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3-2.Motion in Plane
medium
એક કણ એવી રીતે ગતિ કરે છે કે જેથી તેના સ્થાન સદિશ $(x,y) $ નીચે પ્રમાણે મળે છે.
$t=0$ સેકન્ડે $(2\;m,3\;m),$
$t=2 $ સેકન્ડે $(6\;m,7\;m)$ અને
$t=5 $ સેકન્ડે $ (13\;m,14\;m)$
$ t=0$ સેકન્ડથી $t= 5 $ સેકન્ડ સુધીમાં કણનો સરેરાશ વેગ $\vec v_{av}$ કેટલો હશે?
A$\frac{1}{5}\left( {13\hat i + 14\hat j} \right)$
B$\;\frac{7}{3}\left( {\hat i + \hat j} \right)$
C$2\left( {\hat i + \hat j} \right)$
D$\;\frac{{11}}{5}\left( {\hat i + \hat j} \right)$
(AIPMT-2014)
Solution
$\begin{array}{l}
\,\,\,At\,time\,t = 0,\,the\,position\,vector\,of\\
the\,particle\,is\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bar r}_1} = 2\hat i + 3\hat j\\
At\,time\,t = 5\,s,\,the\,position\,vector\,of\\
the\,particle\,is\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bar r}_2} = 13\hat i + 14\hat j\\
Displacement\,from\,\,\,{{\bar r}_1}\,to\,\,{{\bar r}_2}\,is
\end{array}$
$\begin{array}{l}
Displacement\,from\,\,\,{{\bar r}_1}\,to\,\,{{\bar r}_2}\,is\\
\Delta \bar r = {{\bar r}_2} – {{\bar r}_1} = \left( {13\hat i + 14\hat j} \right) – \left( {2\hat i + 3\hat j} \right)\\
= 11\hat i + 11\hat j\\
\therefore \,\,\,Average\,velocity,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bar v}_{av}} = \frac{{\Delta \bar r}}{{\Delta t}} = \frac{{11\hat i + 11\hat j}}{{5 – 0}} = \frac{{11}}{5}\left( {\hat i + \hat j} \right)
\end{array}$
\,\,\,At\,time\,t = 0,\,the\,position\,vector\,of\\
the\,particle\,is\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bar r}_1} = 2\hat i + 3\hat j\\
At\,time\,t = 5\,s,\,the\,position\,vector\,of\\
the\,particle\,is\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bar r}_2} = 13\hat i + 14\hat j\\
Displacement\,from\,\,\,{{\bar r}_1}\,to\,\,{{\bar r}_2}\,is
\end{array}$
$\begin{array}{l}
Displacement\,from\,\,\,{{\bar r}_1}\,to\,\,{{\bar r}_2}\,is\\
\Delta \bar r = {{\bar r}_2} – {{\bar r}_1} = \left( {13\hat i + 14\hat j} \right) – \left( {2\hat i + 3\hat j} \right)\\
= 11\hat i + 11\hat j\\
\therefore \,\,\,Average\,velocity,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\bar v}_{av}} = \frac{{\Delta \bar r}}{{\Delta t}} = \frac{{11\hat i + 11\hat j}}{{5 – 0}} = \frac{{11}}{5}\left( {\hat i + \hat j} \right)
\end{array}$
Standard 11
Physics
