A police party is moving in a jeep at a constant speed $v$. They saw a thief at a distance $x$ on a motorcycle which is at rest. The moment the police saw the thief, the thief started at constant acceleration $\alpha$. Which of the following relations is true if the police is able to catch the thief?
Answer: (C) $v^2 > 2\alpha x$
Concept. The police catch the thief if their positions become equal for some real time $t$.
Step 1. Meeting condition.
$$vt = x + \frac12\alpha t^2 \;\Rightarrow\; \alpha t^2 - 2vt + 2x = 0$$
Step 2. Real root. The discriminant must be non-negative:
$$4v^2 - 8\alpha x \geq 0 \;\Rightarrow\; v^2 \geq 2\alpha x$$
Among the options, the police catch the thief when $v^2 > 2\alpha x$ (at equality they just reach him).
Answer. $v^2 > 2\alpha x$ (Option C).
Solution by Sreeraj P, M.Sc Physics