Question

A car is chasing a motorbike on a straight road. The car is going at its maximum speed $v$ (assumed uniform). The person on motorbike started his ride when the car was at a distance $d$ away from him, and the motorbike starts with constant acceleration $a$. What should be the condition for the speed of the car $v$, so that motorbike will be caught?

• A. $v\le \sqrt{ad}$
• B. $v\le \sqrt{2ad}$
• C. $v\ge \sqrt{2ad}$
• D. None of these

Answer 1

The correct option is C $v\ge \sqrt{2ad}$

Given, speed of the car = $v$
Initial speed of the motorbike, $u=0$
Acceleration of the motorbike = $a$
Initial separation between car and motorbike = d
Let, the motorbike is caught at a time t after it starts. So, the distance travelled by the motorbike during this interval is
$s=ut+\frac{1}{2}a{t}^2=0+\frac{1}{2}a{t}^2=\frac{1}{2}a{t}^2$................(i)
During this interval the car travels a distance $s+d=vt\Rightarrow s=vt-d$.......................(ii)
From eq (i) and (ii) we have
$\frac{1}{2}a{t}^2-vt+d=0\Rightarrow t=\frac{v\pm \sqrt{v^2-2ad}}{a}$
The motorbike will be caught if time is real and positive. So, this is possible only when $v^2\ge 2ad\Rightarrow v\ge \sqrt{2ad}$