Question

the speed (v) of sound in a gas is given by v=kp*x s*y where k is dimensionless constant, p is pressure and s is the density then a) x=½,y= ½ b) x = -½ y= -½ c) x=½,y= -½ d) x = - ½ y=½( choose the correct answer) (ps---x and y are powers)​

Answer 1

Given:

The speed (v) of sound in a gas is given by v=kp*x s*y where k is dimensionless constant, p is pressure and s is the density.

To find:

Value of x and y

Calculation:

Let velocity be represented in terms of pressure and density as follows :

$\therefore \: v \: \propto \: {p}^{x} \: \times \: {s}^{y}$

Representing each of them in terms of basic physical quantities :

$= > \: L{T}^{ - 1} \: \propto \: { \bigg \{M{L}^{ - 1} {T}^{ - 2} \bigg \} }^{x} \: \times \: { \bigg \{M{L}^{ - 3} \bigg \} }^{y}$

$= > \: L{T}^{ - 1} \: \propto \: \bigg \{{M \bigg \}}^{x + y} \times { \bigg \{L \bigg \}}^{ - x - 3y} \times { \bigg \{T \bigg \}}^{ - 2x}$

Comparing exponents of similar terms:

$\therefore \: - 2x = - 1 \\ = > x = \dfrac{1}{2}$

Again ,

$\therefore \: - x - 3y = 1$

$= > y = \dfrac{ - 1}{2}$

So , final answer is :

$\boxed{ \sf{ \green{ x = \dfrac{1}{2} \: \: \: and \: \: \: y = \dfrac{ - 1}{2} }}}$