A body is executing Simple Harmonic Motion. At a displacement x its potential energy is $E_1$ and at a displacement y its potential energy is $E_2$ . The potential energy E at displacement (x + y) is
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$\mathrm{E_1} = \frac{1}{2} \mathrm{K} x^2 \Rightarrow x = \sqrt{\frac{2 \mathrm{E_1}}{\mathrm{K}}}$ , $E_2 = \frac{1}{2} K y^2 \Rightarrow y = \sqrt{\frac{2E_2}{K}}$
and $E = \frac{1}{2} K (x + y)^2 \Rightarrow x + y = \sqrt{\frac{2E}{K}}$
$\Rightarrow \sqrt{\frac{2E_1}{K}} + \sqrt{\frac{2E_2}{K}} = \sqrt{\frac{2E}{K}} \Rightarrow \sqrt{E_1} + \sqrt{E_2} = \sqrt{E}$
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