JEE PYQ: Motion in a Straight Line - Question ID c775fc558079 (JEE Main 2023)

In kinematics, the average speed of an object over a journey is defined as the total distance traveled divided by the total time taken. Mathematically, $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}.$ This definition holds regardless of the number of segments or changes in speed during the journey.

When a journey consists of multiple segments, each with its own speed and distance, the total time is the sum of the times taken for each segment. For a segment of distance $x$ traveled at speed $v$, the time taken is $t = \frac{x}{v}$.

Let’s break down the problem:

1. The car travels a distance $x$ at speed $v_1$. The time taken for this segment is: $t_1 = \frac{x}{v_1}.$
2. The car then travels the same distance $x$ at speed $v_2$. The time taken for this segment is: $t_2 = \frac{x}{v_2}.$
3. The total distance traveled by the car is: $\text{Total Distance} = x + x = 2x.$
4. The total time taken for the entire journey is: $\text{Total Time} = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2}.$
5. Now, apply the definition of average speed: $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{x}{v_1} + \frac{x}{v_2}}.$
6. Simplify the denominator: $\frac{x}{v_1} + \frac{x}{v_2} = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) = x \left( \frac{v_2 + v_1}{v_1 v_2} \right).$
7. Substitute back into the average speed expression: $\text{Average Speed} = \frac{2x}{x \left( \frac{v_1 + v_2}{v_1 v_2} \right)} = \frac{2x \cdot v_1 v_2}{x (v_1 + v_2)}.$
8. The $x$ terms cancel out: $\text{Average Speed} = \frac{2 v_1 v_2}{v_1 + v_2}.$

This matches Option B.

Trap 1: Arithmetic Mean Misconception
Many students mistakenly assume that the average speed is the arithmetic mean of the two speeds, i.e., $\frac{v_1 + v_2}{2}$ (Option D). This is incorrect because the car spends different amounts of time traveling at each speed unless $v_1 = v_2$. The arithmetic mean only works when the time intervals are equal, not the distances.

Trap 2: Harmonic Mean Confusion
Some students recall that average speed for equal distances is the harmonic mean of the speeds, but they misapply the formula. The correct harmonic mean for two speeds is $\frac{2 v_1 v_2}{v_1 + v_2}$, not $\frac{v_1 v_2}{2(v_1 + v_2)}$ (Option A). The latter is half the correct value.

Trap 3: Distance-Time Mismatch
Students may incorrectly compute total time as $\frac{2x}{v_1 + v_2}$ (Option C), which is wrong because it assumes the car travels the entire $2x$ distance at the average of $v_1$ and $v_2$. This ignores the fact that the car’s speed changes after the first segment.

Exam Tip:
Always start by writing down the definition of average speed and compute total distance and total time separately. For equal distances, the average speed is the harmonic mean of the speeds. For equal times, it is the arithmetic mean. Never assume symmetry unless explicitly given!