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**Hint:**To answer this question, we first need to understand about conservation of energy. Energy cannot be made or killed, according to the law of conservation of energy. It can only be transferred from one source of energy to another.Unless energy is added from the outside, a device still has the same amount of energy.

**Complete step by step answer:**

**Kinetic energy:**Kinetic energy is a property of a moving object or particle that is determined by its mass as well as its motion. Translation (movement along a path from one location to another), rotation around an axis, vibration, or any combination of motions are possible. Kinetic energy is a term that refers to the amount of energy.

**Potential energy:**Energy that is contained – or conserved – in an object or material is referred to as potential energy. The stored energy is determined by the object's or substance's location, arrangement, or state. It can be thought of as energy with the 'potential' to do work.

As discussed above also total energy is conserved.

Therefore, Total initial energy = Total final energy.

As described in question $H$ is the maximum height so at this point only potential energy is present as velocity at this point is zero, so kinetic energy is zero. Whereas at height $h$ there is presence of moth potential as well as kinetic energy.

So initial potential energy = final potential energy + final kinetic energy.

Substituting formula of both kinetic and potential energy

$mgH = mgh + \dfrac{1}{2}m{v^2}$

Dividing equation by mg.

We get, $H = h + \dfrac{{{v^2}}}{{2g}}$

**Hence, the final answer is option B.**

**Note:**The theory of conservation of energy states that the energy of interacting bodies or particles in a closed system remains constant. The potential energy is transformed back into kinetic energy as the pendulum swings back down. The sum of potential and kinetic energy is constant at all times.

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