Question

show that the amount of kinetic energy always decreases in perfectly inelastic collision in One dimension​

Answer 1

For explanation , we take two bodies of mass m₁ and m₂ are moving with speed u₁ and u₂ ,after perfectly inelastic collision bodies collapse and formed a single body as shown in attachment , speed of single body is v and mass of it is (m₁ + m₂ ) 

Here it is clear that there is no external force act on bodies , so linear momentum of system of bodies will be conserved .
e.g., initial momentum = final momentum 
m₁u₁ + m₂u₂ = (m₁ + m₂)v 
v = (m₁u₁ + m₂u₂)/(m₁ + m₂) ------(1)

Now, use energy conservation theorem, 
Change in kinetic energy = Kf - ki 
final kinetic energy , Kf = 1/2 (m₁ + m₂) [(m₁u₁ + m₂u₂)/(m₁ + m₂)]² = 1/2(m₁u₁ + m₂u₂)²/(m₁ + m₂)
initial kinetic energy , ki = 1/2m₁u₁² + 1/2m₂u₂² 
Now, change in kinetic energy , ∆K = 1/2 (m₁u₁ + m₂u₂)²/(m₁ + m₂) - 1/2m₁u₁² - 1/2m₂u₂² 
= 1/{2(m₁ + m₂)} [m₁²u₁² + m₂²u₂² + 2m₁m₂u₁u₂ - m₁²u₁² - m₁m₂u₁² - m₂²u₂²- m₁m₂u₂²]
= 1/2 {m₁m₂/(m₁ + m₂)}[ 2u₁u₂ - u₁² - u₂² ] 
= -1/2 {(m₁m₂/(m₁ + m₂)} [u₁ - u₂ ]² ⇒ negative 

Here you can see that change in kinetic energy is negative , hence it is clear that amount of kinetic energy always decreases in perfectly inelastic collision in one dimension .