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.
If the density will increase the speed of sound will :

Options

A. Increase
B. decrease
C. unchanged
D. none of these

Answer

To solve the problem, we need to analyze the relationship between the speed of sound (v), pressure (p), and density (d) using dimensional analysis. The student has proposed a formula of the form:

\[ v = k p^x d^y \]

where \( k \) is a dimensionless constant, and \( x \) and \( y \) are the exponents we need to determine.

### Step-by-Step Solution:

1. **Identify the dimensions of each variable:**
- The dimension of speed \( v \) is \( [L T^{-1}] \) (length per time).
- The dimension of pressure \( p \) is \( [M L^{-1} T^{-2}] \) (mass per length per time squared).
- The dimension of density \( d \) is \( [M L^{-3}] \) (mass per volume).

2. **Express the dimensions in terms of \( k \), \( p \), and \( d \):**
- The dimensions of the right side of the equation can be expressed as:
\[ [v] = [k] [p]^x [d]^y \]
- Since \( k \) is dimensionless, we can ignore it in our dimensional analysis:
\[ [v] = [p]^x [d]^y \]

3. **Substitute the dimensions into the equation:**
- Substitute the dimensions we identified:
\[ [L T^{-1}] = ([M L^{-1} T^{-2}])^x \cdot ([M L^{-3}])^y \]

4. **Expand the right side:**
- This gives:
\[ [L T^{-1}] = [M^x L^{-x} T^{-2x}] \cdot [M^y L^{-3y}] \]
- Combining the terms:
\[ [L T^{-1}] = [M^{x+y} L^{-x-3y} T^{-2x}] \]

5. **Set up equations for each dimension:**
- Now, we can equate the dimensions on both sides:
- For mass (M): \( x + y = 0 \)
- For length (L): \( -x - 3y = 1 \)
- For time (T): \( -2x = -1 \) or \( 2x = 1 \) which gives \( x = \frac{1}{2} \)

6. **Solve for \( y \):**
- Substitute \( x = \frac{1}{2} \) into the mass equation:
\[ \frac{1}{2} + y = 0 \implies y = -\frac{1}{2} \]

7. **Final relationship:**
- Thus, we have:
\[ v = k p^{\frac{1}{2}} d^{-\frac{1}{2}} \]
- This indicates that speed \( v \) is directly proportional to the square root of pressure and inversely proportional to the square root of density.

8. **Conclusion:**
- If the density \( d \) increases, the speed of sound \( v \) will decrease, as they are inversely related in the derived formula.

### Final Answer:
If the density increases, the speed of sound will **decrease**.