Answer:
For no collision of two trains, the relative velocity of
faster train w.r.t slower train \[\left( {{\upsilon }_{\text{1}}}-{{\upsilon
}_{\text{2}}} \right)\] should become zero, while travelling a relative
displacement d with acceleration a.
Here ; \[\text{u}={{\upsilon
}_{\text{1}}}-{{\upsilon }_{\text{2}}};\text{ }\upsilon =0;\text{a}=-\text{a
};\text{S}=\text{d}\].
As. \[{{\upsilon
}^{\text{2}}}={{\text{u}}^{\text{2}}}+\text{2as}\], so, \[0={{\left( {{\upsilon }_{\text{1}}}-\text{
}{{\upsilon }_{\text{2}}} \right)}^{\text{2}}}+\text{ 2}(-\text{a})\text{d}\]or
\[\text{d}={{({{\upsilon }_{\text{1}}}-{{\upsilon }_{\text{2}}})}^{\text{2}}}/\text{2a}\]
There will be no collision if\[\text{d}>{{({{\upsilon
}_{\text{1}}}-{{\upsilon }_{\text{2}}})}^{\text{2}}}/\text{2a}\].
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