The equation of motion of a particle is given by \(x = a \sin\left(50t + \dfrac{\pi}{3}\right)\) cm. The particle will come to rest at time \(t_1\) and it will have zero acceleration at time \(t_2\). The \(t_1\) and \(t_2\) respectively are _______.

The equation of motion is given by \(x = a \sin\left(50t + \dfrac{\pi}{3}\right)\).

Velocity is the rate of change of displacement:
\(v = \dfrac{dx}{dt} = 50a \cos\left(50t + \dfrac{\pi}{3}\right)\)

The particle comes to rest when \(v = 0\). For the first time it comes to rest:
\(\cos\left(50t_1 + \dfrac{\pi}{3}\right) = 0\)

\(50t_1 + \dfrac{\pi}{3} = \dfrac{\pi}{2}\)

\(50t_1 = \dfrac{\pi}{2} - \dfrac{\pi}{3} = \dfrac{\pi}{6}\)

\(t_1 = \dfrac{\pi}{300} \text{ s}\)

Acceleration is the rate of change of velocity:
\(A = \dfrac{dv}{dt} = -2500a \sin\left(50t + \dfrac{\pi}{3}\right)\)

The particle has zero acceleration when \(A = 0\). For the first time it has zero acceleration:
\(\sin\left(50t_2 + \dfrac{\pi}{3}\right) = 0\)

\(50t_2 + \dfrac{\pi}{3} = \pi\)

\(50t_2 = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}\)

\(t_2 = \dfrac{2\pi}{150} = \dfrac{\pi}{75} \text{ s}\)

Answer: \(\dfrac{\pi}{300} \, s, \dfrac{\pi}{75} \, s\)