Related Formula
Vsound = √((Y)/(ρ))$$V_{\text{sound}} = \sqrt{\frac{Y}{\rho}}$$
where Y$Y$ is Young's modulus and ρ$\rho$ is density.
Core Logic
Taking natural logarithm on both sides:
ln V = (1)/(2) ln Y - (1)/(2) ln ρ$$\ln V = \frac{1}{2} \ln Y - \frac{1}{2} \ln \rho$$
Differentiating to find fractional errors/changes:
(Δ V)/(V) = (1)/(2) (Δ Y)/(Y) - (1)/(2) (Δ ρ)/(ρ)$$\frac{\Delta V}{V} = \frac{1}{2} \frac{\Delta Y}{Y} - \frac{1}{2} \frac{\Delta \rho}{\rho}$$
Step 1: Calculate Percentage Change
Multiply by 100 to get percentages:
(Δ V)/(V) × 100 = (1)/(2) ( (Δ Y)/(Y) × 100 ) - (1)/(2) ( (Δ ρ)/(ρ) × 100 )$$\frac{\Delta V}{V} \times 100 = \frac{1}{2} \left( \frac{\Delta Y}{Y} \times 100 \right) - \frac{1}{2} \left( \frac{\Delta \rho}{\rho} \times 100 \right)$$
Given (Δ Y)/(Y) × 100 = 1%$\frac{\Delta Y}{Y} \times 100 = 1\%$ and (Δ ρ)/(ρ) × 100 = 0.5%$\frac{\Delta \rho}{\rho} \times 100 = 0.5\%$.
(Δ V)/(V) × 100 = (1)/(2)(1%) - (1)/(2)(0.5%) = 0.5% - 0.25% = 0.25%$$\frac{\Delta V}{V} \times 100 = \frac{1}{2}(1\%) - \frac{1}{2}(0.5\%) = 0.5\% - 0.25\% = 0.25\%$$
Step 2: Calculate New Speed
(Δ V)/(V) = (0.25)/(100) = (1)/(400)$$\frac{\Delta V}{V} = \frac{0.25}{100} = \frac{1}{400}$$
Δ V = (1)/(400) × 400 = 1 m/s$$\Delta V = \frac{1}{400} \times 400 = 1 \text{ m/s}$$
The final speed is:
Vfinal = V + Δ V = 400 + 1 = 401 m/s$$V_{\text{final}} = V + \Delta V = 400 + 1 = 401 \text{ m/s}$$
Pattern Recognition
For products and quotients of the form A^a / B^b$A^a / B^b$, the small percentage change is ± a(%Δ A) ∓ b(%Δ B)$\pm a(\%\Delta A) \mp b(\%\Delta B)$. Here, velocity scales with Y1/2 ρ-1/2$Y^{1/2} \rho^{-1/2}$, so change is (1)/(2)(%Δ Y) - (1)/(2)(%Δ ρ)$\frac{1}{2}(\%\Delta Y) - \frac{1}{2}(\%\Delta \rho)$.
Chapter Mix
Class 11 Physics: Mechanical Properties of Solids
Class 11 Physics: Waves
