# Two moving points from the origin

Calculus Level 2

Two points $$P$$ and $$Q$$ start at the origin and move from the origin at the same time along the $$x$$-axis. At time $$t>0$$, $$P$$ and $$Q$$ have velocities of $$\sin (\pi t)$$ and $$2\sin (2\pi t)$$, respectively. If $$P$$ and $$Q$$ first meet again after $$\frac{m}{n}$$ seconds, where $$m$$ and $$n$$ are positive co-prime integers, what is the value of $$m+n$$?

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