Question

The velocity of sound increases by 50% when it enters a liquid from air and when it enters another liquid from the air, the increase in velocity is 150%. What is the ratio of % increase in the wavelength in two cases?

• A. 1:1
• B. 1:2
• C. 1:4
• D. 1:3

Answer 1

The correct option is D

1:3

STEP 1: GIVEN DATA

CASE 1 (FIRST LIQUID)

Let the Initial velocity of sound is $v_1$

after that velocity is increased by 50%

so the final velocity of sound is $v_2=$ $v_1+\frac{50}{100}{v}_1=1.5v_1$

${\lambda }_1,{\lambda }_{2}arewavelengthsrespectively$

STEP 2: EXPLANATION

$v\alpha \lambda$

$Velocity=Wavelength\times frequency$

$v=f\times \lambda$
frequency is same

so, $v\alpha \lambda$

$\frac{v_1}{v_2}=\frac{{\lambda }_1}{{\lambda }_2}$

$\frac{v_1}{1.5v_1}=\frac{{\lambda }_1}{{\lambda }_2}$

${\lambda }_2=1.5{\lambda }_1$

$percentage\%increaseinwavelength=\frac{{\lambda }_{2_{}}-{\lambda }_1}{{\lambda }_1}\times 100$

$=\frac{1.5{\lambda }_1-{\lambda }_1}{{\lambda }_1}\times 100$

$=50\%$

CASE(2): ( SECOND LIQUID)

Let the Initial velocity of sound is $v_3$

After that velocity is increased by 50%

So, the final velocity of sound is $v_4=$ $v_3+\frac{150}{100}{v}_3=2.5v_3$

${\lambda }_3,{\lambda }_{4}arewavelengthsrespectively$

STEP 2: EXPLANATION

so, $v\alpha \lambda$

$\frac{v_3}{v_4}=\frac{{\lambda }_3}{{\lambda }_4}$

$\frac{v_3}{2.5v_3}=\frac{{\lambda }_3}{{\lambda }_4}$

${\lambda }_3=1.5{\lambda }_4$

$percentage\%increaseinwavelength=\frac{{\lambda }_3-{\lambda }_4}{{\lambda }_3}\times 100$

$=\frac{2.5{\lambda }_3-{\lambda }_4}{{\lambda }_3}\times 100$

$=150\%$

$Hencetheratioofpercentageincrease\%inwavelengthintwocases$$=\frac{50}{150}=\frac{1}{3}$

option 'D’ is the correct answer