A ball is thrown vertically downwards from a height of 20m with an initial velocity ${{V}_{\circ }}$ . It collides with the ground, loses 50% of its energy in collision and rebounds to the same height. The initial velocity ${{V}_{\circ }}$ is (Take, g =10$m/{{s}^{2}}$).
(A). $10m{{s}^{-1}}$
(B). $14m{{s}^{-1}}$
(C). $20m{{s}^{-1}}$
(D). $28m{{s}^{-1}}$

Hint: As the ball is thrown vertically downwards g is also acting on the ball. At first we need to find the rebound velocity of the ball, then from that we need to calculate the energy after a rebound that the ball contains, we know that there is a 50% loss of energy that happened in the collision, now according to the law of conservation of energy find the initial velocity.

Complete step-by-step answer:
Let us consider rebound velocity of the ball to be ‘v’
$v=\sqrt{2gh}=\sqrt{2\times 10\times 20}$
$=20m/s$
The, energy of the ball just after rebound is,
$E=\dfrac{1}{2}m{{v}^{2}}=200m$
As we know there is 50% loss just before collision that means that just before collision energy was 400m because E=200m.
Now, according to the law of conservation of energy, we have.
$\dfrac{1}{2}mv_{0}^{2}+mgh=400m$
$\Rightarrow \dfrac{1}{2}mv_{0}^{2}+m\times 10\times 20=400m$
$\Rightarrow {{v}_{0}}=20m/s$
Therefore, the correct option is C (20 m/s)

Additional Information:
From the first law of thermodynamics, according to ,the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time.
Energy can be converted but it cannot be destroyed, we can produce kinetic energy from chemical energy(In case of a bomb blast).
Acceleration due to gravity is the free fall acceleration of an object in vacuum without any drag. This steady gain in speed is caused exclusively by the force of gravitational pull.

Note: Do not forget ‘g’ while calculating rebound velocity that is the velocity after hitting the ground. Always remember the law of conservation of energy is required in many cases. There is a 50% loss of velocity after touching the ground, that is its velocity reduces to half.