Material 1 has Young's modulus Y1 and density rho1, material 2 has Young's modulus Y2 and density rho2, and material 3 has Young's modulus Y3 and density rho 3. If Y1 > Y2 > Y3 and if rho1 < rho2 < rho 3, which material has the highest speed of sound? Group of answer choices

v_s, 1 > v_s, 2 > v_s, 3

Step-by-step explanation:

Given:

- Material 1:

modulus of elasticity = E_1

density of material = p_1

- Material 2:

modulus of elasticity = E_2

density of material = p_2

- Material 3:

modulus of elasticity = E_3

density of material = p_3

- E_1 > E_2 > E_3

- p_1 < p_2 < p_3

Find:

- Which material has highest speed of sound from highest to lowest:

Solution:

- The relationship between velocity of sound in a material with its elastic modulus and density is:

v_s = sqrt (E / p)

- Since, v_s is proportional to E^0.5 and inversely proportional to p^0.5, then we have:

E_1 > E_2 > E_3

E_1^0.5 > E_2^0.5 > E_3^0.5

and p_1 < p_2 < p_3

p_1^0.5 < p_2^0.5 < p_3^0.5

Divide the two: (E_1 / p_1) ^0.5 > (E_1 / p_1) ^0.5 > (E_1 / p_1) ^0.5

Hence, v_s, 1 > v_s, 2 > v_s, 3