Elastic Collisions: What Went Wrong in My Calculation?

[fuzzish]

Just to begin, uhh. I'm 75% sure my work makes logical sense [I hope], and that my problem lies in a computational error. But I really don't know why my answers aren't working out. I've now gone through the problem in 3 different ways, and I keep getting the same answer, but it's apparently still incorrect. If it is an arithmetic error, sorry for making you read through all of this to fix such a stupid mistake >.>

Homework Statement

A 12.7g object moving to the right at 26.5cm/s overtakes and collides elastically with a 10.5g object moving in the same direction at 10.7cm/s.

Find the velocity of the faster object after the collision. Answer in units of cm/s.

Find the velocity of the slower object after the collision. Answer in units of cm/s.

Homework Equations

m1v1 + m2v2 = m1v3 + m2v4

.5m1v1^2 + .5m2v2^2 = .5m1v3^2 + .5m2v4^2

And, sorry in advanced for the ugly way that equation appears. And for the work that is likely to look just as messy.

The Attempt at a Solution

m1 = 12.7g ; v1 = 26.5cm/s
m2 = 10.5g ; v2 = 10.7cm/s

I plugged those two into the first equation and got:
448.9 = 12.7v3 + 10.5v4

Rearranged for v3:
v3 = 35.35 - 0.827v4

I then plugged in the m1, v1, m2, and v2 into the second equation and got:
5060.36 = 6.35v3^2 + 5.25v4^2

Plugged in the v3 from the previous equation:
5060.36 = 6.35 ( 35.35 - 0.827x )^2 + 5.25 x^2

AND HEREEEE I FIND MY PROBLEMO.
My TI-89 keeps giving me two answers:
v3 = 28 [in which case v4 = 8.89]
or
v3 = 10.7 [and thus v4 = 29.8]

Problemo Uno: Why are there two answers? ? And both seem to work when plugged back into the original equations? This point makes me most uncomfy and confused.

Problemo Dos: If it's an elastic collision, shouldn't one of the objects be moving in a negative direction now? Perhaps my thought process is wrong here though, but idk.

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Thanks in advanced to whoever offers help yay. And I know I'm a noob leeching off these forums, so just so I don't feel like I'm nomming up the answer and running away, I'll stay around to help someone else if I can xD.

 

[vela]

Your set-up is correct, but neither of your solutions work when I plug them in here.

 

[fuzzish]

Auurrghhghghghhhh. Fifth round of calculator plug and chug, and I still got 2 answers, but the one that worked worse out of the two turned out to be the right pair. This. Is. Really. Weird. Maybe I need a new calculator, idk.

Anyways, thanks for rechecking my set-up :].

 

[vela]

By the way, for one-dimensional elastic collisions, you can combine the two equations to come up with a third one that relates the objects' relative velocities. It's linear so you can avoid the quadratic messiness. Check your notes or your book for its derivation.