This problem is 2.95 of University Physics, 11th edition.
Catching the Bus: A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m/s2.
There are subproblems a,b,c,d,e, and f; which I've all figured out except for f)
f) What is the minimum speed the student must have to just catch up with the bus?
Let the subscript b mean bus, and let the subscript p mean student (p for pupil).
The acceleration of the bus was given, I used calculus to find the velocity and position.
The Attempt at a Solution
I modeled the problem by considering the bus and the student as point particles. The points in time where the student and the bus are at the same place are the intersections of the graphs of the position functions. The position function of the bus is known and given above, but the position function of the student is the integral of the student's velocity, which is constant.
The position function would be of the form sp(t)=v(t)*t+0
I managed to figure out that sp(t) must be the equation of the tangent line of sb(t) which passes through the origin.
How do I find the equation of the tangent line of sb(t)=0.085t2+40 m (And thus, the velocity) that passes through the origin?