How to Solve a Differential Equation Problem?

[Impact215]

[SOLVED] Differential Equation Problem

Hi, I am having a problem with a question in my Differential Equations class.

Homework Statement

Two drivers (A and B) are going to race from a standing start. Both leave at the same time and both have constant accelerations. Driver A covers the last 1/4 of the track in 3 seconds while driver B covers the last 1/3 of the track in 4 seconds. Who wins and by how much?

I already found a solution on this site at:

https://www.physicsforums.com/showthread.php?t=209021

I understand everything in his solution up until i get to this equation

$$\frac{1}{4}x = \sqrt{\frac{3a_ax}{2}}(3) + \frac{1}{2}a_a(9)$$

I do not know how to solve for $$a_a \ in \ term \ of \ x: a_a = 0.0039887x; \ \ \ \ 0.77379x...(5)$$

Can someone show me how this is done?

I was able to figure out how he solved this, I think my problem was that I was substituting a value for x, rather than just leaving it as x.

I used the quad. formula with:
a = 324
b = 252x
c = x^2

Homework Equations

The Attempt at a Solution

The solution to the answer from the book is Driver B wins by $$6\sqrt{3} - 4\sqrt{6}$$ sec which is approximately 0.594 sec

 

[Impact215]

Now that i figured out how he got his answer, is there another way to do this using integration? possibly using $$a = d^2 x / dt^2$$ ?

 

[Dick]

You use integration to derive the result that x(t)=x(0)+x'(0)*t+(1/2)*a*t^2 from the differential equation of which you speak.