Question

A police inspector in a jeep is chasing a pickpocket on a straight road. The jeep is going at its maximum speed v (assumed uniform). The pickpocket rides on the motorcycle of a waiting friend when the jeep is at a distance d away, and the motorcycle starts with a constant acceleration a . Show that the pickpocket will be caught if $v\le \sqrt{2ad}$

Answer 1

For a given quadratic equation,if you have to determine the roots are real,then you use the Discriminant $(D)>=0

Here also,in this case,since we have a quadratic equation in t

$a{t}^2-2vt+2d=0$

And now, for the pick pocket to be caught,this equation should result in a real value of t.

Now for the given quadratic equation,$a{x}^2+bx+c=0$

$D=b^2-4ac$

so, in the given quadratic equation ,$b=-2v$and$a=a$ and $c=2d$

Hence$D=(-2v{)}^2-4\times a\times 2d>=0$

$4v^2-8ad>=0$

$v^2>=2ad$

$v>=\sqrt{2ad}$

Hence proved.