{"id":3891,"date":"2021-01-30T12:24:38","date_gmt":"2021-01-30T12:24:38","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3891"},"modified":"2022-04-04T02:21:07","modified_gmt":"2022-04-03T20:21:07","slug":"application-of-formulae-related-to-acceleration-in-case-of-vertical-motion","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/application-of-formulae-related-to-acceleration-in-case-of-vertical-motion\/","title":{"rendered":"\u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u0997\u09a4\u09bf\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u09b8\u09c2\u09a4\u09cd\u09b0\u09b8\u09ae\u09c2\u09b9\u09c7\u09b0 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997 | Application of formulae related to acceleration in case of vertical motion"},"content":{"rendered":"

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u0995\u09c7 \u0989\u09aa\u09b0 \u09a5\u09c7\u0995\u09c7 \u099b\u09c7\u09a1\u09bc\u09c7 \u09a6\u09bf\u09b2\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09ac\u09b2\u09c7\u09b0 \u09aa\u09cd\u09b0\u09ad\u09be\u09ac\u09c7 \u09ad\u09c2-\u09aa\u09c3\u09b7\u09cd\u09a0\u09c7 \u09aa\u09a1\u09bc\u09c7\u0964 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09ac\u09b2 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09a5\u09be\u0995\u09be\u09af\u09bc \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u098f\u0995\u099f\u09bf \u09b8\u09c1\u09b7\u09ae \u09a4\u09cd\u09ac\u09b0\u09a3 \u09a5\u09be\u0995\u09c7\u0964 \u098f \u09a4\u09cd\u09ac\u09b0\u09a3\u0995\u09c7 \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3\u0964 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u0997\u09a4\u09bf\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0987 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3\u0995\u09c7 <\/span>g<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be \u0995\u09b0\u09c7 \u09a6\u09c7\u0996\u09be \u0997\u09c7\u099b\u09c7 \u09ad\u09c2 \u09aa\u09c3\u09b7\u09cd\u09a0\u09c7 <\/span>g<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 <\/span>9.8\u00a0m\/<\/span>sec<\/span>^2<\/span><\/span> \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be \u098a\u09b0\u09cd\u09a7\u09cd\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0995\u09b0\u09b2\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7 \u09ac\u09b2\u09c7\u09b0 \u09aa\u09cd\u09b0\u09ad\u09be\u09ac\u09c7 \u09ac\u09c7\u0997 \u0995\u09ae\u09a4\u09c7 \u09a5\u09be\u0995\u09c7 \u09ac\u09b2\u09c7 \u09b8\u09c7\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3 <\/span>g<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u09a7\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09e7\u09ec\u09e6\u09e6 \u0996\u09cd\u09b0\u09bf\u09b8\u09cd\u099f\u09be\u09ac\u09cd\u09a6\u09c7\u09b0 \u09b6\u09c7\u09b7\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae \u09ac\u09be\u09b0\u09c7\u09b0 \u09ae\u09a4 \u09ac\u09bf\u099c\u09cd\u099e\u09be\u09a8\u09c0 \u0997\u09cd\u09af\u09be\u09b2\u09bf\u09b2\u09bf\u0993 \u09a4\u09be\u09a4\u09cd\u09a4\u09cd\u09ac\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0\u09c7\u09a8 \u09af\u09c7 \u09ac\u09be\u09af\u09bc\u09c1\u09b9\u09c0\u09a8 \u0995\u09cb\u09a8\u09cb \u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u09ad\u09b0\u09c7\u09b0 \u0993 \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09ac\u09b8\u09cd\u09a4\u09c1 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u099b\u09c7\u09a1\u09bc\u09c7 \u09a6\u09bf\u09b2\u09c7 \u098f\u09b0\u09be \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u09ad\u09c2\u09ae\u09bf\u09a4\u09c7 \u09aa\u09a1\u09bc\u09c7\u0964 \u09ac\u09bf\u099c\u09cd\u099e\u09be\u09a8\u09c0 \u09a8\u09bf\u0989\u099f\u09a8 <\/span>1<\/span> \u09ae\u09bf\u099f\u09be\u09b0 \u09ac\u09be\u09af\u09bc\u09c1\u09b6\u09c2\u09a8\u09cd\u09af \u099f\u09bf\u0989\u09ac\u09c7 \u0997\u09bf\u09a8\u09bf \u0993 \u09aa\u09be\u09b2\u0995 \u09a8\u09bf\u09af\u09bc\u09c7 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be \u0995\u09b0\u09c7 \u09a6\u09c7\u0996\u09be\u09a8 \u09af\u09c7 \u0997\u09bf\u09a8\u09bf \u0993 \u09aa\u09be\u09b2\u0995 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u099b\u09c7\u09a1\u09bc\u09c7 \u09a6\u09bf\u09b2\u09c7 \u099f\u09bf\u0989\u09ac\u09c7\u09b0 \u09a4\u09b2\u09be\u09af\u09bc \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u09aa\u09cc\u0981\u099b\u09c7\u0964<\/span><\/p>\n

\u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a8 (Magnitude of acceleration due to gravity): <\/b><\/h2>\n

C.G.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 <\/span>g<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 <\/span>981<\/span> \u09b8\u09c7.\u09ae\u09bf.\/<\/span>\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1<\/span>^2<\/span><\/span>, <\/span>M.K.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 <\/span>9.81<\/span> \u09ae\u09bf\u099f\u09be\u09b0\/<\/span>\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1<\/span>^2<\/span><\/span> \u098f\u09ac\u0982 <\/span>F.P.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 <\/span>32<\/span> \u09ab\u09c1\u099f\/<\/span>\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1<\/span>^2<\/span><\/span>\u0964 \u09ac\u09bf\u09b7\u09c1\u09ac \u0985\u099e\u09cd\u099a\u09b2\u09c7 <\/span>g<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ae\u09c7\u09b0\u09c1 \u0985\u099e\u09cd\u099a\u09b2 \u09b9\u09a4\u09c7\u0993 \u0995\u09ae\u0964 \u09ae\u09c7\u09b0\u09c1 \u0985\u099e\u09cd\u099a\u09b2\u09c7 \u098f\u09b0 \u09ae\u09be\u09a8 \u09b8\u09ac\u099a\u09c7\u09af\u09bc\u09c7 \u09ac\u09c7\u09b6\u09bf\u0964 \u09ad\u09c2-\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7 \u098f\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c2\u09a8\u09cd\u09af\u0964 \u0986\u09ac\u09be\u09b0, \u09ad\u09c2-\u09aa\u09c3\u09b7\u09cd\u09a0 \u09b9\u09a4\u09c7 \u09af\u09a4 \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u0989\u09a0\u09be \u09b9\u09af\u09bc \u09ac\u09be \u09a8\u09bf\u099a\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09af\u09a4 \u09a8\u09be\u09ae\u09be \u09b9\u09af\u09bc, <\/span>g<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a4\u09a4\u0987 \u0995\u09ae\u09a4\u09c7 \u09a5\u09be\u0995\u09c7\u0964\u00a0<\/span><\/p>\n

\u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u09b0\u09c7\u0996\u09be\u09af\u09bc \u099a\u09b2\u09ae\u09be\u09a8 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u0995\u09c7 <\/span>h<\/span><\/span> \u098f\u09ac\u0982 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3\u0995\u09c7 <\/span>g<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0997\u09a4\u09bf\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0997\u09c1\u09b2\u09bf \u09a8\u09bf\u09ae\u09cd\u09a8\u09b0\u09c2\u09aa:<\/span><\/p>\n\n\n\n\n\n\n\n
\u0989\u099a\u09cd\u099a\u09b8\u09cd\u09a5\u09be\u09a8 \u09b9\u09a4\u09c7 \u0985\u09ac\u09be\u09a7\u09c7 \u09aa\u09a4\u09a8\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7<\/b><\/p>\n

u=0<\/span>\u00a0<\/b><\/td>\n

\u09ad\u09c2\u09ae\u09bf \u09b9\u09a4\u09c7 <\/b>u<\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf \u0996\u09be\u09dc\u09be \u0989\u09aa\u09b0\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09b9\u09b2\u09c7<\/b><\/td>\n\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f <\/b>h<\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be \u09b9\u09a4\u09c7 <\/b>u<\/span> \u09ac\u09c7\u0997\u09c7 \u09a8\u09bf\u099a\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09b9\u09b2\u09c7<\/b><\/td>\nh<\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be \u09b9\u09a4\u09c7 <\/b>u<\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u0989\u09aa\u09b0\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09b9\u09b2\u09c7\u00a0<\/b><\/td>\n<\/tr>\n
v=g t<\/span><\/span><\/td>\nv=u-g t<\/span><\/span><\/td>\nv=u+g t<\/span><\/span><\/td>\nv=-u+g t<\/span><\/span><\/td>\n<\/tr>\n
h=\\frac{1}{2} g t^{2}<\/span><\/span><\/td>\nh=u t-\\frac{1}{2} g t^{2}<\/span><\/span>
\n<\/span><\/td>\n		h=u t+\\frac{1}{2} g t^{2}<\/span><\/span>
\n<\/span><\/td>\n		h=-u t+\\frac{1}{2} g t^{2}<\/span><\/span><\/td>\n<\/tr>\n
v^{2}=2 g h<\/span><\/span><\/td>\nv^{2}=u^{2}-2 g h<\/span><\/span><\/td>\nv^{2}=u^{2}+2 g h<\/span><\/span><\/td>\nv^{2}=u^{2}+2 g h<\/span><\/span><\/td>\n<\/tr>\n
h_{t h}=\\frac{1}{2} g(2 t-1)<\/span><\/span><\/td>\nh_{t h}<\/span><\/span>
\n<\/span><\/i>=u-\\frac{1}{2} g(2 t-1)<\/span><\/span>
\n<\/span><\/td>\n		h_{t h}<\/span> <\/span><\/i>=u+\\frac{1}{2} g(2 t-1)<\/span><\/span><\/td>\nh_{t h}<\/span> <\/i>\u00a0<\/span><\/i>=-u+\\frac{1}{2} g(2 t-1)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u0989\u09aa\u09b0\u09c7 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09b8\u09b0\u09cd\u09ac\u09be\u09a7\u09bf\u0995 \u0989\u099a\u09cd\u099a\u09a4\u09be \u0993 \u0990 \u0989\u099a\u09cd\u099a\u09a4\u09be\u09af\u09bc \u09aa\u09cb\u0981\u099b\u09be\u09b0 \u09b8\u09ae\u09af\u09bc:<\/b><\/h2>\n

u<\/span><\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u0995\u09c7 \u09ad\u09c2\u09ae\u09bf \u09a5\u09c7\u0995\u09c7 \u0996\u09be\u09a1\u09bc\u09be \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0995\u09b0\u09b2\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3 \u09aa\u09cd\u09b0\u09a4\u09bf\u0995\u09c2\u09b2\u09c7 \u0995\u09be\u099c \u0995\u09b0\u09c7 \u09ac\u09b2\u09c7 <\/span>g<\/span><\/span> \u09ae\u09a8\u09cd\u09a6\u09a8\u09c7\u09b0 \u09b8\u09c3\u09b7\u09cd\u099f\u09bf \u09b9\u09af\u09bc \u09ab\u09b2\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u09ac\u09c7\u0997 \u0995\u09cd\u09b0\u09ae\u09b6\u0983 \u0995\u09ae\u09a4\u09c7 \u09a5\u09be\u0995\u09c7\u0964 <\/span>T_{1}<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \u09b8\u09b0\u09cd\u09ac\u09be\u09a7\u09bf\u0995 <\/span>H<\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09af\u09bc \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u09ac\u09c7\u0997 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\\therefore 0=u^{2}-2 g H \\quad\\left[v^{2}=u^{2}-2 g h\\right.<\/span><\/span> \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7]<\/span><\/p>\n

\u09ac\u09be, <\/span>2 g H=u^{2}<\/span><\/span><\/p>\n

\\therefore H=\\frac{u^{2}}{2 g}=<\/span> <\/span>\u09b8\u09b0\u09cd\u09ac\u09be\u09a7\u09bf\u0995 \u0989\u099a\u09cd\u099a\u09a4\u09be\u00a0<\/b><\/p>\n

\u098f\u09ac\u0982 <\/span>0=u-g T_{1} \\quad[\\because v=u-g t]<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>g T_{1}=u<\/span><\/span><\/p>\n

\\therefore T_{1}=\\frac{u}{g}=<\/span><\/span> \u0989\u09a4\u09cd\u09a5\u09be\u09a8\u0995\u09be\u09b2\u00a0\u00a0<\/b><\/p>\n

\u09af\u09a6\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf <\/span>T_{2}<\/span><\/span> \u09b8\u09ae\u09df\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7 \u09aa\u09c1\u09a8\u09b0\u09be\u09df \u09ad\u09c2\u09ae\u09bf\u09a4\u09c7 \u09ab\u09bf\u09b0\u09c7 \u0986\u09b8\u09c7 \u09a4\u09ac\u09c7,<\/span><\/p>\n

H=0+\\frac{1}{2} g T_{2}{ }^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>\\frac{u^{2}}{2 g}=\\frac{1}{2} g T_{2}{ }^{2} \\quad\\left[\\because H=\\frac{u^{2}}{2 g}\\right]<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>g^{2} T_{2}{ }^{2}=u^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>T_{2}{ }^{2}=\\frac{u^{2}}{g^{2}}<\/span><\/span><\/p>\n

\\therefore T_{2}=\\frac{u}{g}=<\/span><\/span> \u09aa\u09a4\u09a8\u0995\u09be\u09b2<\/b><\/p>\n

\u2234<\/span> \u0989\u09a4\u09cd\u09a5\u09be\u09a8\u0995\u09be\u09b2 <\/b>=<\/span> \u09aa\u09a4\u09a8\u0995\u09be\u09b2\u00a0<\/b><\/p>\n

\"\"<\/p>\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be \u0996\u09be\u09dc\u09be \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0995\u09b0\u09b2\u09c7 \u0989\u09a4\u09cd\u09a5\u09be\u09a8\u0995\u09be\u09b2 \u0993 \u09aa\u09a4\u09a8\u0995\u09be\u09b2 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0995\u09b0\u09b2\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa\u09c7\u09b0 \u09aa\u09b0 \u09b9\u09a4\u09c7 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0989\u09a0\u09be\u09b0 \u09aa\u09b0 \u09aa\u09c1\u09a8\u09b0\u09be\u09df \u09ad\u09c2\u09ae\u09bf\u09a4\u09c7 \u09ab\u09bf\u09b0\u09c7 \u0986\u09b8\u09a4\u09c7 \u09af\u09c7 \u09b8\u09ae\u09df \u09b2\u09be\u0997\u09c7 \u09a4\u09be\u0995\u09c7 \u09ac\u09bf\u099a\u09b0\u09a3 \u0995\u09be\u09b2 (Time of flight) \u09ac\u09b2\u09be \u09b9\u09df\u0964\u00a0<\/span><\/p>\n

\u09ae\u09cb\u099f \u09ac\u09bf\u099a\u09b0\u09a3\u0995\u09be\u09b2 <\/span>T<\/span><\/span> \u09b9\u09b2\u09c7, <\/span>T=T_{1}+T_{2}=\\frac{u}{g}+\\frac{u}{g}=\\frac{2 u}{g}<\/span><\/span><\/p>\n

\u2234<\/span> \u09ae\u09cb\u099f \u09ac\u09bf\u099a\u09b0\u09a3\u0995\u09be\u09b2 <\/b>=\\frac{2 u}{g}<\/span><\/span><\/p>\n

\u09ae\u09be\u09a7\u09cd\u09af\u09be\u0995\u09b0\u09cd\u09b7\u09a3\u09c7\u09b0 \u09aa\u09cd\u09b0\u09ad\u09be\u09ac\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u0997\u09a4\u09bf (Vertical motion under gravity)<\/b><\/h2>\n

u<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u0996\u09be\u09dc\u09be \u0989\u09b0\u09cd\u09a7\u09cd\u09ac \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0997\u09a4\u09bf:\u00a0<\/b><\/p>\n

\u0996\u09be\u09dc\u09be \u0989\u09b0\u09cd\u09a7\u09cd\u09ac \u09a6\u09bf\u0995\u0995\u09c7 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09a7\u09b0\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3 <\/span>\\frac{d^{2} s}{d t^{2}}=-g<\/span><\/span>\u0964<\/span><\/p>\n

\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>\\frac{d s}{d t}=-g t+A \\quad ... \\quad ...(1)<\/span><\/span><\/p>\n

\u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09af\u0996\u09a8t=0<\/span><\/span>,<\/span> \u09a4\u0996\u09a8 \u09ac\u09c7\u0997 <\/span>\\frac{d s}{d t}=u<\/span><\/span><\/p>\n

\\therefore (1)<\/span><\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>A=u<\/span><\/span><\/p>\n

\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>(1)<\/span> \u098f\u09b0 <\/span>A=u<\/span><\/span> \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987, <\/span>\\frac{d s}{d t}=-g t+u \\quad ... \\quad ...(2)<\/span><\/span><\/p>\n

\u09af\u09a6\u09bf <\/span>t<\/span><\/span> \u09b8\u09ae\u09df\u09c7 \u0995\u09a3\u09be\u099f\u09bf\u09b0 \u09ac\u09c7\u0997 <\/span>y<\/span><\/span> \u09b9\u09df \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span>\\frac{d s}{d t}=v<\/span><\/span>\u00a0<\/span> \u09b9\u09df, \u09a4\u09ac\u09c7 <\/span>(2)<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>v=u-g t \\quad ... \\quad ...(3)<\/span><\/span>
\n<\/span><\/p>\n

\u098f\u0996\u09a8 <\/span>(t)<\/span><\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>(2)<\/span> \u0995\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>s=u t-\\frac{1}{2} g t^{2}+B \\quad ... \\quad ...(4)<\/span><\/span><\/p>\n

\u0986\u09a6\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09af\u0996\u09a8 <\/span>t=0<\/span>,<\/span> \u09a4\u0996\u09a8 <\/span>s=0<\/span><\/span> \u09ab\u09b2\u09c7 <\/span>B=0<\/span><\/span><\/p>\n

\u2234(4)<\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>s=u t-\\frac{1}{2} g t^{2} \\quad ... \\quad ...(5)<\/span><\/span><\/p>\n

(5)<\/span> \u098f <\/span>s<\/span><\/span> \u098f\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u0989\u099a\u09cd\u099a\u09a4\u09be\u0995\u09c7 <\/span>h<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>h=u t-\\frac{1}{2} g t^{2} \\quad ... \\quad ...(6)<\/span><\/span><\/p>\n

\u098f\u0996\u09a8 \u09a4\u09cd\u09ac\u09b0\u09a3 <\/span>\\frac{d^{2} s}{d t^{2}}=-g \\Rightarrow \\frac{d}{d t}\\left(\\frac{d s}{d t}\\right)=-g \\Rightarrow \\frac{d v}{d t}=-g, \\quad\\left[\\because \\frac{d s}{d t}=v\\right]<\/span><\/span><\/p>\n

\\Rightarrow \\frac{d v}{d s} \\cdot \\frac{d s}{d t}=-g \\Rightarrow \\frac{d v}{d s} \\cdot v=-g \\Rightarrow v d v=-g d s \\quad ... \\quad ...(7)<\/span><\/span><\/p>\n

\u09af\u0996\u09a8 <\/span>s=0<\/span><\/span>, \u09a4\u0996\u09a8 <\/span>v=u<\/span><\/span> \u098f\u09ac\u0982 \u09af\u0996\u09a8 <\/span>s=s<\/span><\/span> \u09a4\u0996\u09a8 <\/span>v=v<\/span><\/span> \u0964 \u098f \u09b8\u09c0\u09ae\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>(7)<\/span> \u098f\u09b0 \u0989\u09ad\u09df \u09aa\u0995\u09cd\u09b7\u0995\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,\u00a0\u00a0<\/span><\/p>\n

\\int_{u}^{v} v d v=-g \\int_{0}^{s} d s<\/span><\/span><\/p>\n

\\Rightarrow\\left[\\frac{v^{2}}{2}\\right]_{u}^{v}=-g[s]_{0}^{s}<\/span><\/span><\/p>\n

\\Rightarrow \\frac{v^{2}}{2}-\\frac{u^{2}}{2}=-g(s-0)<\/span><\/span><\/p>\n

\\therefore v^{2}=u^{2}-2 g s \\quad ... \\quad ...(8)<\/span><\/span><\/p>\n

\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/span>(8)<\/span> \u098f <\/span>s<\/span><\/span> \u098f\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u0989\u099a\u09cd\u099a\u09a4\u09be\u0995\u09c7 <\/span>h<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>v^{2}=u^{2}-2 g h \\quad ... \\quad ...(9)<\/span><\/span>
\n<\/span><\/p>\n

h<\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be \u09b9\u09a4\u09c7 \u0985\u09ac\u09be\u09a7\u09c7 \u09aa\u09a4\u09a8\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0997\u09a4\u09bf:<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be \u09a5\u09c7\u0995\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1 \u09ae\u09be\u09a7\u09cd\u09af\u09be\u0995\u09b0\u09cd\u09b7\u09a3\u099c\u09a8\u09bf\u09a4 \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u09aa\u09cd\u09b0\u09ad\u09be\u09ac\u09c7 \u0985\u09ac\u09be\u09a7\u09c7 \u09aa\u09a4\u09bf\u09a4 \u09b9\u09df\u09c7 <\/span>t<\/span><\/span> \u09b8\u09ae\u09df\u09c7 \u09ad\u09c2\u09ae\u09bf\u09a4\u09c7 <\/span>v<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u0986\u0998\u09be\u09a4 \u0995\u09b0\u09c7\u0964 \u0985\u09ac\u09be\u09a7\u09c7 \u09aa\u09a4\u09a8\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u0986\u09a6\u09bf\u09ac\u09c7\u0997 <\/span>u=0<\/span><\/span><\/p>\n

\\therefore h=0 \\times t+\\frac{1}{2} g t^{2} \\Rightarrow t^{2}=\\frac{2 h}{g} \\Rightarrow t=\\sqrt{\\frac{2 h}{g}} \\ldots .(i)<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 <\/span>v^{2}=0+2 g h \\quad \\therefore v=\\sqrt{2 g h} \\ldots \\ldots (ii)<\/span><\/span>;<\/span> \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09a8\u09bf\u099a\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u0997\u09a4\u09bf\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09ac\u09c7\u0997 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995\u0964\u00a0<\/span><\/p>\n

(i)<\/span><\/span> \u0993 <\/span>(ii)<\/span><\/span> \u09a5\u09c7\u0995\u09c7 \u09ac\u09b2\u09be \u09af\u09be\u09df, <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be \u09b9\u09a4\u09c7 \u0985\u09ac\u09be\u09a7 \u09aa\u09a4\u09a8\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1 <\/span>\\sqrt{2 h \/ g}<\/span><\/span> \u09b8\u09ae\u09df \u09aa\u09b0\u09c7 <\/span>\\sqrt{2 g h}<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u09ad\u09c2\u09ae\u09bf\u09a4\u09c7 \u0986\u0998\u09be\u09a4 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af:<\/b> \u09aa\u09c2\u09b0\u09cd\u09ac\u09c7\u09b0 \u0985\u09a8\u09c1\u099a\u09cd\u099b\u09c7\u09a6\u0997\u09c1\u09b2\u09cb\u09a4\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09c7\u09af\u09bc\u09c7\u099b\u09bf, <\/span>v=u+f t, s=u t+\\frac{1}{2} f t^{2}, v^{2}=u^{2}+2 f s<\/span><\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7 <\/span>f=-g<\/span><\/span> \u0993 <\/span>s=h<\/span><\/span> \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>v=u-g t, h=u t-\\frac{1}{2} g t^{2}, v^{2}=u^{2}-2 g h<\/span><\/span>\u0964\u00a0<\/span><\/p>\n

u<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u0996\u09be\u09a1\u09bc\u09be \u09a8\u09bf\u09ae\u09cd\u09a8 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0997\u09a4\u09bf\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, \u09a4\u09cd\u09ac\u09b0\u09a3 <\/span>=\\frac{d^{2} s}{d t^{2}}=g<\/span><\/span><\/p>\n

\u0989\u09aa\u09b0\u09bf\u0989\u0995\u09cd\u09a4 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09af\u09be\u09ac\u09c7, <\/span>v=u+g t, h=u t+\\frac{1}{2} g t^{2}, v^{2}=u^{2}+2 g h<\/span><\/span>.<\/span><\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7: \u098f\u0995\u099f\u09bf \u09aa\u09be\u09a5\u09b0 \u0995\u09c1\u09af\u09bc\u09be\u09b0 \u09ad\u09bf\u09a4\u09b0 \u09ab\u09c7\u09b2\u09be\u09b0 <\/b>t<\/span><\/span> \u09b8\u09ae\u09af\u09bc \u09aa\u09b0\u09c7 \u09aa\u09be\u09a8\u09bf\u09a4\u09c7 \u098f\u09b0 \u09aa\u09a4\u09a8 \u09b6\u09ac\u09cd\u09a6 \u09b6\u09cb\u09a8\u09be \u0997\u09c7\u09b2\u0964 \u09b6\u09ac\u09cd\u09a6\u09c7\u09b0 \u09ac\u09c7\u0997 <\/b>v<\/span><\/span> \u098f\u09ac\u0982 \u0995\u09c1\u09af\u09bc\u09be\u09b0 \u0997\u09ad\u09c0\u09b0\u09a4\u09be <\/b>h<\/span><\/span> \u09b9\u09b2\u09c7, \u09ac\u09be\u09a4\u09be\u09b8\u09c7\u09b0 \u09ac\u09be\u09a7\u09be \u0985\u0997\u09cd\u09b0\u09be\u09b9\u09cd\u09af \u0995\u09b0\u09c7 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0 \u09af\u09c7,<\/b><\/p>\n

t=\\sqrt{\\frac{2 h}{g}}+\\frac{h}{v}<\/span><\/span>
\n<\/span><\/p>\n

g v^{2} t^{2}-2 g h v+h\\left(g h-2 v^{2}\\right)=0<\/span><\/span><\/p>\n

g t^{2}=2 h\\left(1+\\frac{g t}{v}\\right)<\/span><\/span> ,<\/span> \u09af\u0996\u09a8v>h<\/span><\/span><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8: <\/b>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09be\u09a5\u09b0\u099f\u09bf <\/span>t_{1}<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09c1\u09af\u09bc\u09be\u09b0 \u09aa\u09be\u09a8\u09bf\u09a4\u09c7 \u09aa\u09a4\u09bf\u09a4 \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u09b8\u09c7\u0996\u09be\u09a8 \u09a5\u09c7\u0995\u09c7 \u09aa\u09a4\u09a8 \u09b6\u09ac\u09cd\u09a6 \u0995\u09c1\u09df\u09be\u09b0 \u0989\u09aa\u09b0\u09bf\u09ad\u09be\u0997\u09c7 \u0986\u09b8\u09a4\u09c7 <\/span>t_{2}<\/span><\/span> \u09b8\u09ae\u09af\u09bc \u09b2\u09be\u0997\u09c7\u0964 \u09a4\u09be\u09b9\u09b2\u09c7, <\/span>t=t_{1}+t_{2} \\quad ... \\quad ...(i)<\/span><\/span><\/p>\n

\u09aa\u09be\u09a5\u09b0 \u09aa\u09a4\u09a8\u09c7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, <\/span>h=0 \\times t_{1}+\\frac{1}{2} g t_{1}^{2} \\Rightarrow g t_{1}^{2}=2 h \\Rightarrow t_{1}=\\sqrt{\\frac{2 h}{g}}<\/span><\/span><\/p>\n

\"\u0995\u09c1\u09af\u09bc\u09be\"\u09b6\u09ac\u09cd\u09a6\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, <\/span>h=v t_{2} \\Rightarrow t_{2}=\\frac{h}{v}<\/span><\/span><\/p>\n

(i)<\/span> \u09a8\u0982 \u098f <\/span>t_{1}<\/span><\/span> \u0993 <\/span>t_{2}<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987, <\/span>t=\\sqrt{\\frac{2 h}{g}}+\\frac{h}{v}<\/span><\/span> (Proved)<\/span><\/p>\n

\\Rightarrow t-\\frac{h}{v}=\\sqrt{\\frac{2 h}{g}} \\Rightarrow t^{2}-\\frac{2 h t}{v}+\\frac{h^{2}}{v^{2}}=\\frac{2 h}{g} \\quad ... \\quad ...(i i)<\/span><\/span> [\u09ac\u09b0\u09cd\u0997 \u0995\u09b0\u09c7]<\/span><\/p>\n

\\therefore g v^{2} t^{2}-2 g h v t+g h^{2}=2 h v^{2}<\/span><\/span> (Proved)<\/span><\/p>\n

\u09b6\u09ac\u09cd\u09a6\u09c7\u09b0 \u09ac\u09c7\u0997 <\/span>v>h<\/span><\/span> \u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 <\/span>\\frac{h^{2}}{v^{2}}<\/span><\/span> \u0995\u09c7 \u0985\u09a4\u09bf \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0 \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09c7 \u0985\u0997\u09cd\u09b0\u09be\u09b9\u09cd\u09af \u0995\u09b0\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n

\\therefore t^{2}-\\frac{2 h t}{v}=\\frac{2 h}{g} \\Rightarrow g t^{2}-\\frac{2 g h t}{v}=2 h \\Rightarrow g t^{2}=2 h\\left(1+\\frac{g t}{v}\\right)<\/span><\/span> (Proved)<\/span>\u00a0<\/b><\/p>\n

\u09ad\u09c2\u09ae\u09bf\u09a4\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u09aa\u09a4\u09a8\u09ac\u09c7\u0997:<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ad\u09c2\u09ae\u09bf \u09b9\u09a4\u09c7 <\/span>u<\/span><\/span> \u0986\u09a6\u09bf \u09ac\u09c7\u0997\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0995\u09b0\u09be \u09b9\u09b2\u0964 \u09b8\u09b0\u09cd\u09ac\u09be\u09a7\u09bf\u0995 <\/span>H<\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09af\u09bc \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf\u09b0 \u09ac\u09c7\u0997 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u0985\u09a4\u0983\u09aa\u09b0 \u09a4\u09be\u09ce\u0995\u09cd\u09b7\u09a3\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u099a\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09aa\u09a1\u09bc\u09a4\u09c7 \u09a5\u09be\u0995\u09c7\u0964 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf <\/span>v<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u09ad\u09c2\u09ae\u09bf\u0995\u09c7 \u0986\u0998\u09be\u09a4 \u0995\u09b0\u09b2\u09c7,<\/span><\/p>\n

v^{2}=u^{2}+2 g h<\/span><\/span> \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>v^{2}=0+2 g H=2 g \\times \\frac{u^{2}}{2 g}=u^{2}<\/span><\/span><\/p>\n

\\Rightarrow|v|=u<\/span><\/span><\/p>\n

\u2234<\/span> \u0989\u09a4\u09cd\u09a5\u09be\u09a8 \u09ac\u09c7\u0997 <\/b>=<\/span> |\u09aa\u09a4\u09a8 \u09ac\u09c7\u0997|<\/b><\/p>\n

\u09ad\u09c2\u09ae\u09bf\u09b0 \u0989\u09b0\u09cd\u09a7\u09cd\u09ac\u09c7 <\/b>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be \u09a5\u09c7\u0995\u09c7 <\/b>u<\/span><\/b><\/span> \u09ac\u09c7\u0997\u09c7 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0989\u09aa\u09b0\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be <\/b>t<\/span><\/b><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 <\/b>v<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u09aa\u09a4\u09bf\u09a4 \u09b9\u09b2\u09c7, <\/b>(i) \\quad v=-u+g t<\/span><\/span> (ii) \\quad h=-u t+\\frac{1}{2} g t^{2}<\/span><\/b><\/span><\/b><\/p>\n

\"\u09ad\u09c2\u09ae\u09bf\u09a4\u09c7<\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ad\u09c2\u09ae\u09bf\u09b0 \u098a\u09b0\u09cd\u09a7\u09cd\u09ac\u09c7 <\/span>AB=h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09af\u09bc <\/span>B<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>u<\/span><\/span> \u0986\u09a6\u09bf \u09ac\u09c7\u0997\u09c7 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf <\/span>t_{1}<\/span><\/span> \u09b8\u09ae\u09af\u09bc \u09aa\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 <\/span>C<\/span> \u09a4\u09c7 \u09aa\u09cc\u0981\u099b\u09c7 \u098f\u09ac\u0982 \u09ac\u09c7\u0997 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09df\u0964<\/span><\/p>\n

\\therefore 0=u-g t_{1} \\Rightarrow t_{1}=\\frac{u}{g}<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 <\/span>B C=\\frac{u^{2}}{2 g}<\/span><\/span> [<\/span>\u2235<\/span> \u09b8\u09b0\u09cd\u09ac\u09be\u09a7\u09bf\u0995 \u0989\u099a\u09cd\u099a\u09a4\u09be <\/span>=\\frac{u^{2}}{2 g}<\/span><\/span>]\u00a0<\/span><\/p>\n

\u09a7\u09b0\u09bf, \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf <\/span>B<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>t<\/span><\/span> \u09b8\u09ae\u09af\u09bc \u09aa\u09b0 \u09ad\u09c2\u09ae\u09bf\u09b0 <\/span>A<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0986\u0998\u09be\u09a4 \u0995\u09b0\u09c7\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf <\/span>C<\/span> \u09b9\u09a4\u09c7 \u09b6\u09c2\u09a8\u09cd\u09af \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 <\/span>(t-t_{1})<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 <\/span>CA<\/span> \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\\therefore A C=0+\\frac{1}{2} g\\left(t-t_{1}\\right)^{2}=\\frac{1}{2} g\\left(t-\\frac{u}{g}\\right)^{2}=\\frac{1}{2} g\\left(t^{2}-2 \\frac{u}{g} t+\\frac{u^{2}}{g^{2}}\\right)=\\frac{1}{2} g t^{2}-u t+\\frac{u^{2}}{2 g}<\/span><\/span><\/p>\n

\\therefore h=-u t+\\frac{1}{2} g t^{2}<\/span><\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf <\/span>C<\/span> \u09b9\u09a4\u09c7 \u09b6\u09c2\u09a8\u09cd\u09af \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 <\/span>(t-t_{1})<\/span><\/span> \u09b8\u09ae\u09df \u09aa\u09b0 \u09ad\u09c2\u09ae\u09bf\u0995\u09c7 <\/span>v<\/span><\/span> \u09ac\u09c7\u0997\u09c7 \u0986\u0998\u09be\u09a4 \u0995\u09b0\u09b2\u09c7,\u00a0<\/span><\/p>\n

v=0+g\\left(t-t_{1}\\right)=g\\left(t-\\frac{u}{g}\\right)=g t-u<\/span><\/span>
\n<\/span><\/p>\n

\\therefore v=-u+g t<\/span><\/span>
\n<\/span><\/p>\n

\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0995\u09cb\u09a8\u09cb \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997 \u0993 \u09b8\u09ae\u09df (Time and velocity of a particle to a given height)\u00a0\u00a0<\/b><\/p>\n

O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>u<\/span><\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df <\/span>v<\/span><\/span> \u09ac\u09c7\u0997 \u0985\u09b0\u09cd\u099c\u09a8 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\\therefore v^{2}=u^{2}-2 g h<\/span><\/span> \u09ac\u09be, <\/span>v=\\pm \\sqrt{u^{2}-2 g h}<\/span><\/span><\/p>\n

\\therefore h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df \u0997\u09a4\u09bf\u09ac\u09c7\u0997 <\/span>=\\pm \\sqrt{u^{2}-2 g h}<\/span><\/span>
\n<\/span><\/p>\n

\\pm<\/span> \u099a\u09bf\u09b9\u09cd\u09a8\u09c7\u09b0 \u09a6\u09cd\u09ac\u09be\u09b0\u09be <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df \u0995\u09cb\u09a8 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09ae\u09ae\u09be\u09a8\u09c7\u09b0 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09ae\u09c1\u0996\u09c0 \u09ac\u09c7\u0997 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0989\u09a0\u09a8\u09cd\u09a4 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09ac\u09c7\u0997 <\/span>\\sqrt{u^{2}-2 g h}<\/span><\/span> \u098f\u09ac\u0982 \u09aa\u09dc\u09a8\u09cd\u09a4 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09ac\u09c7\u0997 <\/span>\u00a0-\\sqrt{u^{2}-2 g h}<\/span> \u09ac\u09c1\u099d\u09be\u09df\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>u<\/span><\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u0996\u09be\u09dc\u09be\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09b9\u09b2\u09cb \u098f\u09ac\u0982 \u0989\u09b9\u09be <\/span>g<\/span><\/span> \u09ae\u09a8\u09cd\u09a6\u09a8\u09c7 <\/span>t<\/span><\/span> \u09b8\u09ae\u09df\u09c7 <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df \u09aa\u09cc\u0981\u099b\u09c7\u0964<\/span><\/p>\n

h=u t-\\frac{1}{2} g t^{2}<\/span><\/span> \u09ac\u09be, <\/span>2 h=2 u t-g t^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>g t^{2}-2 u t+2 h=0<\/span><\/span> \u09ac\u09be, <\/span>t=\\frac{2 u \\pm \\sqrt{(-2 u)^{2}-4 . g .2 h}}{2 . g}<\/span><\/span><\/p>\n

=\\frac{2 u \\pm \\sqrt{4\\left(u^{2}-2 g h\\right)}}{2 . g}<\/span><\/span><\/p>\n

=\\frac{u \\pm \\sqrt{u^{2}-2 g h}}{g}<\/span><\/span><\/p>\n

=\\frac{u}{g} \\pm \\frac{\\sqrt{u^{2}-2 g h}}{g}<\/span><\/span><\/p>\n

u^{2}-2 g h>0<\/span><\/span> \u09b9\u09b2\u09c7 <\/span>t<\/span><\/span> \u098f\u09b0 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09a6\u09c1\u099f\u09bf \u09ae\u09be\u09a8 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09ac\u09c7\u0964<\/span><\/p>\n

\u2234<\/span> \u0996\u09be\u09dc\u09be\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1 \u09a6\u09c1\u0987\u09ac\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

t<\/span><\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09ae\u09be\u09a8 <\/span>\\frac{u}{g}-\\frac{\\sqrt{u^{2}-2 g h}}{g}<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf \u0989\u09a0\u09a8\u09cd\u09a4 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df <\/span>t<\/span><\/span> \u098f\u09b0 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae <\/span>\\frac{u}{g}+\\frac{\\sqrt{u^{2}-2 g h}}{g}<\/span> <\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf \u09aa\u09dc\u09a8\u09cd\u09a4 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df <\/span>h<\/span><\/span> \u0989\u099a\u09cd\u099a\u09a4\u09be\u09df \u0997\u09ae\u09a8 \u0995\u09be\u09b2\u0995\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7<\/span><\/p>\n

\u09ac\u09bf\u09b6\u09c7\u09b7 \u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af: <\/b>\u098f\u0995\u099f\u09bf \u09b6\u09c2\u09a8\u09cd\u09af \u0995\u09c2\u09aa\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u098f\u0995\u099f\u09bf \u09a2\u09bf\u09b2 \u09ab\u09c7\u09b2\u09be\u09b0 <\/span>t<\/span><\/span> \u09b8\u09c7. \u09aa\u09b0\u09c7 \u0995\u09c2\u09aa\u09c7\u09b0 \u09a4\u09b2\u09a6\u09c7\u09b6\u09c7 \u09a2\u09bf\u09b2 \u09aa\u09dc\u09be\u09b0 \u09b6\u09ac\u09cd\u09a6 \u09b6\u09cb\u09a8\u09be \u0997\u09c7\u09b2\u0964 \u09b6\u09ac\u09cd\u09a6\u09c7\u09b0 \u09ac\u09c7\u0997 <\/span>v<\/span><\/span> \u09b9\u09b2\u09c7 \u0995\u09c2\u09aa\u09c7\u09b0 \u0997\u09ad\u09c0\u09b0\u09a4\u09be, <\/span>h=\\frac{g t^{2}}{2\\left(1+\\frac{g t}{v}\\right)}<\/span><\/span> (\u09ac\u09b9\u09c1\u09a8\u09bf\u09b0\u09cd\u09ac\u09be\u099a\u09a8\u09c0\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09aa\u09cd\u09b0\u09af\u09cb\u099c\u09cd\u09af)<\/span><\/p>\n

\u09ac\u09be\u09df\u09c1\u09b6\u09c2\u09a8\u09cd\u09af \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u09a4\u09b2\u09c7 \u09aa\u09cd\u09b0\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09af\u09be\u09b0\u09be\u09ac\u09cb\u09b2\u09be (The path of a projectile in vacuous is a parabola)<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>u<\/span><\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09c7\u09b0 \u09b8\u0999\u09cd\u0997\u09c7 <\/span> \\alpha<\/span> \u0995\u09cb\u09a3\u09c7 \u09aa\u09cd\u09b0\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09b9\u09b2\u09cb\u0964 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u0993 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac <\/span>OX<\/span> \u0993 <\/span>OY<\/span> \u09b0\u09c7\u0996\u09be\u09a6\u09cd\u09ac\u09df\u0995\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>x<\/span><\/span>-\u0985\u0995\u09cd\u09b7 \u0993 <\/span>y<\/span><\/span>-\u0985\u0995\u09cd\u09b7 \u09a7\u09b0\u09b2\u09c7 <\/span>t<\/span><\/span> \u09b8\u09ae\u09df\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf <\/span>P(x,y)<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09c7\u0964 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 <\/span>u<\/span><\/span> \u098f\u09b0 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 <\/span>u \\cos \\alpha<\/span><\/span> \u098f\u09ac\u0982 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 <\/span>u \\sin \\alpha<\/span><\/span> \u0964 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3 \u09b6\u09c2\u09a8\u09cd\u09af\u0964<\/span><\/p>\n

\"\u09aa\u09cd\u09b0\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4<\/p>\n

t<\/span><\/span> \u09b8\u09ae\u09df\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09b8\u09b0\u09a3 <\/span>x=u \\cos \\alpha \\cdot t \\quad \\therefore t=\\frac{x}{u \\cos \\alpha}<\/span><\/span>
\n<\/span><\/p>\n

t<\/span><\/span> \u09b8\u09ae\u09df\u09c7 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u09b8\u09b0\u09a3 <\/span>y=u \\sin \\alpha \\cdot t-\\frac{1}{2} g t^{2} \\quad ... \\quad ...(i)<\/span><\/span><\/p>\n

(i)<\/span> \u098f <\/span>t<\/span> <\/span>\u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987,\u00a0<\/span><\/p>\n

y=u \\sin \\alpha \\cdot \\frac{x}{u \\cos \\alpha}-\\frac{1}{2} g \\frac{x^{2}}{u^{2} \\cos ^{2} \\alpha}=x \\tan \\alpha-\\frac{g x^{2}}{2 u^{2} \\cos ^{2} \\alpha} \\quad ... \\quad ...(ii)<\/span><\/span><\/p>\n

u,\\alpha,g<\/span><\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09ac\u09b2\u09c7 <\/span>a=-\\frac{g}{2 u^{2} \\cos ^{2} a}<\/span><\/span> \u098f\u09ac\u0982 <\/span>b=\\tan \\alpha<\/span><\/span> \u09a7\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>y=a x^{2}+b x<\/span><\/span>,<\/span> \u09af\u09be <\/span>t<\/span><\/span> \u09ae\u09c1\u0995\u09cd\u09a4\u0964<\/span><\/p>\n

\u2234<\/span> \u09ac\u09be\u09df\u09c1\u09b6\u09c2\u09a8\u09cd\u09af \u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09a4\u09bf\u09b0\u09cd\u09af\u0995\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5 \u09aa\u09cd\u09af\u09be\u09b0\u09be\u09ac\u09cb\u09b2\u09be\u0964<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4: <\/b>\u09ac\u09be\u09df\u09c1\u09b9\u09c0\u09a8 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u0985\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span>\\alpha<\/span> \u0995\u09cb\u09a3\u09c7 \u09b6\u09c2\u09a8\u09cd\u09af\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09aa\u09be\u09b2\u09cd\u09b2\u09be <\/span>R<\/span> \u09b9\u09b2\u09c7, \u098f\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, <\/span>y=x \\tan \\alpha\\left(1-\\frac{x}{R}\\right)<\/span><\/span><\/p>\n

(ii)<\/span><\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 \u09aa\u09be\u0987,\u00a0<\/span><\/p>\n

y=x \\tan \\alpha-\\frac{g x^{2}}{2 u^{2} \\cos ^{2} \\alpha}=x \\tan \\alpha-\\frac{g x^{2} \\tan \\alpha}{2 u^{2} \\sin a \\cos \\alpha}<\/span><\/span><\/p>\n

=x \\tan \\alpha\\left(1-\\frac{g x}{u^{2} \\sin 2 \\alpha}\\right)=x \\tan \\alpha\\left(1-\\frac{x}{\\frac{u^{2} \\sin 2 \\alpha}{g}}\\right)=x \\tan \\alpha\\left(1-\\frac{x}{R}\\right)<\/span><\/span><\/p>\n

\u09ad\u09c2\u09ae\u09bf \u09b9\u09a4\u09c7 \u0989\u099a\u09cd\u099a \u0995\u09cb\u09a8\u09cb \u09b8\u09cd\u09a5\u09be\u09a8 \u09a5\u09c7\u0995\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09af\u09be\u09b0\u09be\u09ac\u09cb\u09b2\u09be:<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u0995\u09c7 \u09ac\u09be\u09af\u09bc\u09c1\u09b6\u09c2\u09a8\u09cd\u09af \u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0989\u099a\u09cd\u099a\u09a4\u09be\u09af\u09bc <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>u<\/span><\/span> \u0986\u09a6\u09bf\u09ac\u09c7\u0997\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0995\u09b0\u09be \u09b9\u09b2\u09cb\u0964 <\/span>OX<\/span> \u0995\u09c7 <\/span>x<\/span><\/span>-\u0985\u0995\u09cd\u09b7 \u0993 <\/span>OY<\/span> \u0995\u09c7 <\/span>y<\/span><\/span>-\u0985\u0995\u09cd\u09b7 \u09a7\u09b0\u09b2\u09c7 <\/span>t<\/span> <\/span>\u09b8\u09ae\u09af\u09bc\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf <\/span>P(x,-y)<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09c7\u0964 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 <\/span>u<\/span><\/span> \u098f\u09b0 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 <\/span>u \\cos 0^{\\circ}<\/span><\/span>\u09ac\u09be <\/span>u<\/span><\/span> \u098f\u09ac\u0982 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac <\/span>u \\sin 0^{\\circ}<\/span><\/span> \u09ac\u09be <\/span>0<\/span>\u0964 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09a6\u09bf\u0995\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3 \u09b6\u09c2\u09a8\u09cd\u09af\u0964<\/span><\/p>\n

\"\u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995t<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09b8\u09b0\u09a3 <\/span>x=u \\cos 0^{\\circ} \\cdot t=u t<\/span><\/span><\/p>\n

\\therefore t=\\frac{x}{u} \\ldots \\ldots(i)<\/span><\/span><\/p>\n

t<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \u0989\u09b2\u09cd\u09b2\u09ae\u09cd\u09ac \u09b8\u09b0\u09a3 <\/span>-y=+u \\sin 0^{\\circ} t-\\frac{1}{2} g t^{2}<\/span><\/span> \u09ac\u09be, <\/span>-y=0-\\frac{1}{2} g t^{2}<\/span><\/span> \u09ac\u09be, <\/span>y=\\frac{1}{2} g t^{2}<\/span><\/span><\/p>\n

(i)<\/span><\/span> \u09a8\u0982 \u09b9\u09a4\u09c7 <\/span>t<\/span><\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09af\u09bc\u09c7, <\/span>y=\\frac{1}{2} g \\cdot \\frac{x^{2}}{u^{2}}<\/span><\/span> \u09ac\u09be, <\/span>x^{2}=\\frac{2 u^{2}}{g} y<\/span><\/span>
\n<\/span><\/p>\n

u<\/span><\/span> \u098f\u09ac\u0982 <\/span>g<\/span><\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09ac\u09b2\u09c7 <\/span>\\frac{2 u^{2}}{g}=4 a<\/span><\/span> \u09a7\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>x^{2}=4 a y<\/span><\/span>,<\/span> \u09af\u09be \u09aa\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0964\u00a0<\/span><\/p>\n

\u2234<\/span> \u09ad\u09c2\u09ae\u09bf \u09b9\u09a4\u09c7 \u0989\u099a\u09cd\u099a \u0995\u09cb\u09a8\u09cb \u09b8\u09cd\u09a5\u09be\u09a8 \u09b9\u09a4\u09c7 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09af\u09be\u09b0\u09be\u09ac\u09cb\u09b2\u09be\u0964<\/span><\/p>\n

\u09ac\u09bf.\u09a6\u09cd\u09b0. <\/b>\u09ac\u09b9\u09c1\u09a8\u09bf\u09b0\u09cd\u09ac\u09be\u099a\u09a8\u09bf \u09aa\u09cd\u09b0\u09b6\u09cd\u09a8 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0995\u09bf\u099b\u09c1 \u09ac\u09bf\u09b6\u09c7\u09b7 \u0995\u09cc\u09b6\u09b2 \u09a8\u09bf\u09ae\u09cd\u09a8\u09c7 \u09a6\u09c7\u0993\u09df\u09be \u09b9\u09b2\u09cb:\u00a0<\/span><\/p>\n