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JEE PYQ: Motion in a Straight Line - Question ID eacf43d26e4a (JEE Main 2024)

ID: eacf43d26e4aJEE Main 2024Single Correct MCQ

A body starts moving from rest with constant acceleration covers displacement $S_1$ in first $(p-1)$ seconds and $\mathrm{S}_2$ in first $p$ seconds. The displacement $\mathrm{S}_1+\mathrm{S}_2$ will be made in time :

Select Option

Step-by-step Explanation

Core Formula & Concept:

When a body starts from rest and moves with constant acceleration, its displacement $s$ in the first $t$ seconds is given by the kinematic equation: $s = \tfrac{1}{2}\,a\,t^{2}.$ Here $a$ is the constant acceleration.

The problem gives two displacements: $S_{1}$ in the first $(p-1)$ seconds, and $S_{2}$ in the first $p$ seconds. We are asked to find the time $T$ in which the body covers the combined displacement $S_{1}+S_{2}$ .

Step-by-Step Derivation:

1. Express $S_{1}$ and $S_{2}$ in terms of $a$ and $p$ .

S_{1} = \tfrac{1}{2}\,a\,(p-1)^{2}, \quad S_{2} = \tfrac{1}{2}\,a\,p^{2}.

2. Compute the combined displacement $S_{1}+S_{2}$ .

S_{1}+S_{2} = \tfrac{1}{2}\,a\bigl[(p-1)^{2} + p^{2}\bigr] = \tfrac{1}{2}\,a\bigl[p^{2}-2p+1 + p^{2}\bigr] = \tfrac{1}{2}\,a\bigl[2p^{2}-2p+1\bigr] = a\bigl(p^{2}-p+\tfrac{1}{2}\bigr).

3. Let $T$ be the time in which the body covers $S_{1}+S_{2}$ .

Using the same kinematic formula,

S_{1}+S_{2} = \tfrac{1}{2}\,a\,T^{2}.

Equate the two expressions for

S_{1}+S_{2}

\tfrac{1}{2}\,a\,T^{2} = a\bigl(p^{2}-p+\tfrac{1}{2}\bigr).

Cancel

a

(nonzero) and multiply both sides by

2

T^{2} = 2\bigl(p^{2}-p+\tfrac{1}{2}\bigr) = 2p^{2}-2p+1.

Hence

T = \sqrt{2p^{2}-2p+1}.

4. Match with the given options.

The correct expression for

T

\sqrt{2p^{2}-2p+1}\,s

, which corresponds to option D. Common Traps & Exam Tip:

• Many students mistakenly add the times $(p-1)+p$ and try to use that sum directly, ignoring the quadratic nature of displacement under constant acceleration. • Others forget to take the square root at the end, leading them to choose option C instead of D. • Always write down the kinematic formula explicitly and substitute step by step to avoid sign or algebraic errors.

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