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(D) The amplitude of a damped harmonic oscillator at any time $t$ is given by the equation $A(t) = A_0 e^{-bt/2m}$, where $A_0$ is the initial amplitu

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$\frac{A_1 A_2}{A_0}$

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(D) The amplitude of a damped harmonic oscillator at any time $t$ is given by the equation $A(t) = A_0 e^{-bt/2m}$, where $A_0$ is the initial amplitude at $t=0$ and $b$ is the damping constant.At time $t_1$, the amplitude is $A_1 = A_0 e^{-bt_1/2m}$. Thus, $e^{-bt_1/2m} = \frac{A_1}{A_0}$.At time $t_2$, the amplitude is $A_2 = A_0 e^{-bt_2/2m}$. Thus, $e^{-bt_2/2m} = \frac{A_2}{A_0}$.We want to find the amplitude $A$ at time $t = t_1 + t_2$, which is $A(t_1+t_2) = A_0 e^{-b(t_1+t_2)/2m}$.This can be written as $A(t_1+t_2) = A_0 (e^{-bt_1/2m}) (e^{-bt_2/2m})$.Substituting the expressions for the exponential terms, we get $A(t_1+t_2) = A_0 \left(\frac{A_1}{A_0}\right) \left(\frac{A_2}{A_0}\right)$.Simplifying this, we obtain $A(t_1+t_2) = \frac{A_1 A_2}{A_0}$.

If the amplitudes of a damped harmonic oscillator at times $t=0, t_1$ and $t_2$ are $A_0, A_1$ and $A_2$ respectively, then the amplitude of the oscillator at a time of $(t_1+t_2)$ is

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