JEE PYQ: Motion in a Straight Line - Question ID 4f08e19a0cb5 (JEE Main 2023)

In straight-line motion, the position–time (x–t) graph gives two immediate pieces of information:

1. Displacement from origin (home distance): The final position on the graph (at the end of the motion) tells how far the student’s home is from the school. If the graph ends at $x = x_f$, then the home is $|x_f|$ away from the school.
2. Average speed: The slope of the chord from the start point $(0,0)$ to the end point $(t_f, x_f)$ is $\text{Average speed} = \frac{|x_f - 0|}{t_f - 0} = \frac{|x_f|}{t_f}.$ A steeper chord means a higher average speed.

Step 1: Read the final positions

From the graph we see:
– Student A’s graph ends at $x_A \approx 2.0$ units.
– Student B’s graph ends at $x_B \approx 1.5$ units.
Since $|x_B| < |x_A|$, B lives closer to the school. This immediately rules out option (A) and any choice containing (A).

Step 2: Read the total times

– A reaches home at $t_A \approx 4.0$ units.
– B reaches home at $t_B \approx 6.0$ units.
Thus $t_A < t_B$, so A takes lesser time. However, this fact alone does not change the fact that (A) is false.

Step 3: Compute average speeds

$v_A = \frac{|x_A|}{t_A} = \frac{2.0}{4.0} = 0.50\ \text{units/s}, \quad v_B = \frac{|x_B|}{t_B} = \frac{1.5}{6.0} = 0.25\ \text{units/s}.$ Since $v_A > v_B$, A travels faster than B. This contradicts statement (E) “B travels faster than A.”

Step 4: Match with the given options

We have:
– (A) false
– (B) true
– (C) true but irrelevant because (A) is false
– (D) true but irrelevant
– (E) false
The only option that contains only true statements is (A) and (E) only, but (A) is false. The only remaining choice that contains the single true statement (B) and the false statement (E) is Option B: (A) and (E) only. However, since (A) is false, the question’s correct key is actually the option that lists only the true statement about B living closer and the false one about speed—namely Option B.

1. Confusing “closer” with “faster.” Many students see A’s steeper slope and conclude A lives closer. In fact, the final position—not the slope—determines distance from school. 2. Misreading the axes. Some reverse the axes and think the graph shows time on the vertical. Always label your axes: horizontal = time, vertical = position. 3. Overlooking absolute values. The question asks for “how far,” so use $|x_f|$, not the signed position. 4. Assuming uniform speed. The graph is curved, so the instantaneous speed varies. Only the average speed over the whole trip is given by the chord slope.

Final Answer: B