Collisions

An instance of one moving object or person striking violently against another.

Elastic collision:

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms.

If two bodies of masses m₁ and m₂, moving with speeds u₁ and u₂ with u₁ > u₂ the same straight line, collide with the each other, their respective speeds v₁ and v₂ after collision are given by

By the law of conservation of momentum

          m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

          m₁ (u₁ – v₁) = m₂ (v₂ – u₂)        … (1)

By the law of conservation of kinetic energy
       

         m₁ (u₁² – v₁²) = m₂ (v₂² – u₂²)

         m₁ (u₁ + v_­1) (u₁ – v₁) = m₂ (v₂ + u₂) (v₂ – u₂)         … (2)

Divide the equation (2) by equation (1)

We get    

         u₁ + v₁ = v₂ + u₂        … (3)

         V₂ = u₁ + v₁ – u₂

Substitute this in the equation (1) we get

       m₁ (u₁ – v₁) = m₂ (u₁ + v₁ – u₂ – u₂)

       m₁u₁ – m₂u₁ + 2m₂u₂ = m₂v₁ + m₁v₁

     

Now from equation (3) we can write
       v₁ = u₂ + v₂ –u₁

Substitute this in the equation 1 and solve as done above then you will get the value of v₂ as

     

Inelastic collision:

An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. In collisions of macroscopic bodies, all kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.

If two bodies of masses m₁ and m₂, moving with speeds u₁ and u₂ along the same straight line, collide and stick together, the speed v of the composite body is given by

    

The kinetic energy of the system of bodies after collision is less than that before collision. The loss in kinetic energy appears in the form of heat and sound energy.

The coefficient of restitution (e):

The coefficient of restitution (COR) is a measure of the "restitution" of a collision between two objects: how much of the kinetic energy remains for the objects to rebound from one another vs. how much is lost as heat, or work done deforming the objects.

   

Suppose a body is dropped from a height h₀ and it strikes the ground with a speed v₀. Let, after the inelastic collision, the speed with which it rebounds be v₁ and h₁ the height to which it rises, then

    

If after n collisions with the ground, the speed is v_n and the height to which the body rises is h_n, then

       

Height after nth bounces is h_n = e^2π h₀.
Speed of rebound after nth bounce is . 

Total distance travelled before the body comes to rest   

Graphs of simple functions:

CONSTANT FUNCTION:

Let k be a fixed real number. Then a function f (x) given by f (x) = k for all x ϵ R is called a constant function. Sometimes we also call it the constant function k

IDENTITY FUNCTION:

The function defined by I (x) = x for all x ϵ R is called the identity function on R.

RECIPROCAL FUNCTION:

The function that associates each non-zero real number x to its reciprocal 1/x is called the reciprocal function.

The domain and range of the reciprocal function are both equal to R – {0} i.e., the set of all non-zero real numbers.

SQUARE ROOT FUNCTION:

The function that associates every positive real number x to +√x is called the square root function i.e. f(x) = +√x

Negative real numbers do not have real square roots. So, f (x) is not defined when  is a negative real number. Therefore, domain of f is the set of all non-negative real numbers = [0, ∞)

Range (f) = {+√x | x ϵ [0, ∞)} = [0, ∞). 

LOGARITHMIC FUNCTION:

If ‘a’ is a positive real number, then the function that associates every positive real number to log_a x i.e., f (x) = log_a x is called the logarithmic function.

The domain of the logarithmic function is the set of all positive real numbers and the range is the set R of all real numbers.

EXPONENTIAL FUNCTION

If a is positive real number, then the function which associates every real number x to a^x i.e., f (x) = a^x is called the exponential function.

The domain of the exponential function is R and the range is the set of all positive real numbers.

GRAPH OF SQUARE FUNCTION:

A function given by f(x) = x² is called the square function.

The domain of the square function is R and its range is [0, ∞). The equation of the curve represented by the square function is y= x²

GRAPH OF CUBE FUNCTION:

A function given by f(x) = x³ is called the cube function. 

The domain and range of cube function are both equal to R.

Quantum Mechanical Model

The wave mechanical model of atom comes from wave equation. The wave equation describes the electron in motion as a three dimensional wave.

This was given by Schrodinger

x, y, z are Cartesian coordinates m is mass of electron.

V is potential energy of electron

E is total energy of electron

h is Planck’s constant.

Ψ is wave function of electron.

This can be written as

               

               

is Laplacian operator.

Significance of Ψ and Ψ²:

•  Ψ is wave function. It gives the amplitude of the electron wave.
• Ψ may have positive (or) negative value depending upon values of coordinates.
• Ψ² is the probability of finding electron at a point in space is given by Ψ² (x,y,z)
• The wave function “Ψ” should satisfy certain conditions called boundary conditions. They are :

1. Ψ must be continuous
2. Ψ must be finite
3. Ψ must be single valued at any point.
4. The probability of finding the electron over the space from + ∞ to - ∞ must be equal to one.

Quantum numbers:

1. Principle Quantum number (n):-

·         It determines the size of the orbital.

·         It indicates the main energy level to which the electron belongs.

n = 1,2,3,4,5,…… are named as K,L,M,N,O.

·         The maximum number of electrons in K, L, M and N energy levels are 2,8,18 and 32.

2.  Azimuthal Quantum number:-

•  It is denoted by “l”
• It defines the three dimensional shape of the orbital.
• For a given values of n, l can have values ranging from 0 to n – 1
• If n = 1 , l = 0;
  If n = 2, l = 0, 1.
• It also denotes the number of sub shells in a principal shell.
  

  value of l	0	1	2	3	4
  subshell	s	p	d	f	g

  
  
• The number of subshells in a principal shell is equal to the value of n.
• Orbital angular momentum of electron is
  
  

      3. Magnetic Quantum Number:-

·         It is denoted by m (or) m_l

·         It gives the orientation of the orbital with respect to the co – ordinate axis.

·         For any sub – shell it will have 2l + 1 values of m_l
           = -l,-(l-1),-(l-2), -------0, ------ (l-2), (l-1), l

      
    4. Spin Quantum number:-

·         It is denoted by “s” or m_s

·         It denotes the magnetic property of the electron.

·         If  the electron spins in clockwise   the electron spins  in anti-clock wise.

·         It defines the direction of spin of the electron.

·         Spin angular momentum is