A ball is dropped vertically from rest at a height of 12m. After striiking the ground , it bounces back to a height of 9m. What fraction of kinetic energy does it lose on striking the ground?

Here,
Let v = velocity of the ball just before it strikes the ground
We know
v2 = 2gh
Multiplying both sides by m
mv2 = 2mgh
=> ½mv2 = mgh
Now loss in KE =  ½mv12 – ½mv22
=>  ½mv12 – ½mv22 = mgh1 – mgh2
Loss in KE = Loss in PE
Potential energy of the ball at height 12 m is
PE1 = mg12
Potential energy of the ball at height 9 m is
PE2 = mg9
Loss in energy = PE1 – PE2 = mg(12 - 9)
  = 3mg
Fraction of KE lost = 3mg / 12mg
  = ¼ or 25%

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