The largest mass (m) that can be moved by a flowing river depends on velocity (v), density `(rho)` of river water and acceleration due to gravity (g). The correct relation is

Options

A. `mprop(rho^(2)v^(4))/(g^(2))`
B. `mprop(rhov^(6))/(g^(2))`
C. `mprop(rhov^(4))/(g^(3))`
D. `mprop(rhov^(6))/(g^(3))`

Answer

To derive the correct relation for the largest mass (m) that can be moved by a flowing river, we will use dimensional analysis. The mass depends on the velocity (v), density (ρ) of river water, and acceleration due to gravity (g). Let's go through the steps systematically.

### Step 1: Establish the relationship
We start with the assumption that the mass (m) is proportional to the velocity (v), density (ρ), and acceleration due to gravity (g). We can express this relationship as:
\[ m \propto v^x \cdot \rho^y \cdot g^z \]
where x, y, and z are the powers to which each variable is raised.

### Step 2: Write the dimensions
Next, we need to express the dimensions of each variable:
- The dimension of mass (m) is \([M^1 L^0 T^0]\).
- The dimension of velocity (v) is \([L^1 T^{-1}]\).
- The dimension of density (ρ) is \([M^1 L^{-3}]\).
- The dimension of acceleration due to gravity (g) is \([L^1 T^{-2}]\).

### Step 3: Substitute the dimensions into the equation
Now, substituting the dimensions into our proportionality equation:
\[ m \propto (L^1 T^{-1})^x \cdot (M^1 L^{-3})^y \cdot (L^1 T^{-2})^z \]
This expands to:
\[ m \propto M^y \cdot L^{x - 3y + z} \cdot T^{-x - 2z} \]

### Step 4: Equate dimensions
Since both sides of the equation must have the same dimensions, we can equate the dimensions of mass, length, and time:
1. For mass: \(y = 1\)
2. For length: \(x - 3y + z = 0\)
3. For time: \(-x - 2z = 0\)

### Step 5: Solve the equations
From the first equation, we have:
\[ y = 1 \]

Substituting \(y = 1\) into the second equation:
\[ x - 3(1) + z = 0 \]
This simplifies to:
\[ x + z = 3 \quad (1) \]

From the third equation:
\[ -x - 2z = 0 \]
This simplifies to:
\[ x + 2z = 0 \quad (2) \]

Now we can solve these two equations (1) and (2).

From equation (2), we can express \(x\) in terms of \(z\):
\[ x = -2z \]

Substituting \(x = -2z\) into equation (1):
\[ -2z + z = 3 \]
This simplifies to:
\[ -z = 3 \]
Thus:
\[ z = -3 \]

Now substituting \(z = -3\) back into \(x = -2z\):
\[ x = -2(-3) = 6 \]

### Step 6: Write the final relation
Now we have:
- \(x = 6\)
- \(y = 1\)
- \(z = -3\)

Substituting these values back into our original proportionality gives us:
\[ m \propto v^6 \cdot \rho^1 \cdot g^{-3} \]

Thus, the relation can be expressed as:
\[ m = k \cdot v^6 \cdot \rho \cdot g^{-3} \]
where \(k\) is a proportionality constant.

### Final Answer
The correct relation is:
\[ m \propto v^6 \cdot \rho \cdot g^{-3} \]