The correct answer is $\frac{{2{{\text{V}}^2}}}{{\sqrt 3 {\text{R}}}}$.
Lacey’s silt factor is a dimensionless number that is used to estimate the sediment transport capacity of a stream. It is defined as follows:
$$f = \frac{{2{{\text{V}}^2}}}{{\sqrt 3 {\text{R}}}}$$
where $V$ is the mean velocity of the stream in meters per second and $R$ is the hydraulic mean depth of the stream in meters.
The silt factor is a function of the sediment size, the stream slope, and the stream bed material. It is typically determined empirically by comparing the measured sediment transport rate to the predicted sediment transport rate based on the silt factor.
The silt factor is an important parameter in the design of hydraulic structures, such as dams and levees. It is also used in the prediction of sediment transport in rivers and streams.
The other options are incorrect because they do not represent the correct definition of Lacey’s silt factor.