The discharge through a V-notch is a common method of measuring the rate of flow of water. The discharge through a V-notch varies with the head of water above the notch. The correct option for the variation of discharge through a V-notch with respect to the head of water is option 'D', which is H5/2.

Explanation:

The discharge through a V-notch can be determined using the following formula:

Q = 1.84 C d H5/2

Where Q is the discharge, C d is the discharge coefficient, and H is the head of water above the notch.

The discharge coefficient depends on the shape and dimensions of the V-notch. The value of C d can be determined experimentally.

The variation of discharge through a V-notch with respect to the head of water can be explained using the following points:

1. The discharge through a V-notch is proportional to the head of water above the notch raised to the power of 5/2.

2. This means that the discharge increases rapidly as the head of water increases.

3. The reason for this variation is due to the shape of the V-notch. The sharp angle at the bottom of the V-notch causes the water to accelerate as it flows over the notch. This acceleration increases with the head of water, resulting in a higher discharge.

4. The variation of discharge through a V-notch with respect to the head of water is important in many engineering applications, such as the design of spillways and weirs.

In conclusion, the correct option for the variation of discharge through a V-notch with respect to the head of water is option 'D', which is H5/2.