Two cars \(P\) and \(Q\) are moving on a road in the same direction. Acceleration of car \(P\) increases linearly with time whereas car \(Q\) moves with a constant acceleration. Both cars cross each other at time \(t=0\), for the first time. The maximum possible number of crossing(s) (including the crossing at \(t=0)\) is ________.

\(\mathrm{a}_{\mathrm{p}}=\mathrm{kt}, \mathrm{k}\) is constant
\(\mathrm{a}_{\mathrm{Q}}=\mathrm{a}, \mathrm{a}\) is constant
\(\mathrm{a}_{\mathrm{QP}}=\mathrm{a}_{\mathrm{Q}}-\mathrm{a}_{\mathrm{p}}=\mathrm{a}-\mathrm{kt}\)
as initial velocities are not mentioned in question, so will have to assume two cases.
Case-I
\(\mathrm{u}_{\mathrm{OP}}\) and \(\mathrm{a}_{\mathrm{QP}}\) in same direction




Total number of crossing \(=2\) Case-II
\(\mathrm{u}_{\mathrm{QP}}\) and \(\mathrm{a}_{\mathrm{QP}}\) in opposite direction

Total number of crossing \(=3\)