Answer
(a) The distance between the raft and the pier is 2.5 meters.
(b) The woman walked a distance of 4.0 meters relative to the pier.
Work Step by Step
(a) Let $x=0$ be the edge of the pier.
The center of mass of the raft is located at the point $x=3.5~m$
The woman is located at the point $x = 6.5~m$
We can find the location of the center of mass of the raft and the woman:
$x_{com} = \frac{(60.0~kg)(6.5~m)+(120~kg)(3.5~m)}{60.0~kg+120~kg}$
$x_{com} = 4.5~m$
Note that the center of mass is $2.0~m$ from the woman.
When the woman walks to the other end of the raft, by conservation of momentum, the location of the center of mass will remain at the point $x=4.5~m$
Then the woman will be standing at the point $x = 2.5~m$ and the pier will be 2.5 meters away. The distance between the raft and the pier is 2.5 meters.
(b) The woman walked from the point $x=6.5~m$ to the point $x = 2.5~m$, so the woman walked a distance of 4.0 meters relative to the pier.
