Question

Proof of perfectly inelastic collision in one dimension there is a loss of kinetic energy

Answer 1

For collision , you should know two important concept
1. Law of conservation of momentum
2. work energy theorem,

Perfectly inelastic collision :- when two body move with different speed and after some time they collide and be a single body moves with some speed which is different from both Bodies . This type of collision known as perfectly inelastic collision.

Let Two bodies m₁ and m₂ moves with speed v₁ and v₂ , collide after some interval of time and new bodies of mass (m₁ + m₂) form and it moves with v velocity as shown in figure.

now, There is no external force act on Bodies so, linear momentum of bodies are conserved .
so, initial momentum = final momentum
m₁v₁ + m₂v₂ = (m₁ + m₂)v ----(1)

Now, change in kinetic energy = final kinetic energy - intial kinetic energy
= 1/2(m₁ + m₂)v² - 1/2m₁v₁² - 1/2m₂v₂²
Putting equation (1)
= 1/2(m₁ + m₂)[(m₁v₁ + m₂v₂)/(m₁ + m₂)]² - 1/2m₁v₁² - 1/2m₂v₂²
= 1/2[m₁v₁ + m₂v₂]²/(m₁ + m₂) - 1/2m₁v₁² - 1/2m₂v₂²
= 1/2[ (m₁²v₁² + m₂²v₂² + 2m₁m₂v₁v₂ - m₁²v₁² - m₁m₂v₁² - m₂²v₂² - m₁m₂v₂²)]/(m₁+m₂)
= 1/2[2m₁m₂v₁v₂ - m₁m₂v₁² - m₁m₂v₂² ]/(m₁ + m₂)
= -1/2{m₁m₂/(m₁+m₂)}(v₁²+v₂²-2v₁v₂)
= -1/2μ (v₁ - v₂)²

where μ = m₁m₂/(m₁ + m₂)
here negative sign shows that kinetic energy is lost .

Hence, proved //

Answer 2

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Explanation:

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