# What will be the condition of no collision with the second train?

## The drivers of a train moving at a speed ${v}_{1}$ sight another train at a distance $d$ ahead of him on the same line, moving with slower speed ${v}_{2}$. 1st driver pulls the brake to create a retardation $r$.

If the driver of first train is able to bring the velocity of his train down from its velocity ${v}_{1}$ to the velocity ${v}_{2}$ of train ahead in a distance $d$, the condition for no collision is met.
${v}^{2} - {u}^{2} = 2 a s$
${v}_{2}^{2} - {v}_{1}^{2} = 2 \left(- r\right) d$
$\implies r = \frac{{v}_{1}^{2} - {v}_{2}^{2}}{2 d}$