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A particle moves along a straight line to follow the equation ${ax}^{2} + {bv}^{2} = k$ , where a, b and k are constants and x and v are x-coordinate and velocity of the particle respectively. Find the amplitude.

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$\sqrt{\frac{b}{k}}$

$\sqrt{\frac{a}{k}}$

$\sqrt{\frac{k}{b}}$

$\sqrt{\frac{k}{a}}$

answer is D.

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Detailed Solution

${ax}^{2} + {bv}^{2} = k$

${bv}^{2} = k - {ax}^{2}$

$v^{2} = \frac{k}{b} - \frac{a}{b} x^{2}$

Compare with $v^{2} = A^{2} ω^{2} - ω^{2} x^{2}$

$ω^{2} = \frac{a}{b} and A^{2} ω^{2} = \frac{k}{b}$

$A = \sqrt{\frac{A^{2} ω^{2}}{ω^{2}}} = \sqrt{\frac{\frac{k}{b}}{\frac{a}{b}}} = \sqrt{\frac{k}{a}}$

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