Answer
The particles collide a distance of $~~0.75~m~~$ from P's original position.
Work Step by Step
Since the center of mass does not move, the two particles meet at the center of mass.
Let P's original position be $x = 0$
Let Q's original position be $x = 1.0~m$
We can find the x coordinate of the center of mass:
$x_{com} = \frac{(0.10~kg)(0)+(0.30~kg)(1.0~m)}{0.10~kg+0.30~kg}$
$x_{com} = 0.75~m$
The particles collide a distance of $~~0.75~m~~$ from P's original position.
