What is the speed of 1.4-kg cart right after the separation?

[emily081715]

Homework Statement

A system consists of a 5.0-kg cart and a 1.4-kg cart attached to each other by a compressed spring. Initially, the system is at rest on a low-friction track. When the spring is released, an explosive separation occurs at the expense of the internal energy of the compressed spring. The change in the spring's internal energy during the separation is 1.0 kJ.
What is the speed of 1.4-kg cart right after the separation?

Homework Equations

k=1/2m_av²+ 1/2 m_bv²

The Attempt at a Solution

-5v₁=1.4v₂
v₁=-1.4v₂/5

1000J=(m_av₁²)/2 +(m_bv₂²)/2
2000J=5(-1.4v₂/5)+1.4v₂²
v₂=√2000/(24/5)
v₂=20.4 m/s

 

[Doc Al]

1000J=(m_av₁²)/2 +(m_bv₂²)/2
2000J=5(-1.4v₂/5)+1.4v₂²

Redo that second equation, making sure to square things properly.

 

[emily081715]

2000J=5(-1.4v2²/5)+1.4v22

 

[emily081715]

2000J=5(-1.4v2²/5)+1.4v22

i caught my error the answer is 33.4 m/s

 

[Doc Al]

2000J=5(-1.4v2²/5)+1.4v22

I would write it as: 2000J=5(1.4/5)²v₂² + 1.4v₂²