A particle starts its motion at t = 0 and its position is given by...

Given the position of the particle:

$x = 5 \sin(20 t + \pi/3)$

The velocity of the particle is the rate of change of position:

$v = \dfrac{dx}{dt} = 100 \cos(20 t + \pi/3)$

The speed of the particle is maximum when it passes through the mean position ($x = 0$), which corresponds to $|v|$ being maximum, or $|\cos(20 t + \pi/3)| = 1$.

This condition is satisfied when:

$20 t + \dfrac{\pi}{3} = n\pi$

For the first time after $t = 0$, we set $n = 1$:

$20 t + \dfrac{\pi}{3} = \pi$

$20 t = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$

$t = \dfrac{2\pi}{60} = \dfrac{\pi}{30} \text{ s}$