How Do You Solve for the Mass and Velocity of a Cart in an Elastic Collision?

[PaleRider09]

Homework Statement

A cart with mass 340 g moving on a frictionless linear air track at an initial speed of 1.2 m/s undergoes an elastic collision with an initially stationary cart of unknown mass. After the collision, the first cart continues in its original direction at 0.66 m/s.
(a) What is the mass of the second cart?
(b) What is its speed after impact?
(c) What is the speed of the two-cart center of mass?

Homework Equations

I'll use this space to organize variables:
m₁ = 0.34 [kg]
v_1i = 1.2 [m/s]
v_1f = 0.66 [m/s]

The Attempt at a Solution

Okay using the Conservation of Momentum model, I have:

m₁v_1i = m₁v_1f + m₂v_2f

So in order to find the mass of the second cart, I have:

(m₁(v_1i - v_1f))/v_2f = m₂

And now I'm stuck because I don't have the final velocity of the second cart in order to compute its mass.

 

[Delphi51]

Welcome to PF, Palerider.
You are in a common situation - two unknowns and only one equation.
You must find a second equation before you can solve for the two unknowns. The word "elastic" in the question is your clue. If you don't know what it means, look it up! "elastic collision" in Wikipedia will likely work.

 

[PaleRider09]

Thank you for the welcome.
Okay so an elastic collision means that K is conserved throughout the collision.

Therefore:
(.5)m₁v_1i² = (.5)m₁v_1f² + (.5)m₂v_2f²

But that leaves me in the same situation as before doesn't it?

 

[Delphi51]

Looks good. Now you have two equations with two unknowns. You should be able to solve for both the unknown mass and unknown final velocity. Solve the simpler equation for m2, then sub that into the other equation.