When two sound waves travel in the same direction in a medium, the displacements of a particle located at $'x'$ at time $'t'$ is given by

${y_1} = 0.05\,\cos \,\left( {0.50\,\pi x - 100\,\pi t} \right)$

${y_2} = 0.05\,\cos \,\left( {0.46\,\pi x - 92\,\pi t} \right)$

then velocity is..... $m/s$

Question

When two sound waves travel in the same direction in a medium, the displacements of a particle located at $'x'$ at time $'t'$ is given by

${y_1} = 0.05\,\cos \,\left( {0.50\,\pi x - 100\,\pi t} \right)$

${y_2} = 0.05\,\cos \,\left( {0.46\,\pi x - 92\,\pi t} \right)$

then velocity is..... $m/s$

JEE MAIN 2013, Medium

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Solution

Standard equation

$\mathrm{y}(\mathrm{x}, \mathrm{t})=\mathrm{A} \cos \left(\frac{\omega}{\mathrm{V}} \mathrm{x}-\omega \mathrm{t}\right)$

From any of the displacement equation Say $y_1$

$\frac{\omega}{V}=0.50 \pi \text { and } \omega=100 \pi$

$\therefore \frac{100 \pi}{V}=0.5 \pi$

$\therefore \mathrm{V}=\frac{100 \pi}{0.5 \pi}=200 \mathrm{m} / \mathrm{s}$

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