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 Ksystem = ( -mB / (mA + mB) ) * 100
^eqn I need to derive
%Δ = 100 * ( ( measuredfinal - measuredinitial ) / measuredinitial )
And I assume I should use K = 0.5mv^2
The Attempt at a Solution
%ΔKsystem = 100 * ( ( Ksysf - Ksysi ) / Ksysi )
%ΔKsystem = 100 * ( ( KAf + KBf ) - ( KAi + KBi ) ) / ( KAi + KBi )
Since cart B is initially at rest, it has no Ki:
%ΔKsystem = 100 * ( ( KAf + KBf ) - ( KAi ) ) / ( KAi )
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:
%ΔKsystem = 100 * ( mAvAf2 + mBvBf2 - mAvAi2 ) / ( mAvAi2 )
And I'm drawing a blank at this point.