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.

(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 ?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

(c) v²=u²+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.

• The displacement by the body in n^th second 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.  

• 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.

       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:-

• 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 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) t²

(c) v² = 2(g sinθ)s         

(ii) If time taken by the body to reach the bottom is t, then   s = ½ (g sinθ) t²

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= m₀/√(1-v²/c²)

Here, m₀ 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 = u²sin2α/g

Maximum Height, H = u²sin²α/2g

Equation of trajectory, y = xtanα-(gx²/2u²cos²α)

• 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 = u²/2g

• Condition of equilibrium:-

(a)  

(b) |F₁+F₂|≥|F₃|≥| F₁-F₂|

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