JEE PYQ: Motion in a Straight Line - Question ID a9bd9ee36820 (JEE Main 2025)

Core Formula & Concept:

In kinematics, the average velocity over a journey is defined as the total displacement divided by the total time taken. Mathematically, $\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}}.$ When an object moves in segments with different constant velocities, we compute the time taken for each segment separately and then sum them to find the total time.

Step-by-Step Derivation:

Given data:
- First segment: distance $x$, velocity $v_1 = 5\ \text{m/s}$.
- Second segment: distance $\frac{3}{2}x$, velocity $v_2$ (to be found).
- Average velocity over the entire journey: $\frac{50}{7}\ \text{m/s}$.

Step 1: Express the time for each segment.
Time for the first segment: $t_1 = \frac{x}{v_1} = \frac{x}{5}.$ Time for the second segment: $t_2 = \frac{\frac{3}{2}x}{v_2} = \frac{3x}{2v_2}.$

Step 2: Compute the total displacement and total time.
Total displacement: $\Delta s = x + \frac{3}{2}x = \frac{5}{2}x.$ Total time: $T = t_1 + t_2 = \frac{x}{5} + \frac{3x}{2v_2}.$

Step 3: Write the average velocity formula and substitute.
Average velocity: $\frac{50}{7} = \frac{\Delta s}{T} = \frac{\frac{5}{2}x}{\frac{x}{5} + \frac{3x}{2v_2}}.$ Simplify the numerator and denominator by factoring out $x$: $\frac{50}{7} = \frac{\frac{5}{2}}{\frac{1}{5} + \frac{3}{2v_2}}.$

Step 4: Solve for $v_2$.
Invert both sides: $\frac{7}{50} = \frac{\frac{1}{5} + \frac{3}{2v_2}}{\frac{5}{2}}.$ Multiply both sides by $\frac{5}{2}$: $\frac{7}{50} \cdot \frac{5}{2} = \frac{1}{5} + \frac{3}{2v_2} \quad\Longrightarrow\quad \frac{7}{20} = \frac{1}{5} + \frac{3}{2v_2}.$ Subtract $\frac{1}{5}$ from both sides: $\frac{7}{20} - \frac{1}{5} = \frac{3}{2v_2} \quad\Longrightarrow\quad \frac{7}{20} - \frac{4}{20} = \frac{3}{2v_2} \quad\Longrightarrow\quad \frac{3}{20} = \frac{3}{2v_2}.$ Cancel the common factor of $3$: $\frac{1}{20} = \frac{1}{2v_2}.$ Hence, $2v_2 = 20 \quad\Longrightarrow\quad v_2 = 10\ \text{m/s}.$

Common Traps & Exam Tip:

1. Misidentifying total displacement: Students sometimes mistakenly add the distances incorrectly, e.g., $x + \frac{3}{2}x = \frac{5}{2}x$ is crucial. 2. Incorrect time expressions: Forgetting to divide the distance by the velocity for each segment leads to wrong total time. 3. Algebraic slips: When solving the equation for $v_2$, a small arithmetic error can yield an incorrect answer. Always double-check each algebraic manipulation.