A student forgot Newton's formula for speed of sound but he knows there were speed `(upsilon)` pressure (p) and density (d) in the formula. He then starts using dimensional analysis method to find the actual relation. `upsilon = kp^(x)d^(y)`. Where k is a dimensionless constant on the basis of above passage answer the following questions.
The value of x is

Options

A. 1
B. `(1)/(2)`
C. `-(1)/(2)`
D. 2

Answer

To find the value of \( x \) in the equation \( \upsilon = k p^{x} d^{y} \) using dimensional analysis, we will follow these steps:

### Step 1: Identify the dimensions of each variable
1. **Speed (\( \upsilon \))**: The dimension of speed is \( L T^{-1} \) (length per time).
2. **Pressure (\( p \))**: Pressure is defined as force per unit area. The dimension of force is \( M L T^{-2} \), and area is \( L^2 \). Therefore, the dimension of pressure is:
\[ [p] = \frac{[F]}{[A]} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]
3. **Density (\( d \))**: Density is defined as mass per unit volume. The dimension of volume is \( L^3 \). Therefore, the dimension of density is:
\[ [d] = \frac{[M]}{[V]} = \frac{M}{L^3} = M L^{-3} \]

### Step 2: Write the dimensional equation
We can express the dimensions of \( \upsilon \) in terms of \( p \) and \( d \):
\[ [L T^{-1}] = [p^{x}] [d^{y}] \]
Substituting the dimensions we found:
\[ L T^{-1} = (M L^{-1} T^{-2})^{x} (M L^{-3})^{y} \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives:
\[ L T^{-1} = M^{x+y} L^{-x-3y} T^{-2x} \]

### Step 4: Equate the dimensions
Now we equate the dimensions on both sides:
1. For mass (\( M \)):
\[ 0 = x + y \quad \text{(since there is no mass on the left side)} \]
2. For length (\( L \)):
\[ 1 = -x - 3y \]
3. For time (\( T \)):
\[ -1 = -2x \]

### Step 5: Solve the equations
From the time equation:
\[ -1 = -2x \implies x = \frac{1}{2} \]

Now substituting \( x = \frac{1}{2} \) into the mass equation:
\[ 0 = \frac{1}{2} + y \implies y = -\frac{1}{2} \]

### Conclusion
The value of \( x \) is:
\[ \boxed{\frac{1}{2}} \]