A particle executing harmonic motion is having velocities v₁ and v₂ at distances is x₁ and x₂ from the equilibrium position. The amplitude of the motion is

Difficult

A

$\sqrt{\frac{v_{1}^{2} x_{2} - v_{2}^{2} x_{1}^{2}}{v_{1}^{2} + v_{2}^{2}}}$
B

$\sqrt{\frac{v_{1}^{2} x_{1}^{2} - v_{2}^{2} x_{2}^{2}}{v_{1}^{2} + v_{2}^{2}}}$
C

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

$\sqrt{\frac{v_{1}^{2} x_{2}^{2} + v_{2}^{2} x_{1}^{2}}{v_{1}^{2} + v_{2}^{2}}}$

Solution

$v_{1} = ω \sqrt{a^{2} - x_{1}^{2}}, v_{2} = ω \sqrt{a^{2} - x_{2}^{2}}$
$We get, a = \sqrt{\frac{v_{1}^{2} x_{2}^{2} - v_{2}^{2} x_{1}^{2}}{v_{1}^{2} - v_{2}^{2}}}$

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