Equations y = 2A cos^2 omega t and y = A (sin | vedclass

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(C) For the first equation: $y = 2A \cos^2 \omega t$.Using the identity $\cos^2 	heta = \frac{1 + \cos 2	heta}{2}$,we get $y = A(1 + \cos 2\omega t)

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Ratio of maximum speeds is $1 : 1$

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(C) For the first equation: $y = 2A \cos^2 \omega t$.Using the identity $\cos^2 	heta = \frac{1 + \cos 2	heta}{2}$,we get $y = A(1 + \cos 2\omega t)$,which is $y - A = A \cos 2\omega t$.This represents $S.H.M.$ with amplitude $A_1 = A$ and angular frequency $\omega_1 = 2\omega$.The maximum speed is $v_{max1} = A_1 \omega_1 = A(2\omega) = 2A\omega$.For the second equation: $y = A(\sin \omega t + \sqrt{3} \cos \omega t)$.Multiplying and dividing by $2$,we get $y = 2A(\frac{1}{2} \sin \omega t + \frac{\sqrt{3}}{2} \cos \omega t) = 2A \sin(\omega t + \frac{\pi}{3})$.This represents $S.H.M.$ with amplitude $A_2 = 2A$ and angular frequency $\omega_2 = \omega$.The maximum speed is $v_{max2} = A_2 \omega_2 = (2A)\omega = 2A\omega$.The ratio of maximum speeds is $\frac{v_{max1}}{v_{max2}} = \frac{2A\omega}{2A\omega} = 1 : 1$.

Equations $y = 2A \cos^2 \omega t$ and $y = A (\sin \omega t + \sqrt{3} \cos \omega t)$ represent the motion of two particles.

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