Using a simple model of constant acceleration, a student predicts that the velocity of the object, exactly 5 seconds after being released from rest, is $5g{m\; s}^{-1}$. Show how the student has obtained this prediction.

Using a refined model that accounts for air resistance, it is assumed that the object's acceleration, $a{m\; s}^{-2}$, at time $t$ seconds after release is given by

where $v{m\; s}^{-1}$ is the velocity of the object at time $t$. Find an expression for $v$ in terms of $t$.

Compare the behavior of $v$ as $t\to \infty$ for both models.