Trains approaching constant acceleraton problem

[Loppyfoot]

Homework Statement

Two trains face each other on adjacent tracks. They are initially at rest, and their front ends are 40m apart. The train on the left accelerates rightward at 1.12m/s². The train on the right accelerates leftward at 1 m/s². How far does the train on the left travel before the front ends of the trains pass?

The Attempt at a Solution

I found that it takes the left train 9.258s to travel 48m. It takes the right train 9.798s to travel 48m as well. This is useless info I think. I also found their final velocities.

But I also think that is useless info. I need some help on how to do this problem.

Thanks!

 

[RoyalCat]

What is the relative displacement between the two front ends? Or, in other words, how far apart are the two front ends at every point in time?

What is $$\vec x_{12}$$?
Find how long it takes for them to 'meet' and then look at how much the one train has traveled over that time.

$$\vec x_{12}\equiv \vec x_1-\vec x_2$$

 

[Loppyfoot]

How do I find how long it takes them to meet?

 

[RoyalCat]

Write out the motion equations for each of the trains and then use relative motion to find how long it takes for them to meet.

Find:
$$\vec x_1(t)$$ and $$\vec x_2(t)$$