JEE PYQ: Motion in a Straight Line - Question ID 8277c78c6296 (JEE Main 2023)

Core Formula & Concept:

The average speed of an object over a journey is defined as the total distance traveled divided by the total time taken. Mathematically, $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}.$ In this problem, the object moves along three straight segments AB, BC, and CD with constant speeds $v_1, v_2,$ and $v_3$ respectively. The key is to express each segment’s length in terms of a common variable and then compute the total distance and total time.

Step-by-Step Derivation:

Step 1: Assign a common length variable.
Let the length of segment AB be $x$. The problem states that AB = BC, so BC is also $x$. Furthermore, AD = 3AB, so AD = $3x$. Since AD = AB + BC + CD, we have $3x = x + x + \text{CD} \implies \text{CD} = x.$ Thus, all three segments AB, BC, and CD have the same length $x$.

Step 2: Compute the time taken on each segment.
Time is distance divided by speed. Therefore, $t_1 = \frac{x}{v_1}, \quad t_2 = \frac{x}{v_2}, \quad t_3 = \frac{x}{v_3}.$

Step 3: Compute the total distance and total time.
Total distance $D = x + x + x = 3x$.
Total time $T = t_1 + t_2 + t_3 = \frac{x}{v_1} + \frac{x}{v_2} + \frac{x}{v_3}$.

Step 4: Express the average speed.
Average speed $v_{\text{avg}} = \frac{D}{T} = \frac{3x}{\frac{x}{v_1} + \frac{x}{v_2} + \frac{x}{v_3}}$.
Factor out $x$ in the denominator: $v_{\text{avg}} = \frac{3x}{x\left(\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}\right)} = \frac{3}{\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}}.$

Step 5: Simplify the denominator.
The denominator is $\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} = \frac{v_2 v_3 + v_1 v_3 + v_1 v_2}{v_1 v_2 v_3}.$ Therefore, $v_{\text{avg}} = \frac{3}{\frac{v_2 v_3 + v_1 v_3 + v_1 v_2}{v_1 v_2 v_3}} = \frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1}.$

Step 6: Match with the given options.
The expression matches option D: $\boxed{\frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1}}.$

Common Traps & Exam Tip:

1. Misidentifying segment lengths: Students often assume CD is different from AB and BC. The problem explicitly states AD = 3AB and AB = BC, so all three segments must be equal. 2. Confusing average speed with arithmetic mean: Option B tempts students to take the simple average $(v_1 + v_2 + v_3)/3$, which is incorrect because average speed depends on time spent at each speed, not just the speeds themselves. 3. Incorrect algebraic simplification: When combining the reciprocals, ensure the common denominator is correctly formed to avoid sign or factor errors.

Exam Tip: Always assign a common variable to equal lengths and express every quantity in terms of that variable. This reduces complexity and minimizes errors.