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A goods train accelerating uniformly on a straight railway track, approaches an electric pole standing on the side of track. Its engine passes the pole with velocity $u$ and the guard's room passes with velocity $v.$ The middle wagon of the train passes the pole with a velocity.
$\frac{{u + v}}{2}$
$\frac{1}{2}\sqrt {{u^2} + {v^2}} $
$\sqrt {uv} $
$\sqrt {\left( {\frac{{{u^2} + {v^2}}}{2}} \right)} $
Solution
$\begin{array}{l}
Let'S'\,be\,the\,d{\rm{istance}}\,{\rm{between}}\,{\rm{two}}\,{\rm{ends}}\\
{\rm{'a'}}\,{\rm{be}}\,{\rm{the}}\,{\rm{constant}}\,{\rm{accrleration}}\\
{\rm{As}}\,{\rm{we}}\,{\rm{konw}}\,{{\rm{V}}^2} – {u^2} = 2aS\\
or,\,aS = \frac{{{v^2} – {u^2}}}{2}\\
Let\,V\,be\,velocity\,at\,mid\,po{\mathop{\rm int}} .
\end{array}$
$\begin{array}{l}
Therefore,\,V_c^2 – {u^2} = 2a\frac{S}{2}\\
V_c^2 = {u^2} + aS\\
v_c^2 = {u^2} + \frac{{{V^2} – {u^2}}}{2}\\
{V_c} = \sqrt {\frac{{{u^2} + {v^2}}}{2}}
\end{array}$
