Parametric Curve Acceleration

Calculus Level 4

\[\large{x = \sin(t) \quad \quad y = \sin^{2}(t)}\]

A particle in the \(xy\) plane has coordinates that vary with time as shown above. \(|a|\) denotes the magnitude of the particle's acceleration. Consider the following ratio:

\[\frac{| a |_{x = 0}}{| a |_{x = 1}} = \frac{A}{\sqrt{B}}\]

If \(A\) and \(B\) are coprime positive integers, determine \(A+B\).

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