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## 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

_{A}v

_{Af}

^{2 }+ m

_{B}v

_{Bf}

^{2 }- m

_{A}v

_{Ai}

^{2}) / ( m

_{A}v

_{Ai}

^{2})

And I'm drawing a blank at this point.

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