Solution to 1986 Problem 77

The motion of the particle is described by
\begin{align*}x(t) = A \cos \left(2 \pi f t + \delta \right)\end{align*}
So, the particle's velocity is
\begin{align*}x'(t) = -2 \pi f A \sin \left(2 \pi f t + \delta \right)\end{align*}
When x(t) = A/2 , \cos \left(2 \pi f t + \delta \right) = 1/2, so \sin \left(2 \pi f t + \delta \right) = \sqrt{3}/2 or -\sqrt{3}/2. So, the particle's speed is
\begin{align*}\left|x'(t) \right| = 2 \pi f A \cdot \frac{\sqrt{3}}{2} = \boxed{\sqrt{3} \pi f A}.\end{align*}
Hence, answer (B) is correct.

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