Collision

Collision is an important fact in Natural life. In this article we are going to discuss the collision between Natural bodies in physical science major.

Let, 2 point objects say O₁, O₂ with mass m₁, m₂ respectively collide with initial velocity u₁,u₂ and final velocity v₁,v₂ with Coefficient of restitution α = (v₁ - v₂ ) / (u₂- u₁), the ratio of final and initial relative velocities.

Now we will have following 2 equations in hand :-
1.  α = (v₁ - v₂ ) / (u₂- u₁)
2.  m₁v₁ + m₂v₂ = m₁u₁  + m₂u₂ , The Conservation of Moments.

Now solving them we get

v₂  = v₀  + μ₂. α. ( u₂ - u₁)
v₁  = v₀  + μ₁. α. ( u₁ - u₂)

Where
v₀ = ( m₁u₁ + m₂u₂ ) / ( m₁ + m₂) , The velocity of the cm of the system.
μ₂ =   m₂ / ( m₁ + m₂) , Mass partition.

Now, if :-
1)  α = 0 then v₁ =  v_{2 ,}Totally inelastic collision.
2)  α = 1 then v₁ +  u₁  =  v₂ + u_2 ,Totally elastic collision.
3) Else, α ∈ (0,1) Natural Collision. :)

In only Natural and Inelastic collision Kinatic energy will be changed into other energy ( Heat , Internal energy etc ) For amount of mechanical energy loss just calculate

Q = 1/2 ( m₁v₁² + m₂v₂² - m₁u₁² - m₂u₂² )

Markov Chain

Many time we face some problem which can be formulated as a set S = {s_i} of state and a the probability to go from one state to another is given . A subset of that kind of problem is Markov chain .

Def(Markov Chain) :- A process organised on a set of states S = {s_i} where inter state transitional probability only depends on previous and next state i and j. So we can create a Transitional matrix P = {p_ij} consist of conditional probability to go from state j to i.  

We can also say that Pⁿ  is representing the matrix {pⁿij} where pⁿij is the probability to go state i from j in n steps.

Similarly we can say that Lt (n→∞) Pⁿ  = ∏ . 

The Outcome of Maxwell's Equations

The outcome is the establishment of electromagnetic wave and later it has been proved that Light is an EM wave.
Now we get E(x,t) = E(x - ct) . It is a function of  x,t. As natural continuous functions are Fourier transformable so can find Fourier transform of this wave w.r.t time and space.

E(x,t) = E(x) ∫f(w) e^iwt dw

Now if we assume that only one frequency 'n' is there so we get  :-
E(x,t) = E_s(x) e^2iπnt ~ E_s(x) cos(2πnt) [In real domain]

So, we get a frequency relation.

Now we want to determine the frequency velocity relationship.

We can further write that E(x,t) = Σ E₀ cos(2π(x/λ - n.t)) by taking Fourier transform about both x,t .

Now for a specific bi-periodic wave E(x,t) = E₀ cos(2π(x/λ - n.t)) we get that c = nλ .

Maxwell's Equations

From the experimental results on the search of Electromagnetism Maxwell had ultimately been able to create 4 golden rules of Electromagnetism.
They are :-

(i) ∇.E = ρ/ϵ 

(ii) ∇.B = 0 

(iii) ∇×E = -∂B/ ∂t 

(iv) ∇×B = μ(J + ϵ.∂E/ ∂t) 

E and B are electric and magnetic fields respectively at a point and ϵ,μ are electric and magnetic permittivity respectively. J is the electric current density. 

Notice that there is no mention of the reference frame in the above equations because these are true for any inertial reference frame. 

Now in free space J = 0 and ρ = 0 . So 

∇×B = μ₀ϵ₀.∂E/ ∂t  

=> ∇×∇×E = -  μ₀ϵ₀.∂²E/ ∂²t 

=> ∇(∇.E) - ∇²E   = - (1/c²) ∂²E/ ∂²t  

=> ∇²E - (1/c²) ∂²E/ ∂²t = 0 [ As ρ = 0 so from (i) ∇.E = 0 ]

=>( ∇² - 1/c² . ∂²/ ∂²t ) E = 0 

Similarly we can get ( ∇² - 1/c² . ∂²/ ∂²t ) B = 0 

For plane wave along +ve x axis E(x,t) = f(x - ct).

Friction and Viscocity

Friction is force acted on the boundary surface a body when it want to get a relative velocity w.r.t another body contacted by that boundary surface.

Friction F =  μN where N is the Normal Reaction.

Similarly for fluid the viscosity is  applied on a surface of the fluid by another surface.

Classical Mechanics

The mechanics of bodies fur larger than the Hydrogen atom and moving in the velocity fur less than the velocity of light in free space is called Classical Mechanics.

Key Scientists :-
1) Galileo Galilei 
2) Sir Isaac Newton
3) Johannes Kepler
4) Lagrange 
5) Bernoulli 


In this branch mass,velocity,force,energy,power,angular mechanics etc are discussed with the Analytical Mechanics with Generalized coordinates .

This mechanics originated from theories gained simple experiments and from there only mathematical search.

Quantum Mechanics

Quantum mechanics is a branch of physics which deals with physical phenomena at microscopic scales, where the action is on the order of the Planck constant. 

Action : It is defined in the classical physics by time integration Lagrangian L = (T - V) of a system.
           
where T is the kinetic-energy and V is potential energy of the system.

Most commonly, the term is used for a functional S which takes a function of time and (for fields) space as input and returns a scalar. 

                          

I think the wikipedia  is really a good document to have an initial understanding of Classical mechanics and variation calculus. 

Key factors of the Quantum mechanics :-

1) Wave-particle duality :- According to De Broglie hypothesis  Every wave is a particle and vice-verse.

where h is Planck's constant, E and p are Energy and momentum respectively. More λ is the wavelength and f is the frequency. 

2) Light is an accumulation of particle named photon :- Light is a wave and particle according to the De-Broglie hypothesis, the particles of light is called photon. 

The nature can be described as the particle interaction as well as the Wave superposition.

This is a nice video of Quantum Object Nature . :)

http://upload.wikimedia.org/wikipedia/commons/e/e4/Wave-particle_duality.ogv

3) Wave Mechanics :- The behavior of wave particle is been discussed in the wave mechanics. 

Schrödinger Equations paved the root of this mechanics. 

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