Question

A police party 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 are able to catch the dacoit?

• A. $v^2\le \alpha x$
• B. $v^2\le 2\alpha x$
• C. $v^2\ge 2\alpha x$
• D. $v^2\ge \alpha x$

Answer 1

The correct option is C $v^2\ge 2\alpha x$

Net after time ${}^{\prime }{t}^{\prime }$$ , police catches dacoit.

$\Rightarrow$ Distance travelled by Dacoit $=$ Distance travelled $+x$ time ${}^{\prime }{t}^{\prime }$ by police in time ${}^{\prime }{t}^{\prime }$

$\Rightarrow vt=\frac{1}{2}a{t}^2+x\Rightarrow 2vt=a{t}^2+2x\Rightarrow a{t}^2-2vt+2x=0$

For real ${}^{\prime }{t}^{\prime }{,}^{\prime }{D}^{\prime }$ of the quadratic equation should be $\ge 0$

$\Rightarrow D\ge 0\Rightarrow 4v^2-8\alpha x\ge 0\Rightarrow v^2\ge 2\alpha x$