JEE PYQ: Motion in a Straight Line - Question ID 87687f41cdb4 (JEE Main 2024)

Core Formula & Concept:

To determine the average speed of a particle moving in a straight line, we use the fundamental definition:

$\text{Average Speed} = \frac{\text{Total Distance Traveled}}{\text{Total Time Taken}}$

This formula is universally applicable, regardless of how the speed varies during the motion. The key steps involve:

• Breaking the entire journey into distinct segments where the speed is constant or follows a known pattern.
• Calculating the time taken for each segment using the relation $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.
• Summing up the total distance and total time to compute the average speed.

In this problem, the motion is divided into three parts:

1. Half the distance is covered at a constant speed of $6 \mathrm{~m/s}$.
2. The remaining half distance is split into two equal time intervals, with speeds $9 \mathrm{~m/s}$ and $15 \mathrm{~m/s}$ respectively.

--- Step-by-Step Derivation:

Let the total distance traveled by the particle be $D$ meters.

Step 1: First Half of the Distance

The particle covers $\frac{D}{2}$ meters at a constant speed of $6 \mathrm{~m/s}$.
Time taken for this segment: $t_1 = \frac{\frac{D}{2}}{6} = \frac{D}{12} \text{ seconds}$

Step 2: Second Half of the Distance

The remaining $\frac{D}{2}$ meters is covered in two equal time intervals, say each of duration $t_2$.
Let the speeds during these intervals be $9 \mathrm{~m/s}$ and $15 \mathrm{~m/s}$ respectively.
Distance covered in the first time interval: $d_1 = 9 \cdot t_2$ Distance covered in the second time interval: $d_2 = 15 \cdot t_2$ Total distance for the second half: $d_1 + d_2 = 9 t_2 + 15 t_2 = 24 t_2 = \frac{D}{2}$ Solving for $t_2$: $t_2 = \frac{D}{48} \text{ seconds}$

Step 3: Total Time Calculation

Total time taken for the entire journey: $T = t_1 + 2 t_2 = \frac{D}{12} + 2 \cdot \frac{D}{48} = \frac{D}{12} + \frac{D}{24} = \frac{2D + D}{24} = \frac{3D}{24} = \frac{D}{8} \text{ seconds}$

Step 4: Average Speed Calculation

Using the definition of average speed: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{8}} = 8 \mathrm{~m/s}$

--- Common Traps & Exam Tip:

Students often make the following mistakes in this question:

1. Misinterpreting "two equal time intervals": Some assume the second half is split into two equal distances instead of equal time intervals. This leads to incorrect time calculations.
2. Incorrectly adding speeds: A few students try to average the speeds directly (e.g., $(6 + 9 + 15)/3$), which is invalid because average speed depends on time, not just speed values.
3. Algebraic errors in time calculation: Mistakes in simplifying fractions (e.g., $\frac{D}{12} + \frac{D}{24}$) can lead to wrong total time values.

Exam Tip: Always define the total distance as a variable (e.g., $D$) and express all times in terms of $D$. This ensures consistency and avoids confusion when dealing with fractions of the journey.