When the ball is at a height h, its total energy = E₀
= Potential energy + kinetic energy = m g h + 1/2 m 0² = mg h
where m = mass of the ball and g is the acceleration due to gravity.
When the ball reaches the floor, its energy = PE + KE = 0 + KE = KE₀
Because energy is conserved, KE₀ = E₀ = m g h
When the ball bounces back, 3/4 E₀ is lost and only 1/4 E₀ or 1/4 mgh remains.
After rebounding (n =1) the ball reaches a height = h/4, at which its PE = 1/4 mgh
E₁ = 1/4 m g h h₁ = h/4
Next collision (n=2) the energy before collision is E₁ and after = E₂ = 1/4 E₁
E₂ = 1/4 [ 1/4 mg h] = 1/4² m g h
After n th collision with the floor, the ball attains a height of