### Freight Train

The engineer of a train moving at a speed $v_{1}$ sights a freight train a distance d ahead of him on the same track, moving in the same direction with a slower speed $v_{2}.$
He puts on the brakes and gives his train a constant deceleration $a.$

Show that:
At the first moment, the relative speed between the two trains is $v_{rel}=v_{1}-v_{2}.$
If the two trains reach a relative speed of 0 they will crash.

We need the formula: $\frac{v(t)^{2}-v_{0}^{2}}{2a}=x-x_{0}.$
This formula is in your book, but you can also get it by taking $v(t)=v_{0}+at$ and plugging it into the formula $x=x_{0}+v_{0}t+(1/2)at^{2}.$
In our case we take the relative speed between them as v(t), and $v_{0}=0,x_{0}=0,and so \frac{(v_{1}-v_{2})^{2}-0}{2a}=x$ and if $x>d$ the trains will not collide.