Assuming that the mass `m` of the largest stone that can be moved by a flowing river depends upon the velocity `v` of the water , its density `rho` , and the acceleration due to gravity `g` . Then `m` is directly proportinal to

Options

A. `v^(3)`
B. `v^(4)`
C. `v^(5)`
D. `v^(6)`

Explanation

To determine how the mass \( m \) of the largest stone that can be moved by a flowing river depends on the velocity \( v \), the density \( \rho \), and the acceleration due to gravity \( g \), we will use dimensional analysis.

### Step-by-Step Solution:

1. **Identify the Variables**:
We have three variables:
- Velocity of water \( v \)
- Density of water \( \rho \)
- Acceleration due to gravity \( g \)

2. **Assume a Relationship**:
We assume that the mass \( m \) is proportional to these variables raised to some powers:
\[ m \propto v^a \rho^b g^c \]

3. **Introduce a Constant**:
To remove the proportionality sign, we introduce a constant \( k \):
\[ m = k v^a \rho^b g^c \]

4. **Write Dimensions**:
We need to express the dimensions of each variable:
- Mass \( m \): \( [M^1 L^0 T^0] \)
- Velocity \( v \): \( [M^0 L^1 T^{-1}] \)
- Density \( \rho \): \( [M^1 L^{-3} T^0] \)
- Acceleration due to gravity \( g \): \( [M^0 L^1 T^{-2}] \)

5. **Substitute Dimensions**:
Substitute the dimensions into the equation:
\[ [M^1 L^0 T^0] = [M^0 L^1 T^{-1}]^a [M^1 L^{-3} T^0]^b [M^0 L^1 T^{-2}]^c \]

6. **Combine Dimensions**:
This gives us:
\[ [M^1 L^0 T^0] = [M^{0a + 1b + 0c} L^{1a - 3b + 1c} T^{-1a + 0b - 2c}] \]

7. **Equate Powers**:
Now, we equate the powers of \( M \), \( L \), and \( T \):
- For \( M \): \( 1 = 0a + 1b + 0c \) → \( b = 1 \)
- For \( L \): \( 0 = 1a - 3b + 1c \)
- For \( T \): \( 0 = -1a + 0b - 2c \)

8. **Substitute \( b \)**:
Substitute \( b = 1 \) into the equations:
- From \( 0 = 1a - 3(1) + 1c \) → \( 0 = a - 3 + c \) → \( c = 3 - a \)
- From \( 0 = -1a - 2c \) → \( 0 = -a - 2(3 - a) \) → \( 0 = -a - 6 + 2a \) → \( a = 6 \)

9. **Find \( c \)**:
Substitute \( a = 6 \) back into \( c = 3 - a \):
\[ c = 3 - 6 = -3 \]

10. **Final Relationship**:
Now we have \( a = 6 \), \( b = 1 \), and \( c = -3 \). Thus, the relationship becomes:
\[ m \propto v^6 \rho^1 g^{-3} \]

### Conclusion:
The mass \( m \) of the largest stone that can be moved by a flowing river is directly proportional to \( v^6 \) and \( \rho \), and inversely proportional to \( g^3 \).