A man is 45 m behind the bus when the bus start accelerating from rest with acceleration $2.5 {\mathrm{ms}}^{-2}$. With what minimum velocity should the man start running to catch the bus?

Let the man will catch the bus after t second. So, he will cover distance ut.
Similarly, distance travelled by the bus will be $\frac{1}{2}{\mathrm{at}}^2$.

For the given condition,

                     $\left.ut=45+\frac{1}{2}a{t}^2=45+1.25t^2 \text{ (As, }a=2.5{ \mathrm{ms}}^{-2}\right)\frac{ }{ }$

$\Rightarrow u=\frac{45}{t}+1.25t$

To find the minimum" value of u

           $\frac{du}{dt}=0\Rightarrow \frac{du}{dt}=\frac{-45}{t^2}+1.25=0$

So, we get t = 6s, then $u=\frac{45}{6}+(1.25\times 6)=15{ \mathrm{ms}}^{-1}$