer:
CBSE class 7 maths| CBSE class 7 science | CBSE class 11 chemistry
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