Two identical objects (such as billiard balls) have a one-dimensional collision in which one is initially motionless. after the collision, the moving object is stationary and the other moves with the same speed as the other originally had.

show that both momentum and kinetic energy are conserved.

  m₁ v = m₂ v

Explanation:

Let's define a system formed by the two objects, for this system the forces during the crash are internal, action and reaction, so that the memento is conserved,

Let's write the initial moment, before shock

     p₀ = m₁ v + 0

Final after crash

      = 0+ m₂ v

How the moment is preserved

    p₀ =  

    m₁ v = m₂ v

It tells us that peer speed

     m₁ = m₂

Since the two masses are the same, it shows that the moment is preserved in this collision.

The kinetic energy is

Initial

     K₀ = ½ m₁ v² + 0

Final

      = 0 + ½ m₂ v²

     K₀ =  

     ½ m₁ v² = ½ m₂ v²

    m₁ = m₂

which is true because it indicates that the objects are the same.