એક પદાર્થ સ્થિર સ્થિતિમાં રહેલાં બીજા સ્થિર પ | vedclass

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$X $ અક્ષ દિશામાં રેખીય વેગમાનનું સંરક્ષણ  $mu_1 = mv_1cos 	heta _1 + mv2cos 	heta_2  &rArr;  u_1 = v_1cos 	heta_1 + v2cos 	heta_2&

Vedclass · Accepted Answer

$90$

Vedclass · Answer

$X $ અક્ષ દિશામાં રેખીય વેગમાનનું સંરક્ષણ  $mu_1 = mv_1cos 	heta _1 + mv2cos 	heta_2  &rArr;  u_1 = v_1cos 	heta_1 + v2cos 	heta_2     &hellip;&hellip;&hellip;.(i)$

$Y $ - દિશામાં રેખીય વેગમાનનું સંરક્ષણ $0 = mv_1sin 	heta _1 - mv_2sin 	heta _2 &rArr;  0 = v_1sin 	heta_1 - v_2sin 	heta_2   &hellip;&hellip;&hellip;(ii)$

ગતીઉર્જા નું સરક્ષણ $\,\frac{{	ext{1}}}{{	ext{2}}}mu_1^2\,\, = \,\,\frac{1}{2}mv_1^2\,\, + \,\,\frac{1}{2}mv_2^2\,\,\,\,\, \Rightarrow \,\,\,\,\,u_1^2\,\, = \,\,v_1^2\,\, + \,\,v_2^2\,\,\,\,\,.........(iii)$

$(i)^2 + (ii)^2 $ પરથી,

$ \Rightarrow \,\,{u_1}^2\,\, + \,\,0\,\, = \,\,{v_1}^2{\cos ^2}{	heta _1}\, + \,\,{v_2}^2{\cos ^2}{	heta _2}\, + \,\,2{v_1}{v_2}\,\cos {	heta _1}\,\cos {	heta _2}\,\, + \,\,{v_1}^2{\sin ^2}{	heta _1}\, + \,\,{v_2}^2{\sin ^2}{	heta _2}\, - \,\,2{v_1}{v_2}\sin {	heta _1}\sin {	heta _2}$

$ \Rightarrow \,\,{u_1}^2\, = \,\,{v_1}^2({\cos ^2}{	heta _1}\, + \,\,{\sin ^2}{	heta _1})\,\, + \,\,{v_2}^2\,({\cos ^2}{	heta _2}\, + \,\,{\sin ^2}{	heta _2})\,\, + \,\,2{v_1}{v_2}(\cos {	heta _1}\,\cos {	heta _2}\, - \,\,\sin {	heta _1}\,\sin {	heta _2})$

$ \Rightarrow \,u_1^2\, = \,\,v_1^2\, + \,\,v_2^2\, + \,\,2{v_1}{v_2}\,\cos \,({	heta _1}\, + \,\,{	heta _2})\,\,\,\,\,\,\{ \,\,\because \,\,{u_1}^2\, = \,\,{v_1}^2\, + \,\,{v_2}^2\,\,\} $

$ \Rightarrow \,\,\cos \,({	heta _1}\, + \,\,{	heta _2})\,\, = \,\,0\,\, \Rightarrow \,\,{	heta _1}\, + \,\,{	heta _2}\,\, = \,\,{90^ \circ }$

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