c
(c) \({v_1} = \frac{{d{y_1}}}{{dt}} = 0.1 \times 100\pi \cos \left( {100\pi t + \frac{\pi }{3}} \right)\)
\({v_2} = \frac{{d{y_2}}}{{dt}} = - 0.1\pi \sin \pi t = 0.1\pi \cos \left( {\pi t + \frac{\pi }{2}} \right)\)
Phase difference of velocity of first particle with respect to the velocity of  \(2^{nd}\) particle at \(t = 0\) is
\(\Delta \phi = {\phi _1} - {\phi _2} = \frac{\pi }{3} - \frac{\pi }{2} = - \frac{\pi }{6}.\)