A man of mass 100 kg . is standing on a platform of mass 200 kg ....

Consider the situation shown in figure.Suppose the man moves at a speed w towards right and the the platform recoils at a speed ($\mathrm{V}$) towards left, both relative to the ice. Hence, the speed of the man relative to the platform is ($\mathrm{V}+\mathrm{w}$). By the question, ($\mathrm{V}+\mathrm{w}=\mathrm{v}$), or ($\mathrm{w}=\mathrm{v}-\mathrm{V} \ldots$) (i) Taking the platform and the man to be the system, there is no external horizontal force on the system. The linear momentum of the system remains constant. Initially both the man and the platform were at rest. Thus, (0=$\mathrm{MV}-\underset{m v}{\operatorname{mor}}$\mathrm{MV}=\mathrm{m}$(\mathrm{v}-\mathrm{V})[\mathrm{Using}(\mathrm{i})]$) or, (V=$\frac{m v}{M+m}$) Hence, the velocity relative to the ice is (V=$\frac{m v}{M+m}$).