Q.The position-time ($x$-$t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 2.9. Choose the correct entries in the brackets below:

House P is nearer the school than Q, and A's line ends at P, so A lives closer. A's line begins at $t=0$ while B's begins later, so A starts earlier. B's line is steeper, so B is faster. Both lines end at the same time, so they reach home together. B's line crosses A's exactly once, so B overtakes A once.

Concept

For one-dimensional motion, an $x$-$t$ graph encodes everything:

• the height of a point gives the position (distance from O),
• the slope $\dfrac{\Delta x}{\Delta t}$ gives the velocity (steeper $=$ faster),
• where two lines intersect, the two bodies are at the same place at the same time (an overtaking event).

Reading each part

1. Who lives closer? House P is marked nearer to O than house Q. A's line terminates at P and B's at Q, and $OP < OQ$, so A's home is nearer. A lives closer.
2. Who starts earlier? A's line leaves the origin at $t = 0$; B's line meets the time axis at a later instant $t_B > 0$, meaning B is still at the school ($x=0$) until then. A starts earlier.
3. Who walks faster? Speed is the magnitude of the slope. B's line is distinctly steeper than A's, so $|v_B| > |v_A|$. B walks faster.
4. Same or different arrival time? Both straight lines end at the same value of $t$ (their upper end-points lie on the same vertical), so the two children arrive at their respective homes at the same instant.
5. Overtaking. The two lines intersect exactly once. Since B starts behind A in position (B leaves later from O) but is faster, B catches up, and at the crossing B draws level with and then passes A. Hence B overtakes A, once.
   ✓Final answer

   1. A lives closer to the school than B.
   2. A starts from the school earlier than B.
   3. B walks faster than A.
   4. They reach home at the same time. (e) B overtakes A once.

Concept: Verifying a Qualitative Graph-Reading Conclusion with a Concrete Numerical Model

Method: Build a Consistent Numeric Example, Then Check Each Statement by Calculation

The description (steeper/gentler slopes, later start, single crossing, simultaneous arrival) can be read directly off the sketch. This method instead manufactures one concrete, numerically consistent version of that graph and checks every one of the five statements by plugging in numbers — a useful cross-check technique whenever a "read the graph" answer needs to be verified rather than just eyeballed.

Steps

1. Assign consistent numeric values. Let house P (child A's home) be at $x=1\ \text{km}$ and house Q (child B's home, farther) at $x=2\ \text{km}$. Let A leave the school at $t=0$ and both children arrive home together at $t=20\ \text{min}$; let B leave later, at $t=5\ \text{min}$.

2. Write each child's position as a straight line, from these numbers:

3. Check (c): who walks faster? Compare slopes: $v_B = 0.133\ \text{km/min} > v_A = 0.05\ \text{km/min}$. B is faster — confirmed numerically.

4. Check (e): do the lines cross exactly once, and who overtakes whom? Set $x_A(t) = x_B(t)$:

Two straight lines with different slopes intersect at exactly one value of $t$ — algebraically guaranteed, matching the "crosses once" reading. At $t\approx8$, $x_A\approx0.40\ \text{km}$, which lies between $0$ and $1\ \text{km}$ (before A reaches home) and between $0$ and $2\ \text{km}$ (before B reaches home) — a physically valid crossing point. Since B starts behind A but is faster, B is behind before $t\approx8$ and ahead after — B overtakes A, once.

5. Check (a), (b), (d) directly from the assigned numbers: A's home ($1\ \text{km}$) is nearer than B's ($2\ \text{km}$) — A lives closer; A leaves at $t=0 < 5\ \text{min}$ — A starts earlier; both $x_A(20)=1$ and $x_B(20)=2$ are reached at the same $t=20$ — same arrival time.

Why this cross-check is useful

Any numeric assignment consistent with the sketch's qualitative features (P nearer than Q, B starts later, both arrive together) will reproduce the same five conclusions — because those conclusions follow from the geometry (relative slopes, starting points, endpoint), not from the particular numbers chosen. Getting the same answers from an arbitrary consistent example is strong evidence the qualitative reading is correct.

Final Answer

1. A lives closer to school;
2. A starts earlier;
3. B walks faster;
4. they arrive at the same time; (e) B overtakes A once — confirmed by direct computation on a concrete numeric model of the graph.