What is the speed of each train at impact?

[Source]

So, I searched and found one almost identical to this, but no direction towards solution was posted, and I couldn't make sense of the formula in the original post. So...

A red train traveling at 72km/h and a green train traveling at 144km/h are headed toward one another along a straight level track. When they are 950m apart, each engineer sees the other's train and applies the brakes. The brakes slow each train at the rate of 1m/s^2.

a) Is there a collision?
b) If so, what is the speed of each train at impact?

So, I know that they collide, and that the red train has stopped, but I'm not sure where they collide. I know that the red train stops after 20 seconds, and that the green train would stop after 40 seconds if it didn't collide with the red one first, but I'm having a huge mental block and can't remember or seem to find the formulas that would help me out here. The book I'm using doesn't seem to have them either.

Can anyone help me out here? I feel really stupid because I *KNOW* this is relatively simple, I just can't wrap my brain around it right now.

 

[Pyrrhus]

Well finding the time it will take to stop the red train in 950 m and finding the time it will take to stop the green train in the same distance does tell you if there's a collision or not. To find the speeds of the trains at impact you would need to make some considerations with the results from a). Remember to apply uniform acceleration equations obviously.

 

[Source]

Finding the time it takes to stop doesn't, but considering that their closing velocity is 60m/s, and that they're slowing at a combined rate of 2m/s^2... If they didn't slow down at all, they would collide in 15.83 seonds. Bah, I can't remember my sum formulas from HS Calc either... I'll have to look back through my notes to see if I can find any of those since something tells me they would be helpful.

 

[Pyrrhus]

Have you considered the distance traveled by each train when it stopped?, it's a determinant factor in order to find if there was a collision. Model each train as a point-particle individually and devise the conclusions from it.

 

[Source]

Using formula:

d=(v_i*t)+(.5at^2)



d_greentrain = 800m ((40*40)+(.5(-1)(40)^2))=800)
d_redtrain = 200m ((20*20)+(.5(-1)(20)^2))=200)

Since d_g+d_r = 1000 and 1000m > 950m, the trains collide.

As to where... since the red train stops after only 200m, it is a safe assumption that it has come to a complete stop by the time the green train hits. 950-200 = 700m that the green train has to travel. So...

700=(40*t)+(.5(-1)t^2)

If I solve for t, I will get the time it takes for the green train to travel 700m. Once I know how much time that takes, I can plug that into the formula v_final = v_initial + at and solve for the final speed of the green train. Correct?

[edit]

I can't seem to solve for t. I end up with:

d_g/v_i=t-.5t^2

There seems to be no easy way to solve that... unless I'm missing something blatantly obvious?

[edit2] Ok, solved it. Ended up solving for t^2+t-1400=0 using my trusty TI-89. That gave me what I needed. Thanks for the help - you pointed me in the right direction.