I get two different answers for the distance. Which one is right?

Question

This is very basic but i can't see why this is wrong. $a =\frac{dv}{dt}$ $\rightarrow$ $adt = dv$. Integrating both sides yields $v= at$. Inserting in $s=vt$ $\rightarrow$ $s=(at^2)$ which is wrong. I don't understand why it is wrong. I know if I take $v = at$ and multiply by $dt$ on both sides and integrate on both sides i get $s = \frac{(at^2)}{2}$ which is right. But why is the first answer wrong?

Answer 1

When $v$ is varying, we take mean velocity

$$ a = \dfrac {(v_{max}+v_{min})/2}{t} $$

If you mean $v_{max}= v , v_{min}= 0 $

then the factor $2$ explains itself.