Why does elastic collision of identical bumper cars have two solutions?

[photon184739]

In an elastic collision, a 400-kg bumper car collides directly from behind with a second, identical bumper car that is traveling in the same direction. The initial speed of the leading bumper car is 5.60 m/s and that of the trailing car is 6.00 m/s. Assuming that the mass of the drivers is much, much less than that of the bumper cars, what are their final speeds?

Name "v" the unknown final speed of the 'leading' bumper care and "w" to the unknown final speed of the 'following' bumper car.

400kg*5.60 m/s + 400kg*6.00 m/s = 400kg*v + 400kg*w (conservation of momentum)
1/2*(400kg)*(5.60 m/s)^2 +1/2*(400kg)*(6.00 m/s)^2 = 1/2*400kg*v^2 + 1/2*400kg*w^2 (conservation of energy)

5.60 m/s + 6.00 m/s = v + w
5.60 m/s + 6.00 m/s = v^2 + w^2

The above system of 2 equations has the following 2 solutions:

https://www.wolframalpha.com/input/?i=5.6+6.0=v+++w,+5.6^2+++6.0^2+=+v^2+++w^2

solution 1: v=5.6, w=6.0
solution 2: v=6.0, w=5.6

Why are there 2 solutions? What additional constraint is needed to know that the leading bumper car's final speed v = 6.0 and the following bumper car's speed w = 5.6?

 

[Ibix]

All the maths does is find values of $v$ and $w$ so that the total momentum and energy are equal to their initial values. Do you notice anything about one solution for $v$ and $w$ compared to the initial values of velocity?

 

[Dale]

Summary:: conservation of momentum and energy problem

Why are there 2 solutions?

Both energy and momentum are conserved either if they collide elastically or if they miss and don’t collide at all.

 

[Vanadium 50]

In addition to any other responses you get, if you need Wolfram Alpha to solve two equations in two unknowns, you need to up your algebra game if you want to succeed in physics. (There's also at least one more mistake)

 

[Ibix]

(There's also at least one more mistake)

Indeed, although it appears to be a typo since the equations as written have complex roots. @photon184739, maths is a lot easier to understand on forums if you use LaTeX, and hence you are less likely to make typos. Check out the guide linked below the reply box.

 

[kuruman]

Summary:: conservation of momentum and energy problem

Why are there 2 solutions? What additional constraint is needed to know that the leading bumper car's final speed v = 6.0 and the following bumper car's speed w = 5.6?

The additional constraint is imposed by common sense. As long as there has been a collision, the following car can only be "following" if it is moving slower than the "leading" car.