JEE Main 2025 Jan 29 Shift 2, Physics Q25: Motion In Two And Three Dimension

JEE Main2025Jan 29, Shift 2Physics

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 ________ .

Your answer

Solution

Let the accelerations of cars $P$ and $Q$ be

$a_{P} = a_{0} + k t, a_{Q} = c$

Hence, the relative acceleration is

$a = a_{P} - a_{Q} = A + k t$

Integrating, the relative velocity is

$v = A t + \frac{k}{2} t^{2} + B$

Integrating again, the relative position is

$x = \frac{A}{2} t^{2} + \frac{k}{6} t^{3} + B t + C$

Since both cars cross at $t = 0$ ,

$x (0) = 0 \Rightarrow C = 0$

$∴ x = t (B + \frac{A}{2} t + \frac{k}{6} t^{2})$

The factor $t$ gives one crossing at $t = 0$ , while the quadratic factor can have at most two real roots.

Hence, the maximum possible number of crossings (including the crossing at $t = 0$ ) is

$3$

Concept behind this question

Motion In Two And Three DimensionNotes, formulas and examples →

Practice more Physics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2025 Jan 29 Shift 2, Physics Q25: Motion In Two And Three Dimension

Concept behind this question

More previous year questions on Motion In Two And Three Dimension

Practice more Physics