The maximum hydraulic efficiency of a Pelton turbine can be achieved when the blade velocity is half the inlet velocity of the jet. Let's understand why this is the case.

1. Introduction to Pelton Turbine Efficiency:
- The hydraulic efficiency of a Pelton turbine is defined as the ratio of the power output of the turbine to the power input of the water jet.
- The power output is determined by the water jet's kinetic energy, and the power input is determined by the hydraulic energy of the water jet.
- Therefore, maximizing the hydraulic efficiency means maximizing the conversion of hydraulic energy into kinetic energy.

2. Working Principle of a Pelton Turbine:
- A Pelton turbine consists of a set of buckets or blades mounted on a rotating wheel.
- A high-speed jet of water, called a water jet, is directed at the blades.
- The water jet impinges on the blades, causing them to deflect and change the direction of the water flow.
- This change in direction results in the transfer of momentum from the water to the blades, causing the wheel to rotate.

3. Relationship between Blade Velocity and Inlet Velocity:
- The blade velocity refers to the velocity of the outer edge of the blades.
- The inlet velocity refers to the velocity of the water jet just before it impinges on the blades.
- The blade velocity is directly related to the rotational speed of the wheel.

4. Optimum Blade Design for Efficiency:
- To maximize the hydraulic efficiency of the Pelton turbine, the blades should be designed in such a way that they extract maximum kinetic energy from the water jet.
- If the blade velocity is too high, the water jet will not have enough time to transfer its kinetic energy to the blades efficiently.
- On the other hand, if the blade velocity is too low, the water jet will have excessive time to transfer its kinetic energy, leading to inefficient energy conversion.
- It has been found through empirical studies that the optimum blade velocity is approximately half the inlet velocity of the jet.

5. Explanation of the Correct Answer:
- Option A, which states that the blade velocity should be half the inlet velocity of the jet, is the correct answer.
- This is because a blade velocity that is half the inlet velocity allows for efficient transfer of kinetic energy from the water jet to the blades.
- At this velocity, the water jet has sufficient time to transfer its kinetic energy to the blades without excessive energy losses.

In conclusion, to obtain maximum hydraulic efficiency in a Pelton turbine, the blade velocity should be approximately half the inlet velocity of the jet. This ensures efficient transfer of kinetic energy from the water jet to the blades, resulting in maximum power output from the turbine.