Two trains are moving with equal speed in opposite directions along two parallel railway tracks. If the wind is blowing with speed u along the track so that the relative velocities of the trains with respect to the wind are in the ratio $1 :2$, then the speed of each train must be:
a) $3u$
b) $2u$
c) $5u$
d) $4u$
e) $u$

Hint: Relative speed is described as the speed of a traveling object concerning another. When two objects are traveling in the same direction, relative speed is measured as their difference. When the two objects are traveling in opposite directions, relative speed is measured by summing the two speeds.

Complete step-by-step solution:
Let the speed of trains be V.
Two trains are moving but they are in opposite directions and wind is blowing with speed up along the track.
$\text{Relative Velocity} = \dfrac{v_{t} – v_{w}}{ v_{t} + v_{w}}$
$v_{t} = V$ and $v_{w}=u$
And given $\text{Relative Velocity} = \dfrac{1}{2}$.
So, we evaluate u in terms of V by equating the formula of relative speed.
$\dfrac{1}{2} = \dfrac{v_{t} – v_{w}}{ v_{t} + v_{w}}$
$\implies \dfrac{1}{2} = \dfrac{V – u}{ V + u}$
$\implies V + u = 2V -2u$
$\implies 3u = V$
The speed of each train must be $3u$.
Option (a) is correct.

Note: When two bodies are traveling in the opposite direction, the relative speed is measured by summing the speed of both bodies. The relative speed and relative velocity difference are that relative speed is the scalar magnitude, whereas relative velocity is the vector magnitude.