Answer:
Let x be
the total length of the train, V be the velocity of the train while passing a
certain middle point and a be the uniform acceleration of the train. Taking the
motion of the train when middle point is passing from the given point, we have
\[\text{u}=\text{u},\text{
v}=\text{V},\text{s}=\text{x}/\text{2 };\text{a}=\text{a}\]
Using, \[{{\upsilon }^{\text{2}}}=\text{ }{{\text{u}}^{\text{2}}}+\text{2as}\],
we have \[{{\text{V}}^{\text{2}}}={{\text{u}}^{\text{2}}}+\text{2ax}/\text{2}={{\text{u}}^{\text{2}}}+\text{ax}\] ...(i)
Taking the motion of train when the last end of train is
passing from the given point, then
\[\text{u}=\text{u},\text{v}=\text{v},\text{
a}=\text{a},\text{ s}=\text{x}\]
Now, we have \[{{\text{v}}^{\text{2}}}={{\text{u}}^{\text{2}}}+\text{2ax}\]
or \[\text{ax}=\frac{{{\upsilon }^{2}}-{{u}^{2}}}{2}\] Putting this value in
(i), we get
\[{{\text{V}}^{\text{2}}}={{\text{u}}^{\text{2}}}+\frac{{{\upsilon
}^{2}}-{{u}^{2}}}{2}=\frac{{{u}^{2}}+{{\upsilon }^{2}}}{2}\]or \[\text{V}=\sqrt{({{u}^{2}}+{{\upsilon
}^{2}})/2}\]
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