A body is moving with a velocity v, breaks up into two equal parts. One of the part retraces back with velocity v. Then the velocity of the other part is

Let the initial mass of body = m 

Initial linear momentum = mv --------(i)

When it breaks into equal masses then one of the fragment retrace back with same velocity 

$\therefore \text{ Final linear momentum }=\frac{\mathrm{m}}{2}(-\mathrm{v})+\frac{\mathrm{m}}{2}\left({\mathrm{v}}_2\right)$---------------(ii)

By the conservation of linear momentum 

$\Rightarrow \mathrm{mv}=\frac{-\mathrm{mv}}{2}+\frac{{\mathrm{mv}}_2}{2}\Rightarrow {\mathrm{v}}_2=3\mathrm{v}$

i.e. other fragment moves with velocity 3v in forward direction