A body is at rest at $x = 0$. At $t = 0$, it starts moving in the positive $x$-direction with a constant acceleration. At the same instant another body passes through $x = 0$ moving in the positive $x$-direction with a constant speed. The position of the first body is given by $x_1(t)$ after time $t$ and that of the second body by $x_2(t)$ after the same time interval. Which of the following graphs correctly describes $(x_1 - x_2)$ as a function of time $t$?
Answer: (B) see figure
Concept. Write both positions and subtract: $x_1 = \dfrac12at^2$, $x_2 = vt$.
Step 1. Difference.
$$x_1 - x_2 = \frac12at^2 - vt$$
Step 2. Shape. An upward-opening parabola through the origin: negative at first (the second body is ahead), minimum at $t = \dfrac{v}{a}$, zero again at $t = \dfrac{2v}{a}$, then positive.
Step 3. Choose. Only graph (2) dips below the axis and then rises.
Answer. Graph (2) (Option B).
Solution by Sreeraj P, M.Sc Physics
