A man of mass M stands at one end of a plank of length L which lies at rest on a frictionless surface.
The man walks to the other end of the plank. If the mass of the plank is $\frac{\mathrm{M}}{3},$ the distance that the man
moves relative to the ground is

Let plank moves x distance in opposite direction. Then, displacement of man relative to ground will be (L $-$ x). Applying law of conservation of momentum,

           $m_R\frac{x_R}{t}=m_L\frac{x_L}{t}$   or   $M(L-x)=\frac{M}{3}x$

Solving this equation, we get $x = \frac{3L}{4}$

$\therefore$ Displacement of man relative to ground $= L-\frac{3L}{4}=\frac{L}{4}$