a
(a)Initial momentum of particle = \(m{V_0}\) Final momentum of system (particle + pendulum) \(= 2mv\) By the law of conservation of momentum 

\(⇒\) \(m{V_0} = 2mv\)

\(⇒\) Initial velocity of system \(v = \frac{{{V_0}}}{2}\)  

Initial K.E. of the system = \(\frac{1}{2}(2m){v^2}\)=\(\frac{1}{2}(2m){\left( {\frac{{{V_0}}}{2}} \right)^2}\)

If the system rises up to height \(h\) then  \(P.E. =2mgh\) 

By the law of conservation of energy \(\frac{1}{2}(2m){\left( {\frac{{{V_0}}}{2}} \right)^2} = 2mgh\) 

\(⇒\) \(h = \frac{{V_0^2}}{{8g}}\)

= \(\frac{1}{2}(2m){\left( {\frac{{{V_0}}}{2}} \right)^2}\)

If the system rises up to height h then \(P.E. = 2mgh\)

By the law of conservation of energy
\(\frac{1}{2}(2m){\left( {\frac{{{V_0}}}{2}} \right)^2} = 2mgh\) 

\(⇒\) \(h = \frac{{V_0^2}}{{8g}}\)