Two trains $P$ and $Q$ are moving along parallel tracks with same uniform speed of $\frac{20m}{s}$. The driver of train $P$ decides to overtake train $Q$ and accelerates his train by $\frac{1m}{s^2}$. After $50s,$ train $p$ crosses the engine of train $Q$. Find out what was the distance between the two trains initially provided the length of each train is $400m.$

The correct option is D $450m$
Let the initial distance between the two trains be
distance travelled by train $P$ in $50s$.
$S_P=ut+\frac{1}{2}a{t}^2$
$=20\times 50+\frac{1}{2}1\times {50}^2$
$S_P=2250m$
$u=\frac{20m}{s};a=\frac{1m}{s^2};t=50s$
distance travelled by train $Q$ in $50s$
$S_Q=Ut=20\times 50=1000m$
Extra distance which train $P$ must cover to cross train $Q=S_P-S_Q$
and this must be equal to initial distance between the trains plus the sum of the lengths of the two trains
$S_P-S_Q=S+2\times 400$
$2250-1000=S+800$
$S=450m$