Lacey’s equation for calculating flood discharge in rivers, is A. $${\text{V}} = 10.8\,{{\text{R}}^{\frac{1}{2}}}{{\text{S}}^{\frac{1}{2}}}$$ B. $${\text{V}} = 10.8\,{{\text{R}}^{\frac{2}{3}}}{{\text{S}}^{\frac{1}{2}}}$$ C. $${\text{V}} = 10.8\,{{\text{R}}^{\frac{2}{3}}}{{\text{S}}^{\frac{1}{3}}}$$ D. $${\text{V}} = 10.8\,{{\text{R}}^{\frac{1}{3}}}{{\text{S}}^{\frac{2}{3}}}$$

The correct answer is: C. $V = 10.8\,R^{2/3}S^{1/3}$

Lacey’s equation is a simplified form of the Manning equation that is used to calculate the discharge of a river. The equation is:

$$V = 10.8\,R^{2/3}S^{1/3}$$

where:

• $V$ is the discharge (volume of water per unit time)
• $R$ is the hydraulic radius (the cross-sectional area of the river divided by the wetted perimeter)
• $S$ is the slope of the river bed

The equation is based on the assumption that the flow of water in a river is turbulent. The exponents 2/3 and 1/3 are empirical constants that were determined by fitting the equation to experimental data.

Lacey’s equation is a useful tool for estimating the discharge of a river, but it is important to note that it is only a simplified model. The actual discharge of a river may be different from the value calculated using the equation, depending on the specific conditions of the river.

The other options are incorrect because they do not include the correct exponents. Option A does not include any exponents, Option B has the exponents 1/2 and 1/2, and Option D has the exponents 1/3 and 2/3.