JEE PYQ: Motion in a Straight Line - Question ID b71f42f7164b (JEE Main 2022)

Core Formula & Concept:

When a ball is released from rest under gravity, it undergoes uniformly accelerated motion with acceleration $g$ (downward). The key formulas for displacement $s$, initial velocity $u$, time $t$, and acceleration $a$ are:

• Displacement as a function of time: $s = ut + \tfrac{1}{2} a t^2$ Since the ball is released from rest, $u = 0$, so $s = \tfrac{1}{2} g t^2$
• Velocity as a function of time: $v = u + at = gt$

The problem divides the total fall into two equal distances: the first half ($h/2$) and the second half ($h/2$). We must find the times $t_1$ and $t_2$ taken to cover these respective halves.

Step-by-Step Derivation:

Step 1: Express total fall time $T$

Let $T$ be the total time to fall distance $h$. Using $s = \tfrac{1}{2} g t^2$: $h = \tfrac{1}{2} g T^2$ $\Rightarrow T = \sqrt{\frac{2h}{g}}$

Step 2: Express time $t_1$ for first half ($h/2$)

For the first half, distance $s_1 = h/2$. Using the same formula: $\frac{h}{2} = \tfrac{1}{2} g t_1^2$ $\Rightarrow t_1 = \sqrt{\frac{h}{g}}$

Step 3: Express time $t_2$ for second half

The second half starts after time $t_1$ and ends at time $T$. Hence: $t_2 = T - t_1 = \sqrt{\frac{2h}{g}} - \sqrt{\frac{h}{g}}$ Factor out $\sqrt{\frac{h}{g}}$: $t_2 = \sqrt{\frac{h}{g}} \left( \sqrt{2} - 1 \right)$ But from Step 2, $t_1 = \sqrt{\frac{h}{g}}$, so: $t_2 = t_1 \left( \sqrt{2} - 1 \right)$ Rearranging gives the relation: $t_2 = (\sqrt{2} - 1) t_1$

Step 4: Match with given options

The derived relation matches option D exactly.

Common Traps & Exam Tip:

Students often mistakenly assume constant velocity or confuse the order of $t_1$ and $t_2$. Remember:

• The ball accelerates, so the second half (lower portion) is covered faster than the first half.
• Always use the displacement formula $s = \tfrac{1}{2} g t^2$ for free fall from rest.
• Double-check the algebraic manipulation when factoring square roots to avoid sign errors.

By carefully applying the kinematic equations and verifying each step, you can confidently select option D as the correct answer.