1. Formula for velocity of longitudinal wave in a solid rod

For a rod, the speed of longitudinal waves is

$v = \sqrt{\frac{Y}{\rho}}$

where:

• $Y$ = Young's modulus
• $\rho$ = density

2. Substitute the given values

$Y = 3.2 \times 10^{11}\ \text{N m}^{-2}$ $\rho = 8 \times 10^3\ \text{kg m}^{-3}$

So,

$v = \sqrt{\frac{3.2 \times 10^{11}}{8 \times 10^3}}$

3. Simplify inside the square root

$\frac{3.2}{8} = 0.4$

and

$10^{11} \div 10^3 = 10^8$

Hence,

$v = \sqrt{0.4 \times 10^8}$

$v = \sqrt{4 \times 10^7} \div \sqrt{10}$

Better directly:

$0.4 \times 10^8 = 4 \times 10^7$

Therefore,

$v = \sqrt{4 \times 10^7}$

$v = 2 \times 10^{3.5}$

Since

$10^{3.5} = 10^3\sqrt{10} \approx 1000 \times 3.162$

So,

$v = 2 \times 3162 \approx 6324\ \text{m s}^{-1}$

Thus,

$v \approx 6.32 \times 10^3\ \text{m s}^{-1}$

4. Match with the options

This corresponds to:

Option B: $6.32 \times 10^3\ \mathrm{m s}^{-1}$

5. Comparison with stored correct answer

Stored correct answer = B

Our derived answer = B

So, they agree.