JEE PYQ: Motion in a Straight Line - Question ID 29ea37252b45 (JEE Main 2024)

Core Formula & Concept:

When a body falls freely under gravity, its motion is uniformly accelerated with acceleration $g = 10 \, \mathrm{ms}^{-2}$ downward. The key equations for motion in a straight line with constant acceleration are:

• Displacement as a function of time: $s = ut + \tfrac{1}{2} a t^2$
• Velocity as a function of time: $v = u + a t$
• Displacement between two instants $t_1$ and $t_2$: $s(t_2) - s(t_1) = u(t_2 - t_1) + \tfrac{1}{2} a (t_2^2 - t_1^2)$

Here, $u$ is the initial velocity at the starting point (which we take as $t = 0$), and $a = g = 10 \, \mathrm{ms}^{-2}$.

Step-by-Step Derivation:

Step 1: Define variables and coordinate system
Let the starting point be $O$. Point $A$ is at distance $h$ below $O$, and point $B$ is $80 \, \mathrm{m}$ below $A$. We choose downward as positive, so the displacement of $A$ from $O$ is $s_A = h$, and the displacement of $B$ from $O$ is $s_B = h + 80$.

Step 2: Write expressions for $s_A$ and $s_B$
If the body starts from rest at $O$, its velocity at $t = 0$ is $u = 0$. Let $t_A$ be the time it takes to reach $A$, and $t_B = t_A + 2$ be the time to reach $B$. Then $s_A = \tfrac{1}{2} g t_A^2 = 5 t_A^2, \quad s_B = \tfrac{1}{2} g t_B^2 = 5 t_B^2.$ Since $s_B - s_A = 80$, we have $5 t_B^2 - 5 t_A^2 = 80 \quad\Longrightarrow\quad t_B^2 - t_A^2 = 16.$

Step 3: Use the time difference
We know $t_B = t_A + 2$. Substitute: $(t_A + 2)^2 - t_A^2 = 16 \quad\Longrightarrow\quad 4 t_A + 4 = 16 \quad\Longrightarrow\quad t_A = 3\, \mathrm{s}.$

Step 4: Compute $h$
Now substitute $t_A = 3$ into $s_A = 5 t_A^2$: $h = 5 \times 3^2 = 45\, \mathrm{m}.$

Common Traps & Exam Tip:

1. Sign convention: Many students take upward as positive and get confused with signs. Consistently choosing downward as positive avoids errors. 2. Assuming the body starts from rest at A: The question says “a body falling under gravity,” implying it started from rest at some point above A. Do not assume $u = 0$ at A. 3. Algebraic slip: When expanding $(t_A + 2)^2$, students sometimes forget the cross term $2 t_A \times 2$. Double-check the expansion.

Exam tip: Always write down the knowns ($g = 10$, $\Delta s = 80$, $\Delta t = 2$) and the unknown ($h$) before jumping into equations. This keeps the derivation clean and reduces mistakes.