JEE PYQ: Motion in a Straight Line - Question ID 4a58e7cbaca6 (JEE Main 2022)

In straight-line motion, the average velocity over a journey is defined as the total displacement divided by the total time taken. Mathematically, $\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}}.$ When the path is straight and the motion is along the same line, displacement equals the distance travelled, so we may write $\text{Average velocity} = \frac{\text{Total distance}}{\text{Total time}}.$

If the journey is split into segments of equal length, each segment’s time is $t_i = \frac{\text{Segment distance}}{v_i}.$ Summing these times gives the total time.

Given data:

• Total distance AB = $D$.
• First one-third: $v_1 = 11\ \text{ms}^{-1}$.
• Second one-third: $v_2 = 2v_1 = 22\ \text{ms}^{-1}$.
• Last one-third: $v_3 = 3v_1 = 33\ \text{ms}^{-1}$.

Step 1: Express each segment’s distance.
Each segment is one-third of the total distance: $d_1 = d_2 = d_3 = \frac{D}{3}.$

Step 2: Compute the time for each segment.
$t_1 = \frac{d_1}{v_1} = \frac{D/3}{11} = \frac{D}{33},$ $t_2 = \frac{d_2}{v_2} = \frac{D/3}{22} = \frac{D}{66},$ $t_3 = \frac{d_3}{v_3} = \frac{D/3}{33} = \frac{D}{99}.$

Step 3: Sum the times to get the total time.
$T = t_1 + t_2 + t_3 = \frac{D}{33} + \frac{D}{66} + \frac{D}{99}.$ To add these fractions, find a common denominator. The least common multiple of 33, 66, and 99 is 198. $T = \frac{6D}{198} + \frac{3D}{198} + \frac{2D}{198} = \frac{11D}{198} = \frac{D}{18}.$

Step 4: Compute the average velocity.
$\text{Average velocity} = \frac{\text{Total distance}}{\text{Total time}} = \frac{D}{D/18} = 18\ \text{ms}^{-1}.$ Common Traps & Exam Tip:

1. Mistaking average speed for average velocity: Some students compute the arithmetic mean of the speeds $\tfrac{v_1+v_2+v_3}{3}$, which is incorrect because the times spent at each speed differ. 2. Incorrect fraction addition: A frequent arithmetic error is failing to find a common denominator when summing the segment times. 3. Ignoring the equal-distance split: One must remember that each segment is exactly one-third of the total distance, not arbitrary lengths.

Exam tip: Always write down the definition of average velocity first, then break the journey into segments, compute each time, sum them, and finally divide total distance by total time.