Derive % change Kinetic energy eqn for inelastic collision

[ArtVandelay]

Homework Statement

We did a lab in my PHYS with Caclulus I class involving a collision of cart A (given an initial push) and cart B (initially at rest) on a relatively smooth surface. *At the moment of the collision, the two carts become attached, providing a completely inelastic collision*. One of the post-lab questions asks the following:

Derive the equation below from first principle (don't work backwards from the answer). Start from the definition of %Δ and then plug in what you know about this type of collision (you may include that mass B starts from rest).

%Δ K_system = ( -m_B / (m_A + m_B) ) * 100

^eqn I need to derive

Homework Equations

%Δ = 100 * ( ( measured_final - measured_initial ) / measured_initial )

And I assume I should use K = 0.5mv^2

The Attempt at a Solution

%ΔK_system = 100 * ( ( K_sysf - K_sysi ) / K_sysi )

%ΔK_system = 100 * ( ( K_Af + K_Bf ) - ( K_Ai + K_Bi ) ) / ( K_Ai + K_Bi )

Since cart B is initially at rest, it has no Ki:

%ΔK_system = 100 * ( ( K_Af + K_Bf ) - ( K_Ai ) ) / ( K_Ai )

I will omit all of the one-halves (0.5's) as I replace all "K"s with (0.5)mv^2, since they will all clearly cancel out:

%ΔK_system = 100 * ( m_Av_Af² + m_Bv_Bf² - m_Av_Ai² ) / ( m_Av_Ai² )

And I'm drawing a blank at this point.

 

[lewando]

Use the conservation of momentum equation.

 

[ArtVandelay]

Use the conservation of momentum equation.

Worked like a charm, thank you.