Find the tangent line that passes through the origin

[Gamerex]

Homework Statement

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?

Homework Equations

Let the subscript b mean bus, and let the subscript p mean student (p for pupil).

s_p(0)=0 m
s_b(0)=40 m

v_p(t)=Unknown

The acceleration of the bus was given, I used calculus to find the velocity and position.
s_b(t)=0.085t²+40 m
v_b(t)=0.17t m/s
a_b(t)=0.17 m/s²

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 s_p(t)=v(t)*t+0

I managed to figure out that s_p(t) must be the equation of the tangent line of s_b(t) which passes through the origin.

How do I find the equation of the tangent line of s_b(t)=0.085t²+40 m (And thus, the velocity) that passes through the origin?

 

[Xisune]

Hmm, I didn't have to use calculus to solve this question. Set the displacements of the student and bus equal, then work from there.

 

[Gamerex]

I don't know the displacement without knowing the velocity...

 

[Xisune]

I mean, take the equations of the displacement of the student and the bus, then set them equal.

 

[Gamerex]

You mean like so?:
s_p(t)=s_b(t)

-> v(t)*t=0.085t²+40,
-> v(t) =(0.085t²+40)/t


This tells me what the velocity would be at my intersection point, but I still don't have the time, t.

 

[Xisune]

Yes, now what kind of equation do you have from there?

 

[Gamerex]

It's a rational equation that gives me the velocity of the student.. If I set that equal to the velocity function of the bus, I can find the time where they intersect!

0.17t=(0.085t2+40)/t

t=21.7 s.. Thank you so much!

 

[Xisune]

Ha, I was actually going with a quadratic equation and setting √(v² - 4ad) = 0, but your method works as well. =)