Let us say the student catch the bus t seconds after the bus start travel.
In this time student runs at 5.5m/s constant speed and bus accelerates at 0.179m/s^2.
When she catches the bus let us say bus has gone xm from its initial position. That means the student has run (42+x) to catch the bus.
Using equations of motion;
`S = ut+1/2at^2`
`x = 0+1/2xx0.179t^2----(1)`
`42+x = 5.5t ---(2)`
`-42 = 0.179/2t^2-5.5t`
`-84 = 0.179t^2-11t`
`0 = 0.179t^2-11t+84`
Solving the above quadratic equation give you;
`t = (11+-sqrt((-11)^2-4xx0.179xx84))/(2xx0.179)`
t = 52.51 OR t = 8.93
There are two answers. Both are valid since they are positive. We must be tactical here. Initially student has more speed than bus because it start from rest. So the student can easiliy overtake the bus and go beyond. But since the bus is accelerating at some time it will overtake the student again. So the required answer is t = 8.93s.