The Driver Of A Train Moving At A Speed at V1, sights another train at a distance d, ahead of him moving in the same direction with a slower speed V2. He Applies The Brakes And Gives A Constant retardation a to his train. Show That there will be no collision If d>(V1-V2)*2/2a.

For the train A,

Initial velocity = v_{1}

Retardation = a

Distance it has to travel to avoid collision is = d

For train B,

Velocity of travel = v_{2}

Now,

The relative velocity of A with respect to B is, u = v_{1} – v_{2}.

The relative velocity becomes zero for the trains to avoid collision. That is, final relative velocity is, v = 0 [at this moment the trains have equal velocity]

Now, using, v^{2} = u^{2} – 2as [negative sign appears since a is retardation]

=> 0 = (v_{1} – v_{2})^{2} – 2ad

=> d = (v_{1} – v_{2})^{2}/(2d)

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