JEE PYQ: Motion in a Straight Line - Question ID 62e6b08b1cf2 (JEE Main 2021)

In problems involving motion with constant acceleration (or deceleration), the fundamental kinematic equations are used. The key formulas relevant here are:

• Velocity as a function of time: $v = u + at$, where $u$ is the initial velocity, $a$ is the acceleration, and $t$ is the time.
• Displacement as a function of time: $s = ut + \frac{1}{2}at^2$. However, in this problem, displacement is not directly required.

Since the scooter starts from rest, the initial velocity $u = 0$. The scooter first accelerates with $a_1$ for time $t_1$, reaching a maximum velocity $v$. Then, it decelerates with $a_2$ for time $t_2$, coming to rest. The key insight is that the final velocity after deceleration is zero, and the velocity just before deceleration is the same as the velocity just after acceleration.

Let’s break down the motion into two phases:

The scooter starts from rest ($u = 0$) and accelerates at $a_1$ for time $t_1$. Using the velocity formula:

The scooter now decelerates at $a_2$ for time $t_2$ and comes to rest. Here, the initial velocity for this phase is $v$ (from Phase 1), and the final velocity is $0$. Using the velocity formula again:

From Equation 1 and Equation 2, both equal $v$, so we can set them equal to each other:

Rearrange the equation to isolate $\frac{t_1}{t_2}$:

This matches option C.

Students often make the following mistakes in this question:

1. Incorrect Sign Handling: Some students forget that deceleration is negative acceleration and write $v = a_2 t_2$ instead of $v = -a_2 t_2$. However, since we are dealing with magnitudes, the negative sign cancels out, and the final result remains correct. Still, it’s important to understand the direction of acceleration.
2. Misapplying Formulas: Some students try to use the displacement formula unnecessarily, complicating the problem. This question only requires the velocity-time relationship.
3. Confusing Ratios: Students may invert the ratio $\frac{t_1}{t_2}$ and choose $\frac{a_1}{a_2}$ (Option D) instead of $\frac{a_2}{a_1}$. Always double-check which variable is in the numerator and denominator.

Exam Tip: For problems involving two phases of motion (acceleration followed by deceleration), always:

• Write the velocity equation for each phase.
• Equate the velocities at the transition point (since the velocity doesn’t change instantaneously).
• Solve for the required ratio or variable.