Solve the following examples.

If the energy of a ball falling from a height of 10 metres is reduced by 40%, how high will it rebound?

when a ball is kept at some height it gains potential energy, which is its total energy and when it is released just before hitting the ground all its potential energy is converted to kinetic energy, now when it collides with ground if no loss of energy is there , kinetic energy of ball remains same as of before collision but direction of velocity is reversed , now ball goes upward and its magnitude of velocity keeps on decreasing due to gravitational force in downward direction and finally ball stops attaining original height, where again all the kinetic energy is converted to potential energy but if energy is lost during collision , the initial speed with which ball starts moving upward is reduced as kinetic energy is reduced and ball attains less height than before

Ball is initially at a height , h = 10 m above the ground

After collision let the height of ball above the ground be x

Situation has been shown in the figure

Let the mass of ball be m now we know ,The Relation for Potential energy of an object of mass m due to earth’s gravity(g is acceleration due to gravity) held at an height h above ground is

P.E. = mgh

Let initial potential energy of object be P_i, so putting value of h we get initial potential object of ball as

P_i = mg × 10m = 10mg

Now all the potential energy is converted to kinetic energy so kinetic energy of ball just before collision is

K_i = 10mg

After collision kinetic energy or total energy is reduced by 40% , i.e. the kinetic energy just after collision K_f is 60% of the initial energy i.e.

K_f = 60/100 × 10mg = 6mg

Now ball will move up with this kinetic energy upto a height x till all its kinetic energy is converted to potential energy and comes to rest momentarily before again falling downwards

Now final potential energy of the ball at a height x above the groung will be

P_f = mgx

Now, equating final potential energy equal to the kinetic energy of the ball just after collision i.e.

P_f = K_f

i.e. mgx = 6mg

solving we get

x = 6m

So the ball rebounds upto a height of 6m above the ground