A raft of mass M with a man of mass m aboard stays motionless on the surface of a lake. The man moves a distance `l^'` relative to the raft with velocity `v^'(t)` and then stops. Assuming the water resitance to be negligible, find:
(a) the displacement of the raft 1 relative to the shore,
(b) the horizontal component of the force with which the man acted on the raft during the motion.

Explanation

(a) Since the resistance of water is negligibly small, the resultant of all external forces acting on the system "a man and a raft" is equal to zero. This means that the position of the C.M. of the given does not change in the process of motion.
i.e. `vec(r_C)=const ant` or, `Deltavec(r_C)=0` i.e. `summ_iDeltavec(r_i)=0`
or, `m(Deltavecr_(mM)+Deltavecr_M)+MDeltavecr_M=0`
Thus, `m(overset(rarr')l+overset(rarr)l)+Mvecl=0`, or, `vecl=(mvecl)/(m+M)`
(b) As net external force on "man-raft" system is equal to zero, therefore the momentum of this system does not change,
So, `0=m[overset(rarr')v(t)+vecv_2(t)]+Mvecv_2(t)`
or, `vecv_2(t)=-(mvecv(t))/(m+M)` (1)
As `overset(rarr')v(t)` or `vecv_2(t)` is along horizontal direction, thus the sought force on the raft
`=(Mdvecv_2(t))/(dt)=-(Mm)/(m+M)(doverset(rarr')(t))/(dt)`
Note: we may get the result of part (a), if we integrate Eq. (1) over the time of motion of man or raft.