c
\(X \) અક્ષ દિશામાં રેખીય વેગમાનનું સંરક્ષણ  \(mu_1 = mv_1cos \theta _1 + mv2cos \theta_2  ⇒  u_1 = v_1cos \theta_1 + v2cos \theta_2     ……….(i)\)

\(Y \) - દિશામાં રેખીય વેગમાનનું સંરક્ષણ \(0 = mv_1sin \theta _1 - mv_2sin \theta _2 ⇒  0 = v_1sin \theta_1 - v_2sin \theta_2   ………(ii)\)

ગતીઉર્જા નું સરક્ષણ \(\,\frac{{\text{1}}}{{\text{2}}}mu_1^2\,\, = \,\,\frac{1}{2}mv_1^2\,\, + \,\,\frac{1}{2}mv_2^2\,\,\,\,\, \Rightarrow \,\,\,\,\,u_1^2\,\, = \,\,v_1^2\,\, + \,\,v_2^2\,\,\,\,\,.........(iii)\)

\((i)^2 + (ii)^2 \) પરથી,

\( \Rightarrow \,\,{u_1}^2\,\, + \,\,0\,\, = \,\,{v_1}^2{\cos ^2}{\theta _1}\, + \,\,{v_2}^2{\cos ^2}{\theta _2}\, + \,\,2{v_1}{v_2}\,\cos {\theta _1}\,\cos {\theta _2}\,\, + \,\,{v_1}^2{\sin ^2}{\theta _1}\, + \,\,{v_2}^2{\sin ^2}{\theta _2}\, - \,\,2{v_1}{v_2}\sin {\theta _1}\sin {\theta _2}\)

\( \Rightarrow \,\,{u_1}^2\, = \,\,{v_1}^2({\cos ^2}{\theta _1}\, + \,\,{\sin ^2}{\theta _1})\,\, + \,\,{v_2}^2\,({\cos ^2}{\theta _2}\, + \,\,{\sin ^2}{\theta _2})\,\, + \,\,2{v_1}{v_2}(\cos {\theta _1}\,\cos {\theta _2}\, - \,\,\sin {\theta _1}\,\sin {\theta _2})\)

\( \Rightarrow \,u_1^2\, = \,\,v_1^2\, + \,\,v_2^2\, + \,\,2{v_1}{v_2}\,\cos \,({\theta _1}\, + \,\,{\theta _2})\,\,\,\,\,\,\{ \,\,\because \,\,{u_1}^2\, = \,\,{v_1}^2\, + \,\,{v_2}^2\,\,\} \)

\( \Rightarrow \,\,\cos \,({\theta _1}\, + \,\,{\theta _2})\,\, = \,\,0\,\, \Rightarrow \,\,{\theta _1}\, + \,\,{\theta _2}\,\, = \,\,{90^ \circ }\)