Question

Two ends of a rain moving with a constant acceleration pass a point with velocities $u$ & $v$. Show that the velocity through which the middle point of the train passes the name points $\sqrt{\frac{u^2+v^2}{2}}$

Answer 1

Let length of train be $l$, so for traveling l distance train has acceleration,

$a=\frac{v^2-u^2}{2l}$, since engine passes a point with $u$ and end passes with$v$

similarly we can find velocity of mid point , (say $v^{\prime }$

$(v^{\prime }{)}^2=u^2+2al/2$, since train covers l/2 distance

$(v^{\prime }{)}^2=u^2+\frac{v^2-u^2}{2l}\times l$

$(v^{\prime }{)}^2=\frac{2u^2+v^2-u^2}{2}=\frac{v^2+u^2}{2}$

this gives , $v^{\prime }=\sqrt{\frac{u^2+v^2}{2}}$