JEE Main · 2025 · 29 Jan 2025 · Shift 2 · Kinematics

Let $x(t)$ denote the position of Q relative to P, with $x(0)=0$ marking the crossing at $t=0$. Car Q has constant acceleration $a$ and car P has acceleration increasing linearly, $a_P=kt$, so the relative acceleration is


The relative velocity is

where $u$ is the (unspecified) relative velocity at $t=0$ — a quadratic in $t$, so it can vanish at no more than 2 values of $t$.

Integrating again, the relative position is


This is a cubic in $t$ with $x(0)=0$ built in. Dividing out the factor of $t$ leaves a quadratic,

which has at most 2 real roots. So $x(t)=0$ has at most 3 roots in total — the one at $t=0$ plus at most 2 more, occurring whenever $u$ and the changing sign of $a_{rel}(t)$ make the relative displacement reverse direction twice after the initial crossing.

The maximum possible number of crossings, including the one at $t=0$, is 3.