All surfaces shown in figure are smooth. Find velocity of wedge (of mass M) when the block (of mass m) reaches the bottom of the wedge.

Explanation

When the block slides down on the smooth wedge, the wedge moves backwards.
In the horizontal direction there is no external force.
`vec(F)_(x)=0`
`therefore vec(P)_(x)=` constant
`vec(P)_(f)=vec(P)_(i)` (along x - axis) `" " m vec(u)+M vec(V)=vec(0)`
`x_(1)=` forward distance moved by the block along x -axis.
`x_(2)=` backward distance moved by the wedge along x - axis.
`m vec(u)=-M vec(V), m(x_(1))/(t)=M(x_(2))/(t)`
`mx_(1)=Mx_(2) x_(1)=(ML)/(M+m)=(M l cos theta)/(M+m)`
`x_(2)=(mL)/(M+m)=(ml cos theta)/(M+m)`
L can also be written as l cos `theta`.