JEE PYQ: Motion in a Straight Line - Question ID 30fc8ec165d4 (JEE Main 2023)

Core Formula & Concept:

In kinematics, the average speed of a moving object over a journey is defined as the total distance traveled divided by the total time taken for the journey. Mathematically, it is expressed as: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$ This formula is fundamental and applies regardless of changes in speed during the journey. It is crucial to note that average speed is not the arithmetic mean of the individual speeds unless the time intervals for each segment are equal.

Step-by-Step Derivation:

Step 1: Identify the given data
The vehicle travels two segments:

• First segment: Distance, $d_1 = 4 \text{ km}$, Speed, $v_1 = 3 \text{ km/h}$
• Second segment: Distance, $d_2 = 4 \text{ km}$, Speed, $v_2 = 5 \text{ km/h}$

Step 2: Calculate the time taken for each segment
Time is calculated using the formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

• Time for first segment: $t_1 = \frac{d_1}{v_1} = \frac{4 \text{ km}}{3 \text{ km/h}} = \frac{4}{3} \text{ hours}$
• Time for second segment: $t_2 = \frac{d_2}{v_2} = \frac{4 \text{ km}}{5 \text{ km/h}} = \frac{4}{5} \text{ hours}$

Step 3: Compute the total distance and total time

• Total distance: $d_{\text{total}} = d_1 + d_2 = 4 \text{ km} + 4 \text{ km} = 8 \text{ km}$
• Total time: $t_{\text{total}} = t_1 + t_2 = \frac{4}{3} + \frac{4}{5}$

To add the times, find a common denominator (15): $t_{\text{total}} = \frac{4 \times 5}{15} + \frac{4 \times 3}{15} = \frac{20}{15} + \frac{12}{15} = \frac{32}{15} \text{ hours}$

Step 4: Calculate the average speed
Using the core formula: $\text{Average Speed} = \frac{d_{\text{total}}}{t_{\text{total}}} = \frac{8 \text{ km}}{\frac{32}{15} \text{ hours}} = 8 \times \frac{15}{32} = \frac{120}{32} = 3.75 \text{ km/h}$

Step 5: Match with the given options
The calculated average speed is $3.75 \text{ km/h}$, which corresponds to Option A.

Common Traps & Exam Tip:

Students often make the following mistakes in such problems:

1. Arithmetic Mean Fallacy: They incorrectly assume that average speed is the arithmetic mean of the two speeds, i.e., $\frac{3 + 5}{2} = 4 \text{ km/h}$. This is wrong because the time spent at each speed is different.
2. Incorrect Time Calculation: Misapplying the formula for time, such as inverting distance and speed, leads to wrong time values and thus incorrect average speed.
3. Unit Confusion: Not ensuring consistent units (e.g., mixing km and meters) can lead to calculation errors. Here, all units are in km and hours, so no conversion is needed.

Exam Tip: Always remember that average speed depends on total distance and total time, not just the speeds. Break the journey into segments, compute time for each, and then apply the formula.