Sponsor Area

Motion In Straight Line

Question
CBSEENPH11017204

Two trains on the same line are approaching each other with velocity u1, and u2When there is distance of x between them, both apply brakes and brakes give the retardation of at1 and a2. Prove that it is just possible to avoid the collision if

space space space straight u subscript 1 squared straight a subscript 2 plus straight u subscript 2 squared straight alpha subscript 1 equals 2 straight alpha subscript 1 straight alpha subscript 2 straight x

Solution
Let u1, be velocity of train A and a1, be retardation.
Let x, be the distance travelled by the train A before it comes at rest.


Using the third equation of motion, 
 space space space space space space space space straight s equals fraction numerator straight v squared minus straight u squared over denominator 2 straight a end fraction w e space g e t comma

space space space space space space space space straight x subscript 1 equals fraction numerator 0 squared minus straight u subscript 1 squared over denominator negative 2 straight alpha subscript 1 end fraction equals fraction numerator straight u subscript 1 squared over denominator 2 straight alpha subscript 1 end fraction
Similarly, if x2 is the distance travelled by train B before it comes at rest then
straight x subscript 2 equals fraction numerator 0 squared minus straight u subscript 2 squared over denominator negative 2 straight a subscript 2 end fraction equals fraction numerator straight u subscript 2 squared over denominator 2 straight a subscript 2 end fraction
Collision can be avoided, if the total distance travelled by both the trains is x.
i.e.   straight x equals straight x subscript 1 plus straight x subscript 2

space equals space fraction numerator straight u subscript 1 squared over denominator 2 straight alpha subscript 1 end fraction plus fraction numerator straight u subscript 2 squared over denominator 2 straight alpha subscript 2 end fraction
equals fraction numerator straight u subscript 1 squared straight alpha subscript 2 plus straight u subscript 2 squared straight a subscript 1 over denominator 2 straight alpha subscript 1 straight alpha subscript 2 end fraction

rightwards double arrow space straight u subscript 1 squared straight alpha subscript 2 plus straight u subscript 2 squared straight alpha subscript 1 equals 2 straight a subscript 1 straight a subscript 2 straight x