Physics - Sound Wave Question with Solution | TestHub

The velocity of sound in a gas is given by the formula:

$v=\sqrt{\frac{\gamma RT}{M}}$

For hydrogen gas ($H_2$):
${\gamma }_1=\frac{7}{5}$
$M_1=2$

For helium gas ($He$):
${\gamma }_2=\frac{5}{3}$
$M_2=4$

The ratio of the velocity of sound in hydrogen to that in helium at the same temperature is:

$\frac{v_1}{v_2}=\sqrt{\frac{{\gamma }_1}{{\gamma }_2}\times \frac{M_2}{M_1}}$

Substituting the given values:

$\frac{v_1}{v_2}=\sqrt{\frac{7/5}{5/3}\times \frac{4}{2}}$

$\frac{v_1}{v_2}=\sqrt{\frac{21}{25}\times 2}$

$\frac{v_1}{v_2}=\sqrt{\frac{42}{25}}=\frac{\sqrt{42}}{5}$

The ratio is $\sqrt{42}:5$.

Answer: $\sqrt{42}:5$