In a laboratory experiment using a scaled down model to measure scour at a bridge pier, the Froude number is important. The ratio of the prototype length to the model length is 100. If the velocity of the model is 1 m/s, the velocity (in m/s) of the prototype is:

The correct option is 10.

Understanding Froude Number Similarity:
In hydraulic model testing, particularly for open-channel flows and studies involving free surface effects such as scour at bridge piers, gravitational forces dominate. Therefore, dynamic similarity between the model (m) and the prototype (p) is achieved by equating their Froude numbers (Fr).

The Froude number is defined as:

$$Fr=\frac{V}{\sqrt{gL}}$$

where:
V = velocity of flow
g = acceleration due to gravity
L = characteristic length scale

Deriving the Velocity Ratio:
For dynamic similarity, we equate the Froude number of the prototype to that of the model:

$$F{r}_p=F{r}_m$$

$$\frac{V_p}{\sqrt{g_p{L}_p}}=\frac{V_m}{\sqrt{g_m{L}_m}}$$

Since the model is tested in the same gravitational field as the prototype, the acceleration due to gravity is equal for both (g_p = g_m = g). This simplifies the relation to:

$$\frac{V_p}{\sqrt{L_p}}=\frac{V_m}{\sqrt{L_m}}$$

Rearranging the equation to solve for the velocity of the prototype (V_p):

$$V_p=V_m\times \sqrt{\frac{L_p}{L_m}}$$

Step-by-Step Calculation:
We are given the following values:
- Length ratio of prototype to model, L_p / L_m = 100
- Velocity of the model, V_m = 1 m/s

Substitute these values into the derived equation:

$$V_p=1\times \sqrt{100}$$

$$V_p=1\times 10=10\text{m/s}$$

Thus, the velocity of the prototype is 10 m/s.