{"id":3849,"date":"2021-01-30T12:30:08","date_gmt":"2021-01-30T12:30:08","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3849"},"modified":"2022-04-04T03:26:35","modified_gmt":"2022-04-03T21:26:35","slug":"motion-of-particles-in-plane","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/motion-of-particles-in-plane\/","title":{"rendered":"\u09b8\u09ae\u09a4\u09b2\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0997\u09a4\u09bf | Motion of Particles in Plane"},"content":{"rendered":"

\u09ac\u09b2\u09ac\u09bf\u09a6\u09cd\u09af\u09be\u09b0 \u09af\u09c7 \u09b6\u09be\u0996\u09be\u09af\u09bc \u0997\u09a4\u09bf\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0993\u09aa\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09ac\u09b2\u0997\u09c1\u09b2\u09bf\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af \u09a8\u09bf\u09af\u09bc\u09c7 \u0986\u09b2\u09cb\u099a\u09a8\u09be \u0995\u09b0\u09be \u09b9\u09af\u09bc \u09a4\u09be\u0987 \u0997\u09a4\u09bf\u09ac\u09bf\u09a6\u09cd\u09af\u09be\u0964 \u098f\u0987 \u0985\u09a7\u09cd\u09af\u09be\u09af\u09bc\u0995\u09c7 \u099a\u09be\u09b0 \u09ad\u09be\u0997\u09c7 \u09ad\u09be\u0997 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964<\/span><\/p>\n

\u09b8\u09b0\u09a3 (Displacement)<\/b>:<\/span><\/h2>\n

\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u0995\u09c7 \u0990 \u09a6\u09bf\u0995\u09c7 \u0990 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u09b8\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u0993 \u09b6\u09c7\u09b7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u09af\u09c7 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b0\u09c8\u0996\u09bf\u0995 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09a4\u09be\u0987 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf\u09b0 \u09b8\u09b0\u09a3\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09aa\u0964<\/span><\/p>\n

\u09af\u09a6\u09bf \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1 \u09b8\u09b0\u09b2 \u0985\u09a5\u09ac\u09be \u09ac\u0995\u09cd\u09b0\u09aa\u09a5\u09c7 \u0995\u09cb\u09a8\u09cb \u098f\u0995 \u09b8\u09ae\u09af\u09bc\u09c7 <\/span>A<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 <\/span>B<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09af\u09be\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 \u09b8\u09a6\u09bf\u0995 \u09b0\u09c7\u0996\u09be\u0982\u09b6 <\/span>\\overrightarrow{A B}<\/span><\/span> \u098f\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09b9\u099a\u09cd\u099b\u09c7 \u09b8\u09c7\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u099f\u09bf\u09b0 \u09b8\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a8, \u098f\u09ac\u0982 \u098f\u09b0 \u09a6\u09bf\u0995 <\/span>A<\/span> \u09a5\u09c7\u0995\u09c7 <\/span>B<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7\u0964<\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 <\/span>\\overrightarrow{A B}<\/span><\/span> \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964 <\/span>M.K.S.\u00a0,\u00a0C.G.S.<\/span> \u098f\u09ac\u0982 <\/span>F.P.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09b0\u09a3\u09c7\u09b0 \u098f\u0995\u0995 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 \u09ae\u09bf\u099f\u09be\u09b0, \u09b8\u09c7\u09a8\u09cd\u099f\u09bf\u09ae\u09bf\u099f\u09be\u09b0 \u098f\u09ac\u0982 \u09ab\u09c1\u099f\u0964\u00a0<\/span><\/p>\n

\"\u09b8\u09b0\u09a3\"<\/p>\n

\u09ac\u09c7\u0997 (Velocity)<\/b><\/h2>\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09b8\u09b0\u09a3\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7\u09b0 \u09b9\u09be\u09b0\u0995\u09c7 \u09ac\u09c7\u0997 \u09ac\u09b2\u09c7\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09af\u09bc\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a6\u09bf\u0995\u09c7 \u099a\u09b2\u09ae\u09be\u09a8 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09c7\u09b0 \u09b9\u09be\u09b0\u0987 \u09b9\u09b2\u09cb \u09ac\u09c7\u0997\u0964 \u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1 \u0995\u09a3\u09be\u09b0 <\/span>t<\/span> \u09b8\u09ae\u09af\u09bc\u09c7 <\/span>s<\/span> \u09b8\u09b0\u09a3 \u09b9\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09c7\u0997, <\/span>v=\\frac{s}{t}<\/span><\/span>, \u09af\u09a6\u09bf \u0997\u09a4\u09bf\u09b6\u09c0\u09b2 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09c7 \u09a4\u09be\u09b9\u09b2\u09c7 \u09b8\u09c7\u0987 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997\u0995\u09c7 \u09b8\u09c1\u09b7\u09ae \u09ac\u09c7\u0997 \u09ac\u09be \u09b8\u09ae\u09ac\u09c7\u0997, \u0986\u09b0 \u09af\u09a6\u09bf \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09ac\u09be \u0989\u09ad\u09af\u09bc\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0998\u099f\u09c7 \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09b8\u09c7\u0987 \u09ac\u09c7\u0997\u0995\u09c7 \u0985\u09b8\u09ae \u09ac\u09c7\u0997 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be \u09af\u09c7 \u09ac\u09c7\u0997 \u09a8\u09bf\u09af\u09bc\u09c7 \u09af\u09be\u09a4\u09cd\u09b0\u09be \u09b6\u09c1\u09b0\u09c1 \u0995\u09b0\u09c7 \u09a4\u09be\u0995\u09c7 \u0986\u09a6\u09bf\u09ac\u09c7\u0997 \u098f\u09ac\u0982 \u09b6\u09c7\u09b7 \u09ae\u09c1\u09b9\u09c2\u09b0\u09cd\u09a4\u09c7\u09b0 \u09ac\u09c7\u0997\u0995\u09c7 \u09b6\u09c7\u09b7 \u09ac\u09c7\u0997 \u09ac\u09b2\u09c7\u0964 \u0986\u09a6\u09bf \u09ac\u09c7\u0997\u0995\u09c7 <\/span>u<\/span><\/span> \u09ac\u09be <\/span>v_{0}<\/span><\/span> \u098f\u09ac\u0982 \u09b6\u09c7\u09b7 \u09ac\u09c7\u0997\u0995\u09c7 <\/span>v<\/span><\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0 <\/span>\\delta t<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5\u09c7 \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 <\/span>\\delta s<\/span><\/span> \u09b9\u09b2\u09c7,<\/span><\/p>\n

\u0990 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u09ac\u09c7\u0997 \u09ac\u09be \u09a4\u09be\u09ce\u0995\u09cd\u09b7\u09a3\u09bf\u0995 \u09ac\u09c7\u0997 <\/span>\\displaystyle{\\lim _{\\delta t \\rightarrow 0} \\frac{\\delta s}{\\delta t}=\\frac{d s}{d t}}<\/span><\/span>
\n<\/span><\/p>\n

\u09ac\u09c7\u0997 \u098f\u0995\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b0\u09be\u09b6\u09bf \u098f\u0995\u09c7 \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 <\/span>u,\u00a0v<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 <\/span>M.K.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u098f\u09b0 \u098f\u0995\u0995 \u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964<\/span><\/p>\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be <\/span>5<\/span> \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a6\u09bf\u0995\u09c7 <\/span>40<\/span> \u09ae\u09bf\u099f\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09b2\u09c7 \u09ac\u09be <\/span>40<\/span> \u09ae\u09bf\u099f\u09be\u09b0 \u09b8\u09b0\u09a3 \u09b9\u09b2\u09c7,<\/span><\/p>\n

\u09ac\u09c7\u0997 <\/span>=<\/span>\\frac{40}{5}<\/span> <\/span>\u09ae\u09bf\u099f\u09be\u09b0<\/span>\/<\/span>\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1<\/span>=8<\/span> \u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964\u00a0<\/span><\/p>\n

\u09ac\u09c7\u0997 \u09a6\u09c1\u0987 \u09aa\u09cd\u09b0\u0995\u09be\u09b0: <\/b>\u09af\u09a5\u09be:<\/span><\/p>\n

(i)<\/span> \u09b8\u09ae\u09ac\u09c7\u0997<\/strong><\/p>\n

(ii)<\/span> \u0985\u09b8\u09ae\u09ac\u09c7\u0997<\/strong><\/p>\n

\u09b8\u09ae\u09ac\u09c7\u0997 (Uniform Velocity):<\/b><\/h2>\n

<\/b>\u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09ae\u09af\u09bc\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09c7 \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u09ac\u09c7\u0997\u0995\u09c7 \u09b8\u09ae\u09ac\u09c7\u0997 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964\u00a0<\/span><\/p>\n

\u09b8\u09cd\u09a5\u09bf\u09b0\u09be\u09ac\u09b8\u09cd\u09a5\u09be \u09a5\u09c7\u0995\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf <\/span>1,\u00a02,\u00a03,\u00a04,\u00a05,\u00a06<\/span> \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>0.25,\u00a00.50,\u00a00.75,\u00a01.00,\u00a01.25,\u00a01.50<\/span> \u09ae\u09bf\u099f\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964 \u09b8\u09ae\u09af\u09bc \u09ac\u09a8\u09be\u09ae \u09b8\u09b0\u09a3 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0\u09c7 <\/span>7<\/span> \u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be <\/span>1<\/span> \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1 \u09aa\u09b0\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09ac\u09b0\u09be\u09ac\u09b0 \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u0997\u09a4\u09bf\u09b6\u09c0\u09b2 \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf \u09aa\u09cd\u09b0\u09a4\u09bf \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u09c7 <\/span>0.25<\/span> \u09ae\u09bf\u099f\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09b8\u09ae\u09be\u09a8 \u09b8\u09ae\u09af\u09bc\u09c7 \u09b8\u09ae\u09be\u09a8 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\u0995\u09be\u099c\u09c7\u0987 \u098f\u099f\u09bf \u09b8\u09ae\u09ac\u09c7\u0997 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u098f\u09ac\u0982 \u09b8\u09ae\u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 <\/span>0.25<\/span> \u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964<\/span><\/p>\n

\"\u09b8\u09ae\u09ac\u09c7\u0997\"<\/p>\n

\u0985\u09b8\u09ae\u09ac\u09c7\u0997 (Variable Velocity):<\/b><\/h2>\n

<\/b>\u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997 \u09af\u09a6\u09bf \u09b8\u09ae\u09af\u09bc\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 \u09a4\u09be\u0995\u09c7 \u0985\u09b8\u09ae\u09ac\u09c7\u0997 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09b8\u09cd\u09a5\u09bf\u09b0\u09be\u09ac\u09b8\u09cd\u09a5\u09be \u09a5\u09c7\u0995\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf \u09aa\u09cd\u09b0\u09a5\u09ae \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u09c7 <\/span>4<\/span> \u09ae\u09bf\u099f\u09be\u09b0, \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u09c7 <\/span>9<\/span> \u09ae\u09bf\u099f\u09be\u09b0, \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u09c7 <\/span>7<\/span> \u09ae\u09bf\u099f\u09be\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964 \u098f\u0996\u09be\u09a8\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf \u09b8\u09ae\u09be\u09a8 \u09b8\u09ae\u09af\u09bc\u09c7 \u09b8\u09ae\u09be\u09a8 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u099b\u09c7 \u09a8\u09be\u0964 \u0995\u09be\u099c\u09c7\u0987 \u098f\u0987 \u09ac\u09c7\u0997 \u09b9\u09ac\u09c7 \u0985\u09b8\u09ae \u09ac\u09c7\u0997\u0964 \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 \u0985\u09b8\u09ae \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0 \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964<\/span><\/p>\n

\"\u0985\u09b8\u09ae\u09ac\u09c7\u0997\"C.G.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u098f\u0995\u0995 \u09b8\u09c7\u09a8\u09cd\u099f\u09bf\u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964<\/span><\/p>\n

M.K.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u098f\u0995\u0995 \u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964<\/span><\/p>\n

F.P.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u098f\u0995\u0995 \u09ab\u09c1\u099f\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964<\/span><\/p>\n

\u09a4\u09cd\u09ac\u09b0\u09a3 (Acceleration)<\/b><\/h2>\n

\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997 \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf\u09b0 \u09b9\u09be\u09b0\u0995\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3 \u09ac\u09b2\u09c7\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cb\u09a8\u09cb \u099a\u09b2\u09ae\u09be\u09a8 \u09ac\u09b8\u09cd\u09a4\u09c1 \u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997 \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf\u09b0 \u09b9\u09be\u09b0\u0987 \u09b9\u09b2\u09cb \u09a4\u09cd\u09ac\u09b0\u09a3\u0964 \u09af\u09a6\u09bf \u0997\u09a4\u09bf\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09ac\u09c7\u0997 \u09b9\u09cd\u09b0\u09be\u09b8 \u09aa\u09be\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09c7\u0997 \u09b9\u09cd\u09b0\u09be\u09b8\u09c7\u09b0 \u09b9\u09be\u09b0\u0995\u09c7 \u09ae\u09a8\u09cd\u09a6\u09a8 \u09ac\u09b2\u09c7\u0964 \u0986\u09ac\u09be\u09b0 \u09af\u09a6\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09c7 \u09a4\u09be\u09b9\u09b2\u09c7 \u09b8\u09c7\u0987 \u09a4\u09cd\u09ac\u09b0\u09a3\u0995\u09c7 \u09b8\u09c1\u09b7\u09ae \u09a4\u09cd\u09ac\u09b0\u09a3 \u098f\u09ac\u0982 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09ac\u09be \u0989\u09ad\u09af\u09bc\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0998\u099f\u09c7 \u09a4\u09be\u09b9\u09b2\u09c7 \u09b8\u09c7\u0987 \u09a4\u09cd\u09ac\u09b0\u09a3\u0995\u09c7 \u0985\u09b8\u09ae \u09a4\u09cd\u09ac\u09b0\u09a3 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n

\u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0 <\/span>\\delta t<\/span><\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5\u09c7\u09b0 \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ac\u09c7\u0997 \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 <\/span>\\delta v<\/span><\/span> \u09b9\u09b2\u09c7 \u0990 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 \u09a4\u09cd\u09ac\u09b0\u09a3 \u09ac\u09be \u09a4\u09be\u09ce\u0995\u09cd\u09b7\u09a3\u09bf\u0995 \u09a4\u09cd\u09ac\u09b0\u09a3 <\/span>\\displaystyle{\\lim _{\\delta t \\rightarrow 0} \\frac{\\delta v}{\\delta t}=\\frac{d v}{d t}}<\/span><\/span>\u00a0<\/span> \u0964 \u09a4\u09cd\u09ac\u09b0\u09a3 \u098f\u0995\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b0\u09be\u09b6\u09bf \u098f\u0995\u09c7 \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 <\/span>f<\/span> \u09ac\u09be <\/span>a<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af: <\/b>\u09b8\u09c1\u09b7\u09ae \u09ac\u09c7\u0997\u09c7 \u099a\u09b2\u09ae\u09be\u09a8 \u09ac\u09b8\u09cd\u09a4\u09c1 \u0995\u09a3\u09be\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3 \u09b6\u09c2\u09a8\u09cd\u09af\u0964 \u09a4\u09cd\u09ac\u09b0\u09a3 \u0993 \u09ae\u09a8\u09cd\u09a6\u09a8 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u09ae\u09c1\u0996\u09c0\u0964<\/span><\/p>\n

\u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3 (Uniform acceleration):<\/b><\/h2>\n

<\/b>\u09af\u09a6\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09c7 \u09a4\u09ac\u09c7 \u09b8\u09c7\u0987 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u09a4\u09cd\u09ac\u09b0\u09a3\u0995\u09c7 \u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/p>\n

\u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u099f\u09bf\u09b0 <\/span>0,\u00a01,\u00a02,\u00a03<\/span> \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1 \u09aa\u09b0 \u09ac\u09c7\u0997 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>2,\u00a04,\u00a06,\u00a08<\/span> \u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964 \u098f\u0996\u09be\u09a8\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 <\/span>2<\/span> \u09ae\u09bf\u099f\u09be\u09b0\/\u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1, \u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09b8\u09ae\u09af\u09bc <\/span>(t)<\/span><\/span> \u09ac\u09a8\u09be\u09ae \u09ac\u09c7\u0997 <\/span>(v)<\/span><\/span> \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0\u099f\u09bf \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\"\u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3\"\u0985\u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3 (Variable acceleration): <\/b><\/h2>\n

\u09af\u0996\u09a8 \u09b8\u09ae\u09af\u09bc\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09b9\u09af\u09bc \u09a4\u09be\u0995\u09c7 \u0985\u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09b8\u09ae\u09af\u09bc \u09ac\u09a8\u09be\u09ae \u09ac\u09c7\u0997 \u09b2\u09c7\u0996\u099a\u09bf\u09a4\u09cd\u09b0\u09c7 \u09ac\u0995\u09cd\u09b0\u09b0\u09c7\u0996\u09be \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0985\u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3 \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964<\/span><\/p>\n

C.G.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u098f\u0995\u0995 <\/span>\u09b8\u09c7<\/span>.<\/span>\u09ae\u09bf<\/span>.\/<\/span>\u09b8\u09c7<\/span>.<\/span>^2<\/span><\/span>
\n<\/span>M.K.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u098f\u0995\u0995 <\/span>\u09ae\u09bf\u099f\u09be\u09b0<\/span>\/<\/span>\u09b8\u09c7<\/span>.<\/span>^2<\/span><\/span>
\n<\/span>F.P.S.<\/span> \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7\u09b0 \u098f\u0995\u0995\u00a0 <\/span>\u09ab\u09c1\u099f<\/span>\/<\/span>\u09b8\u09c7<\/span>.<\/span>^2<\/span><\/span><\/p>\n

\"\u0985\u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3\"\u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf (Resultant of multiple velocities)<\/b><\/h2>\n

\u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf (Resultant of Velocities): <\/b>\u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09c7\u0997 \u0995\u09be\u09b0\u09cd\u09af\u09b0\u09a4 \u09b9\u09b2\u09c7, \u098f\u09a6\u09c7\u09b0 \u09b8\u09ae\u09cd\u09ae\u09bf\u09b2\u09bf\u09a4 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2 \u09af\u09a6\u09bf \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u09ac\u09c7\u0997\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u0990 \u098f\u0995\u099f\u09bf\u09ae\u09be\u09a4\u09cd\u09b0 \u09ac\u09c7\u0997\u0995\u09c7 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2\u09c7 \u098f\u09ac\u0982 \u098f\u0995\u09be\u09a7\u09bf\u0995 \u09ac\u09c7\u0997\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf\u0995\u09c7 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u0985\u0982\u09b6\u0995 \u09ac\u09be \u0989\u09aa\u09be\u0982\u09b6 \u09ac\u09b2\u09c7\u0964\u00a0<\/span><\/p>\n

\u09af\u09a6\u09bf \u098f\u0995\u099f\u09bf \u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 <\/span>\\vec{u}<\/span><\/span> \u0993 <\/span>\\vec{v}<\/span><\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2, \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a6\u09bf\u0995\u09c7 <\/span>\\vec{w}<\/span><\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 <\/span>\\vec{w}<\/span><\/span> \u0995\u09c7\u00a0<\/span>\\vec{u}<\/span> \u0993\u00a0<\/span>\\vec{v}<\/span>\u00a0 \u098f\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u0982\u0995\u09c7\u09a4\u09c7 \u09b2\u09c7\u0996\u09be \u09b9\u09af\u09bc <\/span>\\vec{w}=\\vec{u}+\\vec{v}<\/span><\/span> \u0964 \u098f\u0996\u09be\u09a8\u09c7\u00a0<\/span>\\vec{u}<\/span> \u0993\u00a0<\/span>\\vec{v}<\/span>\u00a0 \u09b9\u09b2\u00a0<\/span>\\vec{w}<\/span>\u00a0 \u098f\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u0985\u0982\u09b6\u0995\u0964<\/span><\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0<\/p>\n

\u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09af\u09bc \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf:<\/b><\/p>\n

\"\u098f\u0995\u0987<\/p>\n

\"\u098f\u0995\u0987\u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cb\u09a8\u09cb \u09ac\u09b8\u09cd\u09a4\u09c1\u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997 <\/span>u<\/span> \u0993 <\/span>v<\/span> \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09b9\u09b2\u09c7 \u098f\u09a6\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09ac\u09c7\u0997 <\/span>u+v<\/span><\/span> \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09ac\u09c7\u0964 <\/span>u<\/span> \u0993 <\/span>v<\/span> \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09af\u09bc \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09b2\u09c7 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 <\/span>u \\sim v<\/span>,<\/span> \u098f\u09ac\u0982 \u098f\u09b0 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09b0 \u09ac\u09c7\u0997\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0 \u098f\u0995\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09b0\u09c7\u0996\u09be\u09af\u09bc \u09ac\u09be \u09a6\u09bf\u0995\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09b2\u09c7 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09b2\u09ac\u09cd\u09a7\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\n

\u09ac\u09c7\u0997\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0 (Parallelogram law of velocity):<\/b><\/h3>\n

\u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997 \u098f\u0995\u099f\u09bf \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09b8\u09a8\u09cd\u09a8\u09bf\u09b9\u09bf\u09a4 \u09ac\u09be\u09b9\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ae\u09be\u09a8\u09c7 \u0993 \u09a6\u09bf\u0995\u09c7 \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u0990 \u09b8\u09a8\u09cd\u09a8\u09bf\u09b9\u09bf\u09a4 \u09ac\u09be\u09b9\u09c1\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u0995\u09b0\u09cd\u09a3 \u09ae\u09be\u09a8\u09c7 \u0993 \u09a6\u09bf\u0995\u09c7 \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09c7\u0997 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09ac\u09c7\u0964<\/span><\/p>\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a3: <\/b>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 <\/span>u,\u00a0v<\/span> \u09ae\u09be\u09a8\u09c7\u09b0 \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09af\u09bc \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>OACB<\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 <\/span>OA<\/span> \u098f\u09ac\u0982 <\/span>OB<\/span> \u09ac\u09be\u09b9\u09c1\u09a6\u09cd\u09ac\u09af\u09bc \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ae\u09be\u09a8\u09c7 \u0993 \u09a6\u09bf\u0995\u09c7 \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09af\u09bc\u0964 <\/span>O,\u00a0C<\/span> \u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0<\/p>\n

\u0995\u09b2\u09cd\u09aa\u09a8\u09be \u0995\u09b0\u09bf, \u0995\u09a3\u09be\u099f\u09bf <\/span>u<\/span> \u09ac\u09c7\u0997\u09c7 <\/span>OA<\/span> \u09ac\u09be\u09b9\u09c1 \u09ac\u09b0\u09be\u09ac\u09b0 \u099a\u09b2\u09c7 \u098f\u09ac\u0982 \u098f\u0995\u0987 \u09b8\u09be\u09a5\u09c7 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u099f\u09bf \u09b8\u09b0\u09cd\u09ac\u09a6\u09be <\/span>OB<\/span> \u098f\u09b0 \u0989\u09aa\u09b0 \u09b0\u09c7\u0996\u09c7 <\/span>OA<\/span> \u09ac\u09be\u09b9\u09c1\u099f\u09bf \u0995\u09a3\u09be\u099f\u09bf\u09b8\u09b9 \u09a8\u09bf\u099c\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2\u09c7 <\/span>v<\/span> \u09ac\u09c7\u0997\u09c7 <\/span>OB<\/span> \u09ac\u09b0\u09be\u09ac\u09b0 \u099a\u09b2\u09c7\u0964 \u098f\u0995\u0995 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09a3\u09be\u099f\u09bf <\/span>u<\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u0995\u09be\u09b0\u09a3\u09c7 <\/span>A<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u09ac\u0982 <\/span>OA<\/span> \u09b0\u09c7\u0996\u09be\u099f\u09bf <\/span>v<\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u0995\u09be\u09b0\u09a3\u09c7 \u09a8\u09bf\u099c\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2\u09c7 <\/span>BC<\/span> \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u09aa\u09cc\u0981\u099b\u09ac\u09c7\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u0995\u09a3\u09be\u099f\u09bf \u098f\u0995\u0995 \u09b8\u09ae\u09af\u09bc \u09aa\u09b0\u09c7 <\/span>C<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u0995\u09a3\u09be\u099f\u09bf \u098f\u0995\u0995 \u09b8\u09ae\u09af\u09bc\u09c7\u09b0 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0 \u09ad\u0997\u09cd\u09a8\u09be\u0982\u09b6 <\/span>t<\/span> \u09b8\u09ae\u09af\u09bc\u09c7 <\/span>P<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09aa\u09cc\u0981\u099b\u09c7\u0964 <\/span>P Q \\| E O<\/span><\/span> \u0986\u0981\u0995\u09bf\u0964 \u0995\u09a3\u09be\u099f\u09bf \u09b8\u09ae\u09ac\u09c7\u0997\u09c7 \u099a\u09b2\u09c7 \u09ac\u09b2\u09c7 <\/span>OQ=ut,\u00a0QP=OE=vt;<\/span> \u09aa\u09c1\u09a8\u09b0\u09be\u09df <\/span>OA=u<\/span> \u098f\u09ac\u0982 <\/span>OB=v<\/span><\/p>\n

\\therefore \\frac{O Q}{Q P}=\\frac{u t}{v t}=\\frac{u}{v}=\\frac{O A}{O B}<\/span><\/span><\/p>\n

\\therefore \\triangle O Q P<\/span><\/span> \u0993 <\/span>\\triangle O A C<\/span><\/span> \u09b8\u09a6\u09c3\u09b6\u0995\u09cb\u09a3\u09c0\u0964<\/span><\/p>\n

\\angle C O A=\\angle P O Q<\/span> \u00a0<\/span> \\therefore O P<\/span><\/span> \u0993 <\/span>O C<\/span><\/span> \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09df \u09a5\u09be\u0995\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0 <\/span>\\frac{O P}{O C}=\\frac{O Q}{O A}=\\frac{u t}{u}=t \\quad \\therefore O P=t . O C<\/span><\/span><\/p>\n

\u098f\u0995\u0995 \u09b8\u09ae\u09df\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce t=1\u00a0 \u09b9\u09b2\u09c7, OP=OC\u00a0\u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u0995\u09a3\u09be\u099f\u09bf \u098f\u0995\u0995 \u09b8\u09ae\u09df\u09c7 C \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u0995\u09b0\u09ac\u09c7\u0964<\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u09c7\u09b0 <\/span>O C<\/span><\/span> \u0995\u09b0\u09cd\u09a3\u0987 \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

\u098f\u0995\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc<\/b><\/h3>\n

\u09e7\u09ae \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf:<\/b><\/p>\n

\"\u098f\u0995\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0<\/p>\n

\"\u098f\u0995\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0<\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \\alpha<\/span><\/span> \u0995\u09cb\u09a3\u09c7 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 <\/span>u<\/span> \u0993 <\/span>v<\/span> \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>OA<\/span> \u0993 <\/span>OB<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ae\u09be\u09a8\u09c7 \u0993 \u09a6\u09bf\u0995\u09c7 \u09b8\u09c2\u099a\u09bf\u09a4\u0964 <\/span>OACB<\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u099f\u09bf \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf \u098f\u09ac\u0982 <\/span>O,\u00a0C<\/span> \u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 <\/span>OC<\/span> \u0995\u09b0\u09cd\u09a3\u099f\u09bf \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09ac\u09c7\u0964 \u09a7\u09b0\u09bf, \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>w<\/span>, \u09af\u09be <\/span>OA<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \\theta<\/span> <\/span>\u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7 \u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span>\\angle A O C=\\theta<\/span><\/span><\/p>\n

OB<\/span> \u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \u0993 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09ac\u09b2\u09c7, <\/span>AC<\/span> \u098f\u0995\u0987 \u09ac\u09c7\u0997 <\/span>v<\/span> \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09ac\u09c7\u0964 <\/span>C<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 \u09ac\u09b0\u09cd\u09a7\u09bf\u09a4 <\/span>OA<\/span> (\u09e7\u09ae \u099a\u09bf\u09a4\u09cd\u09b0\u09c7) \u09ac\u09be <\/span>OA<\/span> (\u09e8\u09af\u09bc \u099a\u09bf\u09a4\u09cd\u09b0\u09c7) \u098f\u09b0 \u0989\u09aa\u09b0 <\/span>CD<\/span> \u09b2\u09ae\u09cd\u09ac \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\u09e7\u09ae \u099a\u09bf\u09a4\u09cd\u09b0\u09c7, <\/span>A D=A C \\cos C A D=Q \\cos \\alpha<\/span> \u098f\u09ac\u0982 C D=A C \\sin C A D=Q \\sin \\alpha<\/span><\/span><\/p>\n

\\therefore O D=O A+A D=u+v \\cos \\alpha<\/span><\/span>
\n<\/span><\/p>\n

\u09e8\u09df \u099a\u09bf\u09a4\u09cd\u09b0\u09c7, <\/span>A D=A C \\cos C A D=v \\cos (\\pi-\\alpha)=-v \\cos \\alpha<\/span><\/span><\/p>\n

\u098f\u09ac\u0982 <\/span>C D=A C \\sin C A D=v \\sin (\\pi-\\alpha)=v \\sin \\alpha<\/span><\/span><\/p>\n

\\therefore O D=O A-A D=u-(-v \\cos \\alpha)=u+v \\cos \\alpha<\/span><\/span>
\n<\/span><\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u0989\u09ad\u09df \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 <\/span>OCD<\/span> \u09b8\u09ae\u0995\u09cb\u09a3\u09c0 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c \u09b9\u09a4\u09c7 \u09aa\u09be\u0987, <\/span>O C^{2}=O D^{2}+C D^{2}<\/span><\/span><\/p>\n

\\Rightarrow w^{2}=(u+v \\cos \\alpha)^{2}+(v \\sin \\alpha)^{2}<\/span><\/span>
\n<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>=u^{2}+v^{2}\\left(\\sin ^{2} \\alpha+\\cos ^{2} \\alpha\\right)+2 u v \\cos \\alpha=u^{2}+v^{2}+2 u v \\cos \\alpha<\/span><\/span><\/p>\n

\\therefore w=\\sqrt{u^{2}+v^{2}+2 u v \\cos \\alpha}<\/span><\/span> , \u09af\u09be \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8\u0964<\/span><\/p>\n

\u098f\u0996\u09a8, <\/span>\\tan \\theta=\\frac{C D}{O D}=\\frac{v \\sin \\alpha}{u+v \\cos \\alpha}<\/span><\/span> ;<\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/span>u+v \\cos \\alpha \\neq 0<\/span><\/span><\/p>\n

\\therefore \\theta=\\tan ^{-1} \\frac{v \\sin \\alpha}{u+v \\cos \\alpha}<\/span><\/span> , \u09af\u09be \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09a6\u09bf\u0995\u0964\u00a0<\/span><\/p>\n

\u09e8\u09df \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf:<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 <\/span> \\alpha<\/span> \u0995\u09cb\u09a3\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997 <\/span>\\vec{u}=\\overrightarrow{O A}<\/span><\/span> \u0993 <\/span>\\vec{v}=\\overrightarrow{O B}<\/span><\/span>.\u00a0\u00a0OACB<\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u099f\u09bf \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09c7 <\/span>O,\u00a0C<\/span> \u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u09b8\u09be\u09ae\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c1\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 <\/span>\\overrightarrow{O C}=\\vec{w}<\/span><\/span> \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09ac\u09c7\u0964 \u09a7\u09b0\u09bf, \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>\\vec{w}<\/span><\/span>,\u00a0 <\/span>\\vec{u}<\/span><\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span> \\theta<\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n

 <\/p>\n

\"\u098f\u0995\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0<\/p>\n

A C=O B<\/span><\/span> \u098f\u09ac\u0982 <\/span>A C \\| O B<\/span><\/span> \u09ac\u09b2\u09c7 <\/span>\\vec{v}=\\overrightarrow{A C}<\/span><\/span>.\u00a0<\/span><\/p>\n

\u098f\u0996\u09be\u09a8\u09c7, <\/span>\\angle O A C=\\pi-\\alpha<\/span><\/span> \u098f\u09ac\u0982 <\/span>\\angle A C O=\\alpha-\\theta<\/span><\/span><\/p>\n

0<\\alpha<\\pi<\/span><\/span> \u09b8\u09c0\u09ae\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 <\/span> \u098f\u09b0 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ae\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af, <\/span>O A C<\/span><\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7 \u0995\u09cb\u09b8\u09be\u0987\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

O C^{2}=O A^{2}+A C^{2}-2 . O A \\cdot A C \\cos (\\pi-\\alpha) \\Rightarrow w^{2}=u^{2}+v^{2}+2 u v \\cos \\alpha<\/span><\/span><\/p>\n

\\therefore w=\\sqrt{u^{2}+v^{2}+2 u v \\cos \\alpha}<\/span><\/span> , \u09af\u09be \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8\u0964\u00a0<\/span><\/p>\n

O A C<\/span><\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7 \u09b8\u09be\u0987\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09df\u09cb\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>\\frac{O A}{\\sin O C A}=\\frac{A C}{\\sin A O C} \\Rightarrow \\frac{u}{\\sin (\\alpha-\\theta)}=\\frac{v}{\\sin \\theta}<\/span><\/span>
\n<\/span><\/p>\n

\\Rightarrow u \\sin \\theta=v(\\sin \\alpha \\cos \\theta-\\sin \\theta \\cos \\alpha) \\Rightarrow(u+v \\cos \\alpha) \\sin \\theta=v \\sin \\alpha \\cos \\theta<\/span><\/span> \u00a0 <\/span>
\n<\/span><\/p>\n

\\Rightarrow \\tan \\theta=\\frac{v \\sin \\alpha}{u+v \\cos \\alpha} \\quad \\therefore \\theta=\\tan ^{-1} \\frac{v \\sin \\alpha}{u+v \\cos \\alpha}<\/span><\/span> \u09af\u09be \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09a6\u09bf\u0995\u0964<\/span><\/p>\n

\u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0995\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc<\/b><\/h3>\n

(i) \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09b2\u09ac\u09cd\u09a7\u09bf: <\/b>\u09af\u0996\u09a8 <\/span>\\alpha=0^{\\circ}<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09df \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be \u0995\u09b0\u09c7, \u09a4\u0996\u09a8<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>w^{2}=u^{2}+v^{2}+2 u v \\cos 0^{\\circ}=u^{2}+v^{2}+2 u v=(u+v)^{2}<\/span><\/span>
\n<\/span><\/p>\n

\u2234<\/span> \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09ae \u09b2\u09ac\u09cd\u09a7\u09bf, <\/span>w_{\\max }=u+v<\/span><\/span> \u00a0<\/span> \u098f\u09ac\u0982 \u09a6\u09bf\u0995 <\/span>u<\/span> \u0993 <\/span>v<\/span> \u098f\u09b0 \u09a6\u09bf\u0995 \u09ac\u09b0\u09be\u09ac\u09b0\u0964\u00a0<\/span><\/p>\n

(ii) \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b2\u09ac\u09cd\u09a7\u09bf: <\/b>\u09af\u0996\u09a8 <\/span>\\alpha=180^{\\circ}<\/span><\/span> \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09df \u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be \u0995\u09b0\u09c7, \u09a4\u0996\u09a8<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>w^{2}=u^{2}+v^{2}+2 u v \\cos 180^{\\circ}=u^{2}+v^{2}-2 u v=(u-v)^{2} \\quad\\left[\\because \\cos 180^{\\circ}=-1\\right]<\/span><\/span><\/p>\n

\u2234<\/span> \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09ae \u09b2\u09ac\u09cd\u09a7\u09bf, <\/span>w_{\\min }=u-v \\quad[u>v]<\/span><\/span> \u098f\u09ac\u0982 \u09a6\u09bf\u0995 \u09ac\u09c3\u09b9\u09a4\u09cd\u09a4\u09b0 \u09ac\u09c7\u0997\u09c7\u09b0 \u09a6\u09bf\u0995 \u09ac\u09b0\u09be\u09ac\u09b0\u0964\u00a0<\/span><\/p>\n

(iii) \u09b8\u09ae\u0995\u09cb\u09a3\u09c7 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b0\u09a4 \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf: <\/b>\u09af\u0996\u09a8 <\/span>\\alpha=90^{\\circ}<\/span><\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>w^{2}=u^{2}+v^{2}+2 u v \\cos 90^{\\circ}=u^{2}+v^{2} \\quad\\left[\\because \\cos 90^{\\circ}=0\\right]<\/span><\/span><\/p>\n

\u2234<\/span> \u09b2\u09ac\u09cd\u09a7\u09bf, <\/span>w=\\sqrt{u^{2}+v^{2}}<\/span><\/span><\/p>\n

\\tan \\theta=\\frac{v \\sin 90^{\\circ}}{u+v \\cos 90^{\\circ}}=\\frac{v .1}{u+v .0}=\\frac{v}{u} \\Rightarrow \\tan ^{-1} \\frac{v}{u}<\/span><\/span>\u00a0<\/span>
\n<\/span><\/p>\n

\u2234<\/span> \u09b2\u09ac\u09cd\u09a7\u09bf \u09ac\u09c7\u0997 <\/span>u<\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \\theta<\/span> <\/span> \u0995\u09cb\u09a3\u09c7 \u0986\u09a8\u09a4\u0964<\/span><\/p>\n

(iv) \u09b8\u09ae\u09be\u09a8 \u09b8\u09ae\u09be\u09a8 \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf:<\/b><\/p>\n

v=u<\/span><\/span> \u09b9\u09b2\u09c7, <\/span>w^{2}=u^{2}+u^{2}+2 u . u \\cos \\alpha=2 u^{2}+2 u^{2} \\cos \\alpha<\/span><\/span>
\n<\/span><\/p>\n

=2 u^{2}(1+\\cos \\alpha)=2 u^{2} \\cdot 2 \\cos ^{2} \\frac{\\alpha}{2}=4 u^{2} \\cos ^{2} \\frac{\\alpha}{2} <\/span><\/span><\/p>\n

\u2234<\/span> \u09b2\u09ac\u09cd\u09a7\u09bf, <\/span>w=2 u \\cos \\frac{\\alpha}{2}<\/span><\/span> \u098f\u09ac\u0982 <\/span>\\tan \\theta=\\frac{u \\sin \\alpha}{u+u \\cos \\alpha}=\\frac{\\sin \\alpha}{1+\\cos \\alpha}=\\frac{2 \\sin \\frac{\\alpha}{2} \\cos \\frac{\\alpha}{2}}{2 \\cos ^{2} \\frac{\\alpha}{2}}=\\tan \\frac{\\alpha}{2} \\quad \\therefore \\theta=\\frac{\\alpha}{2}<\/span><\/span><\/p>\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09b8\u09ae\u09be\u09a8 \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u098f\u09a6\u09c7\u09b0 \u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b0\u09c7\u0996\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3\u0995\u09c7 \u09b8\u09ae\u09a6\u09cd\u09ac\u09bf\u0996\u09a8\u09cd\u09a1\u09bf\u09a4 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\u09ac\u09c7\u0997\u09c7\u09b0 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c \u09b8\u09c2\u09a4\u09cd\u09b0 (Law of Triangle of velocities):<\/b><\/p>\n

\u09ac\u09b0\u09cd\u09a3\u09a8\u09be: <\/b>\u098f\u0995\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09be\u09b0\u09cd\u09af\u09b0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0995\u09cb\u09a8\u09cb \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u098f\u0995\u0987 \u0995\u09cd\u09b0\u09ae\u09c7 \u0997\u09c3\u09b9\u09c0\u09a4 \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09be\u09b9\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6\u09bf\u09a4 \u09b9\u09b2\u09c7, \u09a4\u09be\u09a6\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0990 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u0995\u09cd\u09b0\u09ae\u09c7 \u0997\u09c3\u09b9\u09c0\u09a4 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09ac\u09be\u09b9\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a3:<\/b> \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u098f\u0995\u099f\u09bf \u0995\u09a3\u09be\u09b0 \u0989\u09aa\u09b0 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 <\/span>\\underline{u}<\/span><\/span> \u0993 <\/span>\\underline{v}<\/span><\/span> \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>ABC<\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 <\/span>AB<\/span> \u0993 <\/span>BC<\/span> \u09ac\u09be\u09b9\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ae\u09be\u09a8\u09c7 \u0993 \u09a6\u09bf\u0995\u09c7 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/span><\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0<\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0ABCD<\/span> \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995\u099f\u09bf \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

A D=B C<\/span><\/span> \u098f\u09ac\u0982 <\/span>A D \\| B C<\/span><\/span> \u09ac\u09b2\u09c7, <\/span>\\overrightarrow{A D}=\\overrightarrow{B C}<\/span><\/span><\/p>\n

\\underline{u}+\\underline{v}=\\overrightarrow{A B}+\\overrightarrow{B C}=\\overrightarrow{A B}+\\overrightarrow{A D}=\\overrightarrow{A C}<\/span><\/span>,<\/span> (\u09ac\u09c7\u0997\u09c7\u09b0 \u09b8\u09be\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09a6\u09cd\u09ac\u09be\u09b0\u09be\u0964)<\/span><\/p>\n

\u0985\u09a4\u098f\u09ac,\u00a0<\/span>\\underline{u}<\/span> \u0993\u00a0<\/span>\\underline{v}<\/span> \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 <\/span>ABC<\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u0995\u09cd\u09b0\u09ae\u09c7 \u0997\u09c3\u09b9\u09c0\u09a4 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09ac\u09be\u09b9\u09c1 <\/span>AC<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af:<\/b> ABC<\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09ac\u09be\u09b9\u09c1 <\/span>AC<\/span> \u09ae\u09be\u09a4\u09cd\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09c7 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1<\/span> \\underline{u}<\/span> \u0993 <\/span>\\underline{v}<\/span> \u09ac\u09c7\u0997 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09c7\u0996\u09be\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09c7\u0996\u09be \u0997\u09ae\u09a8 \u0995\u09b0\u09ac\u09c7\u0964<\/span><\/p>\n

\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4:<\/b> \u098f\u0995\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09be\u09b0\u09cd\u09af\u09b0\u09a4 \u09a4\u09bf\u09a8\u099f\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u0995\u09cb\u09a8\u09cb \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u098f\u0995\u0987 \u0995\u09cd\u09b0\u09ae\u09c7 \u0997\u09c3\u09b9\u09c0\u09a4 \u09ac\u09be\u09b9\u09c1 \u09a4\u09bf\u09a8\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09b2\u09c7, \u09a4\u09be\u09a6\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n

\u09ac\u09bf.\u09a6\u09cd\u09b0.: <\/b>\u0985\u09b7\u09cd\u099f\u09ae \u0985\u09a7\u09cd\u09af\u09be\u09af\u09bc\u09c7 \u09ac\u09b2 \u09ac\u09bf\u09ad\u09be\u099c\u09a8, \u09ac\u09b2\u09c7\u09b0 \u09b8\u09be\u0987\u09a8 \u09b8\u09c2\u09a4\u09cd\u09b0, \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 \u09ac\u09bf\u09b6\u09a6 \u0986\u09b2\u09cb\u099a\u09a8\u09be \u0995\u09b0\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964 \u098f\u0996\u09be\u09a8\u09c7, <\/span>P,\u00a0Q,\u00a0R<\/span> \u098f\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>u,\u00a0v,\u00a0w<\/span> \u09b2\u09bf\u0996\u09c7 \u09ac\u09c7\u0997\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0985\u09a8\u09c1\u09b0\u09c2\u09aa \u09b8\u09c2\u09a4\u09cd\u09b0\u0997\u09c1\u09b2\u09bf \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n

\u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6:<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>XY<\/span> \u09b8\u09ae\u09a4\u09b2\u09c7 <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 <\/span>w<\/span> \u09ac\u09c7\u0997 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09af\u09be <\/span>OC<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09b9\u09af\u09bc \u098f\u09ac\u0982 <\/span>x<\/span>-\u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \\theta<\/span> <\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964 <\/span>C A \\perp O X<\/span><\/span> \u098f\u09ac\u0982 <\/span>C B \\perp O Y<\/span><\/span> \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964<\/span><\/p>\n

\u09a4\u09be\u09b9\u09b2\u09c7 <\/span>OACB<\/span> \u098f\u0995\u099f\u09bf \u0986\u09af\u09bc\u09a4\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u0964<\/span><\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0\u09a7\u09b0\u09bf, <\/span>O A=u, O B=v<\/span><\/span><\/p>\n

OX<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7 <\/span>w<\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 <\/span>O A=O C \\cos \\theta=w \\cos \\theta<\/span><\/span><\/p>\n

OY<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7 <\/span>w<\/span> \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 <\/span>O B=O C \\sin \\theta=w \\sin \\theta<\/span><\/span><\/p>\n

\u09ac\u09bf\u09b6\u09c7\u09b7 \u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af: <\/b>\u09a6\u09c1\u099f\u09bf \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09b0 \u09ac\u09c7\u0997\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b2\u09ae\u09cd\u09ac \u09b9\u09b2\u09c7, \u09b2\u09ac\u09cd\u09a7\u09bf\u09b0 \u09ae\u09be\u09a8 \u0993 \u09a6\u09bf\u0995 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc:<\/span><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>u<\/span> \u0993 <\/span>v\u00a0(u>v)<\/span> \u09ac\u09c7\u0997\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>w<\/span> \u098f\u09ac\u0982 \u09a4\u09be <\/span>v<\/span> \u098f\u09b0 \u09a6\u09bf\u0995\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span>90^{\\circ}<\/span><\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964 <\/span>u<\/span> \u0993 <\/span>v<\/span> \u098f\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u0997\u09a4 \u0995\u09cb\u09a3\u09c7\u09b0 \u09ae\u09be\u09a8 \\alpha<\/span><\/span><\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0<\/p>\n

\u098f\u0996\u09a8, <\/span>v<\/span> \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09c7\u0997\u0997\u09c1\u09b2\u09cb\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 \u09a8\u09bf\u09af\u09bc\u09c7 \u09aa\u09be\u0987,\u00a0<\/span><\/p>\n

v+u \\cos \\alpha=w \\cos 90^{\\circ}=0<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>u \\cos \\alpha=-v \\quad \\therefore \\alpha=\\cos ^{-1}\\left(-\\frac{v}{u}\\right)<\/span><\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, <\/span>w^{2}=u^{2}+v^{2}+2 u v \\cos \\alpha<\/span><\/span><\/p>\n

\u00a0 <\/span>=u^{2}+v^{2}+2 v(-v) \\quad[\\because u \\cos \\alpha=-v]<\/span><\/span><\/p>\n

\u00a0 <\/span>=u^{2}-v^{2}<\/span><\/span><\/p>\n

\\therefore w=\\sqrt{u^{2}-v^{2}}<\/span><\/span><\/p>\n

\u098f\u0995\u0987 \u09a4\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u0987 \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u0982\u0996\u09cd\u09af\u0995 \u09ac\u09c7\u0997\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc:<\/b><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>O<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b6\u09c0\u09b2, <\/span>u_{1}, u_{2}, u_{3} \\ldots \\ldots u_{n}<\/span><\/span> \u09ac\u09c7\u0997\u0997\u09c1\u09b2\u09bf <\/span>OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <\/span>\\alpha_{1}, \\alpha_{2}, \\alpha_{3} \\ldots \\ldots \\alpha_{n}<\/span><\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7 \u098f\u09ac\u0982 \u098f\u09a6\u09c7\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf <\/span>w,\u00a0OX<\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \\theta<\/span> <\/span> \u0995\u09cb\u09a3 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7\u0964\u00a0<\/span><\/p>\n

\"\u09ac\u09c7\u0997\u09c7\u09b0OX<\/span> \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09c7\u0997\u0997\u09c1\u09b2\u09bf\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 \u09a8\u09bf\u09af\u09bc\u09c7,<\/span><\/p>\n

w \\cos \\theta=u_{1} \\cos \\alpha_{1}+u_{2} \\cos \\alpha_{2}+u_{3} \\cos \\alpha_{3}+ \\cdots+u_{n} \\cos \\alpha_{n}=w_{x}<\/span><\/span>\u00a0(\u09a7\u09b0\u09bf) <\/span> \u2026\u2026(i)<\/span><\/p>\n

OY<\/span> \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09c7\u0997\u0997\u09c1\u09b2\u09bf\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6 \u09a8\u09bf\u09af\u09bc\u09c7,<\/span><\/p>\n

w \\sin \\theta=u_{1} \\sin \\alpha_{1}+u_{2} \\sin \\alpha_{2}+u_{3} \\sin \\alpha_{3}+\\cdots+u_{n} \\sin \\alpha_{n}=w_{y}<\/span><\/span> (\u09a7\u09b0\u09bf)\u00a0 <\/span> \u2026\u2026\u2026(ii)<\/span><\/p>\n

(i)<\/span> \u09a8\u0982 \u0993 <\/span>(ii)<\/span> \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ac\u09b0\u09cd\u0997 \u09af\u09cb\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, <\/span>w^{2}\\left(\\cos ^{2} \\alpha+\\sin ^{2} \\alpha\\right)=w_{x}^{2}+w_{y}^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>w^{2}=w_{x}^{2}+w_{y}^{2}<\/span><\/span><\/p>\n

\u09ac\u09be, <\/span>w=\\sqrt{w_{x}^{2}+w_{y}^{2}}<\/span><\/span><\/p>\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09ac\u09c7\u0997\u0997\u09c1\u09b2\u09bf\u09b0 \u09b2\u09ac\u09cd\u09a7\u09bf, <\/span>OX<\/span> \u0993 <\/span>OY<\/span> \u09ac\u09b0\u09be\u09ac\u09b0 \u09b2\u09ae\u09cd\u09ac\u09be\u0982\u09b6\u09c7\u09b0 \u09ac\u09b0\u09cd\u0997\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2\u09c7\u09b0 \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\n

\u0986\u09ac\u09be\u09b0, <\/span>(ii)<\/span><\/span> \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 <\/span>(i)<\/span><\/span> \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n

\\frac{w \\sin \\theta}{w \\cos \\theta}=\\frac{w_{y}}{w_{x}} \\quad<\/span><\/span> \u09ac\u09be, <\/span>\\tan \\theta=\\frac{w_{y}}{w_{x}} \\quad \\therefore \\theta=\\tan ^{-1} \\frac{w_{y}}{w_{x}}<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u09ac\u09b2\u09ac\u09bf\u09a6\u09cd\u09af\u09be\u09b0 \u09af\u09c7 \u09b6\u09be\u0996\u09be\u09af\u09bc \u0997\u09a4\u09bf\u09b6\u09c0\u09b2 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0993\u09aa\u09b0 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0\u09a4 \u09ac\u09b2\u0997\u09c1\u09b2\u09bf\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af \u09a8\u09bf\u09af\u09bc\u09c7 \u0986\u09b2\u09cb\u099a\u09a8\u09be \u0995\u09b0\u09be \u09b9\u09af\u09bc \u09a4\u09be\u0987 \u0997\u09a4\u09bf\u09ac\u09bf\u09a6\u09cd\u09af\u09be\u0964 \u098f\u0987 \u0985\u09a7\u09cd\u09af\u09be\u09af\u09bc\u0995\u09c7 \u099a\u09be\u09b0 \u09ad\u09be\u0997\u09c7 \u09ad\u09be\u0997 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7\u0964 \u09b8\u09b0\u09a3 (Displacement): \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09ae\u09af\u09bc\u09c7 \u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u0995\u09c7 \u0990 \u09a6\u09bf\u0995\u09c7 \u0990 \u09ac\u09b8\u09cd\u09a4\u09c1\u09b0 \u09b8\u09b0\u09a3 \u09ac\u09b2\u09be \u09b9\u09af\u09bc<\/p>\n

Read More<\/a><\/p>\n","protected":false},"author":11,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4274,3038,50,3026],"tags":[2742,2740,2743,1121,1120,2744,2745,2741,2739,1117],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3849"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=3849"}],"version-history":[{"count":24,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3849\/revisions"}],"predecessor-version":[{"id":12620,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3849\/revisions\/12620"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3849"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3849"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3849"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}