A bullet fired into a wooden block loses half of its velocity after penetrating $$40\,cm.$$  It comes to rest after penetrating a further distance of

Solution :
For first part of penetration, by equation of motion
$${\left( {\frac{u}{2}} \right)^2} - {\left( u \right)^2} = 2aS\,\,{\text{or}}\,\,a = - \frac{{3{u^2}}}{{8S}}\,......\left( {\text{i}} \right)$$
For latter part of penetration
$$\eqalign{ & {\left( 0 \right)^2} - {\left( {\frac{u}{2}} \right)^2} = 2aS',\,S' = - \frac{{{u^2}}}{{8a}} \cr & S' = - \frac{{{u^2}}}{8}\left( {\frac{{8S}}{{ - 3{u^2}}}} \right)\,\,\left( {{\text{Using}}\,\left( {\text{i}} \right)} \right) \cr & S' = \frac{S}{3}\,\,{\text{or}}\,\,S' = \frac{{40}}{3}cm \cr} $$