# Collisions Deriving some equations for specific situations in the most general forms.

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Collisions Deriving some equations for specific situations in the most general forms

Momentum KE

Solving for the simplest case of equal masses and a stationary target, lets try to find the final velocity of the target as a function of the original speed v1 of the car which strikes it and the masses. If all masses are the same m1 = m2 = m mv1 = mvf1 + mvf2Factor and cancel the mass m v1 = vf1 + vf2 So in this special case, the mass doesn’t matter! Now I’d like to vf2 as a function of v1 only. So I want to eliminate vf1, by using conservation of kinetic energy (elastic case). KE initial = KE final ½ mv1 2 = ½ mvf1 2 + ½ mvf2 2 Factor out and divide away the ½ and m v1 2 = vf1 2 + vf2 2

v1 = vf1 + vf2 Solve simultaneously to eliminate Vf2 We solve the left equation and plug into the right or vice versa. Which do you think will be less work? V1= vf1+ √v1 2 - vf1 2 v1 2 = { vf1+ √v1 2 - vf1 2 v1 2 - vf1 2 = vf2 2 √v1 2 - vf1 2 = vf2 Take square root and substitute To group like terms, square both sides } x {vf1+ √ v1 2 - vf1 2 } Now FOIL v1 2 = vf1 2 + 2vf1 √ v1 2 - vf1 2 + (v1 2 –vf1 2 ) 0 = 2vf1 √(v1 2 - vf1 2 ) v1 2 = vf1 2 + 2vf1 √v1 2 - vf1 2 + (v1 2 –vf1 2 ) Which can only happen if Vf1 = 0 v1 = vf1 + vf2 v1 = vf2 This means that the incoming car stops on collision and the target car goes off with all the speed the incoming car had.

What the CoM is doing? It is moving rightward at constant speed.

What if the masses aren’t equal?

The above equations will work for any elastic collision where v2 starts at rest as a stationary target, no matter what the initial masses.

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