Relative motion based problems

Question

A policeparty is chasing a dacoit in a jeep which is moving at a constant speed v. The dacoit is on a motorcycle. When he is at a distance x from the jeep, he accelerates from rest at a constant rate. Which of the following relations is true if the police is able to catch the dacoit?

Moderate

Solution

If police is able to catch the dacoit after time t, then

$\mathrm{vt}=\mathrm{x}+\frac{1}{2}{\mathrm{\alpha t}}^{2}\text{. This gives}\frac{\mathrm{\alpha}}{2}{\mathrm{t}}^{2}-\mathrm{vt}+\mathrm{x}=0\phantom{\rule{0ex}{0ex}}\mathrm{t}=\frac{\mathrm{v}\pm \sqrt{{\mathrm{v}}^{2}-2\mathrm{\alpha x}}}{\mathrm{\alpha}}\phantom{\rule{0ex}{0ex}}\text{For}\mathrm{t}\text{to be real,}{\mathrm{v}}^{2}\ge 2\mathrm{ax}$

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