You can set up two equations in two unknowns, then solve for those unknowns. First you have the fact that the student and bus must be at the same place at the same time. Let's call the student's velocity 'v' and the time it takes to catch the bus 't'  the equation that has them at the same place at the same time is:
vt = 20 + (1/2)at^2
The second condition is that the student just barely catch the bus before it accelerates away  this is equivalent to saying that at the moment the student reaches the bus the velocity of the student = the velocity of the bus:
v = at
You know the value for 'a', so now you can solve for 'v' and 't'.
