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(D) Let the length of the pendulum be $L$. Initially,the bob is at a horizontal position,so the initial angle $	heta_0 = 90^\circ$. The height of the

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$4$

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(D) Let the length of the pendulum be $L$. Initially,the bob is at a horizontal position,so the initial angle $	heta_0 = 90^\circ$. The height of the bob relative to the lowest point is $h_0 = L(1 - \cos 90^\circ) = L$.After $n$ collisions,the amplitude is $	heta_n$. The height is $h_n = L(1 - \cos 	heta_n)$.The coefficient of restitution $e$ relates the velocities before and after collision. Since the energy loss is related to the height,the ratio of heights after $n$ collisions is given by $\frac{h_n}{h_0} = e^{2n}$.Given $e = \frac{2}{\sqrt{5}}$,we have $e^2 = \frac{4}{5}$.We want the amplitude $	heta_n Thus,$\frac{h_n}{h_0} Substituting the relation: $(e^2)^n For $n=1$: $0.8 > 0.5$.For $n=2$: $0.64 > 0.5$.For $n=3$: $0.512 > 0.5$.For $n=4$: $0.4096 Thus,after $4$ collisions,the amplitude becomes less than $60^\circ$.

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