JEE PYQ: Motion in a Straight Line - Question ID 235a1db5d4ec (JEE Main 2023)

In kinematics, the average velocity ($V_{\text{avg}}$) over a journey is defined as the total displacement divided by the total time taken. Mathematically, $V_{\text{avg}} = \frac{\text{Total Displacement}}{\text{Total Time Taken}}.$

When a person travels two equal distances ($x$ each) with different velocities ($v_1$ and $v_2$), the total displacement is $2x$, but the total time taken is the sum of the individual times for each segment: $t_1 = \frac{x}{v_1}, \quad t_2 = \frac{x}{v_2}.$ Thus, the average velocity $V$ is: $V = \frac{2x}{t_1 + t_2}.$ This concept is crucial for solving problems involving non-uniform motion over equal distances.

Let’s derive the relationship systematically:

1. Define the journey segments:
   • First segment: Distance = $x$, Velocity = $v_1$.
   • Second segment: Distance = $x$, Velocity = $v_2$.
2. Calculate time for each segment: $t_1 = \frac{\text{Distance}_1}{\text{Velocity}_1} = \frac{x}{v_1}, \quad t_2 = \frac{x}{v_2}.$
3. Compute total displacement and total time:
   • Total displacement = $x + x = 2x$.
   • Total time = $t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2}$.
4. Express average velocity ($V$): $V = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{2x}{\frac{x}{v_1} + \frac{x}{v_2}}.$
5. Simplify the expression: Factor out $x$ from the denominator: $V = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}}.$
6. Rearrange to match the given options: Take the reciprocal of both sides: $\frac{1}{V} = \frac{\frac{1}{v_1} + \frac{1}{v_2}}{2} \implies \frac{2}{V} = \frac{1}{v_1} + \frac{1}{v_2}.$

This matches Option D.

Students often confuse average velocity with average speed or assume arithmetic mean applies here. Key mistakes include:

• Using arithmetic mean (Option B): Assuming $V = \frac{v_1 + v_2}{2}$ is incorrect because average velocity depends on time spent at each velocity, not just the velocities themselves.
• Ignoring equal distances: If distances were unequal, the formula would differ. Here, equal distances simplify the problem.
• Reciprocal confusion: Students may misapply the harmonic mean. Remember: For equal distances, average velocity is the harmonic mean of the two velocities.

Exam Tip: Always write down the definition of average velocity first. For equal distances, the formula $\frac{2}{V} = \frac{1}{v_1} + \frac{1}{v_2}$ is a direct consequence of the definition.