JEE PYQ: Motion in a Straight Line - Question ID 04fa630b57cd (JEE Main 2023)

Core Formula & Concept:

In problems involving uniformly accelerated motion in a straight line, the relationship between velocity, acceleration, and time is governed by the first equation of kinematics:

$v = u + a t$

where:

• $v$ = final velocity (in $\mathrm{ms}^{-1}$)
• $u$ = initial velocity (in $\mathrm{ms}^{-1}$)
• $a$ = uniform acceleration (in $\mathrm{ms}^{-2}$)
• $t$ = time taken (in seconds)

This equation assumes that acceleration is constant and motion occurs along a straight line.

Step-by-Step Derivation:

Given data from the question:

• Initial velocity, $u = 10.0 \mathrm{~ms}^{-1}$
• Acceleration, $a = 2.0 \mathrm{~ms}^{-2}$
• Final velocity, $v = 60.0 \mathrm{~ms}^{-1}$

We are to find the time $t$ taken to reach $v = 60.0 \mathrm{~ms}^{-1}$.

Start with the kinematic equation:

$v = u + a t$

Rearrange to solve for $t$:

$v - u = a t$ $t = \frac{v - u}{a}$

Substitute the known values:

$t = \frac{60.0 - 10.0}{2.0}$ $t = \frac{50.0}{2.0}$ $t = 25.0 \text{ seconds}$

Thus, the time taken by the particle to reach $60.0 \mathrm{~ms}^{-1}$ is 25 seconds.

Common Traps & Exam Tip:

Students often make the following mistakes in such problems:

• Incorrect sign handling: If acceleration is in the same direction as velocity, it should be taken as positive. Some students mistakenly use negative acceleration, leading to wrong time values.
• Unit confusion: Mixing up units (e.g., using $\mathrm{cm/s}$ instead of $\mathrm{m/s}$) can lead to incorrect numerical results. Always ensure all units are consistent.
• Misapplying formulas: Using $v^2 = u^2 + 2 a s$ (which involves displacement) instead of $v = u + a t$ is a common error when time is the unknown. Choose the correct kinematic equation based on the given and required quantities.
• Arithmetic errors: Simple subtraction or division errors (e.g., $60 - 10 = 40$, then $40 / 2 = 20$) can lead to selecting option B (6s) or C (3s) instead of the correct D (25s). Always double-check calculations.

Exam Tip: Always write down the known values and the formula before substituting. This helps in avoiding confusion and ensures clarity in problem-solving.