JEE Physics 2025 Question with Solution

Given: Car P has acceleration increasing linearly with time, so $a_P = kt$. Car Q has constant acceleration, so $a_Q = c$. Both cars are at the same position at $t = 0$.

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

Let the initial velocities at $t = 0$ be $u_P$ and $u_Q$.

For car P:

$$a_P = kt$$

Integrating,

$$v_P = u_P + \int kt \, dt = u_P + \frac{1}{2}kt^2$$

Again integrating,

$$x_P = \int v_P \, dt = u_P t + \frac{1}{6}kt^3$$

For car Q:

$$a_Q = c$$

So,

$$v_Q = u_Q + ct$$

and

$$x_Q = u_Q t + \frac{1}{2}ct^2$$

The cars cross whenever their positions are equal:

$$x_P = x_Q$$

Therefore,

$$u_P t + \frac{1}{6}kt^3 = u_Q t + \frac{1}{2}ct^2$$

Rearranging,

$$\frac{1}{6}kt^3 - \frac{1}{2}ct^2 + (u_P - u_Q)t = 0$$

Factor out $t$:

$$t\left(\frac{1}{6}kt^2 - \frac{1}{2}ct + (u_P - u_Q)\right) = 0$$

One root is $t = 0$, which is the given first crossing.

The remaining crossings are determined by the quadratic equation

$$\frac{1}{6}kt^2 - \frac{1}{2}ct + (u_P - u_Q) = 0$$

A quadratic can have at most two real roots. Hence, besides $t = 0$, there can be at most two more crossings.

So the maximum possible total number of crossings is $3$.

Therefore, the required numerical answer is $3$.

Given: The position equality condition decides the crossings.

Find: The maximum number of times the two cars can be at the same position.

Since acceleration of car P varies as $t$, its position contains a $t^3$ term. Since car Q has constant acceleration, its position contains a $t^2$ term. Thus, the equation obtained by setting their positions equal is a cubic in $t$.

From the working,

$$x_P - x_Q = \frac{1}{6}kt^3 - \frac{1}{2}ct^2 + (u_P - u_Q)t$$

So the crossing condition is

$$\frac{1}{6}kt^3 - \frac{1}{2}ct^2 + (u_P - u_Q)t = 0$$

This cubic already has one known root, $t = 0$, because the cars cross there.

After factoring out $t$,

$$t\left(\frac{1}{6}kt^2 - \frac{1}{2}ct + (u_P - u_Q)\right) = 0$$

The bracket is a quadratic expression, and a quadratic can contribute at most two real roots.

Hence the total number of real crossing times can be at most

$$1 + 2 = 3$$

including the crossing at $t = 0$.

Therefore, the maximum possible number of crossings is $3$.