The volume V of a liquid crossing through a tube is related to the area of cross-section A, velocity v and time t as `V alpha A^a v^b t^c` which of the following is correct ( given `a != 1` )

Options

A. `a != b != c`
B. `a = b = c`
C. `a != b = c`
D. `a = b != c`

Answer

To solve the problem, we need to analyze the relationship given in the question using dimensional analysis. The relationship is given as:

\[ V \propto A^a v^b t^c \]

Where:
- \( V \) is the volume,
- \( A \) is the area of cross-section,
- \( v \) is the velocity,
- \( t \) is the time,
- \( a, b, c \) are the exponents we need to determine.

### Step 1: Write down the dimensions of each quantity.

- The dimension of volume \( V \) is \( [V] = L^3 \) (length cubed).
- The dimension of area \( A \) is \( [A] = L^2 \) (length squared).
- The dimension of velocity \( v \) is \( [v] = L T^{-1} \) (length per time).
- The dimension of time \( t \) is \( [t] = T \) (time).

### Step 2: Write the dimensional equation based on the given relationship.

Using the dimensions, we can express the relationship dimensionally:

\[ L^3 = (L^2)^a \cdot (L T^{-1})^b \cdot (T)^c \]

### Step 3: Simplify the right side of the equation.

Now, simplifying the right side:

\[ L^3 = L^{2a} \cdot L^b \cdot T^{-b} \cdot T^c \]

Combining the dimensions gives:

\[ L^3 = L^{2a + b} \cdot T^{c - b} \]

### Step 4: Equate the dimensions on both sides.

For the equation to hold true, the dimensions on both sides must be equal. This gives us two equations:

1. For length:
\[ 2a + b = 3 \]

2. For time:
\[ c - b = 0 \quad \Rightarrow \quad c = b \]

### Step 5: Substitute \( c \) into the first equation.

Since \( c = b \), we can substitute \( b \) into the first equation:

\[ 2a + b = 3 \]

Now, we can express \( a \) in terms of \( b \):

\[ 2a = 3 - b \quad \Rightarrow \quad a = \frac{3 - b}{2} \]

### Step 6: Analyze the relationships between \( a, b, c \).

From the equations we have:
- \( c = b \)
- \( a = \frac{3 - b}{2} \)

Given that \( a \neq 1 \), we can analyze the relationships:
- If \( b = 0 \), then \( a = \frac{3 - 0}{2} = 1.5 \) and \( c = 0 \).
- If \( b \neq 0 \), then \( a \) will not equal \( b \) because \( a \) will always be \( \frac{3 - b}{2} \).

### Conclusion: Identify the correct option.

From the analysis, we find that:
- \( a \) is not equal to \( b \) (since \( a = \frac{3 - b}{2} \)),
- \( b \) is equal to \( c \).

Thus, the correct option is:

**Option 3: \( A \) is not equal to \( B \) and \( B \) is equal to \( C \).**