Two cars P and Q are moving on a road in the same direction. The 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 ______.
Numerical value type. Enter your answer.
Answer: 3
The relative acceleration of P with respect to Q is linear in time, $a_{rel} = \alpha + \beta t$. Integrating twice (with zero relative position at $t = 0$) gives
$$x_{rel}(t) = u\,t + \frac{\alpha t^2}{2} + \frac{\beta t^3}{6}$$
The cars cross whenever $x_{rel} = 0$. This cubic has the root $t = 0$ and at most two more real roots, so there can be at most 3 crossings.
Solution by Sreeraj P, M.Sc Physics