Find the speed and the length of a train being given that it traveled with a constant speed past a stationary observer for $7$ seconds and needed $25$ seconds to pass a platform $378$ meter long with the same speed.
Length of the platform $= 378$ m
Let the length of the train $= \ell$ m
$\therefore \;$ Total distance covered by the train in crossing the platform $= \left(378 + \ell\right)$ m
Time taken to cross the platform $= 25$ s
$\therefore \;$ Speed of train $= \left(\dfrac{378 + \ell}{25}\right)$ m/s $\;\;\; \cdots \; (1)$
The train crosses a stationary observer in $7$ s
$\therefore \;$ Speed of train $= \left(\dfrac{\ell}{7}\right)$ m/s $\;\;\; \cdots \; (2)$
Since the train is traveling with the same speed, we have from equations $(1)$ and $(2)$
$\dfrac{378 + \ell}{25} = \dfrac{\ell}{7}$
i.e. $\;$ $2646 + 7 \ell = 25 \ell$
i.e. $\;$ $18 \ell = 2646$ $\implies$ $\ell = 147$ m
$\therefore \;$ Length of the train $= \ell = 147$ m
Substituting the value of $\ell$ in equation $(2)$ gives
Speed of train $= \dfrac{147}{7} = 21$ m/s $= \dfrac{21 \times 3600}{1000}$ kmph $= 75.6$ kmph