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A particle is making simple harmonic motion along the x-axis. If at a distance $x_{1} & x_{2}$ from the mean position, the velocities of the particles are $v_{1} & v_{2}$ respectively, then the amplitude of its oscillation is given as

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$A = \sqrt{\frac{v_{1}^{2} x_{1}^{2} - v_{2}^{2} x_{2}^{2}}{v_{1} - v_{2}}}$

$A = \sqrt{\frac{v_{2}^{2} x_{1}^{2} - v_{1}^{2} x_{2}^{2}}{v_{1}^{2} - v_{2}^{2}}}$

$A = \sqrt{\frac{v_{2} x_{1} - v_{1} x_{2}}{v_{1} - v_{2}}}$

$A = \sqrt{\frac{v_{1}^{2} x_{2}^{2} - v_{2}^{2} x_{1}^{2}}{v_{1}^{2} - v_{2}^{2}}}$

answer is D.

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Detailed Solution

$V_{1} = ω \sqrt{A^{2} - x_{1}^{2}}, V_{2} = ω \sqrt{A^{2} - x_{2}^{2}}$

$\frac{V_{1}}{V_{2}} = \sqrt{\frac{A^{2} - x_{1}^{2}}{A^{2} - x_{2}^{2}}}, A = \sqrt{\frac{v_{1}^{2} x_{2}^{2} - v_{2}^{2} x_{1}^{2}}{v_{1}^{2} - v_{2}^{2}}}$

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