collisions deriving some equations for specific situations in the most general forms
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CollisionsDeriving some equations for specific situations in the most general forms
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Momentum
KE
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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 + mvf2 Factor 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 mv12 = ½ mvf12 + ½ mvf22 Factor out and divide away the ½ and m
v12 = vf12 + vf22
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v12 = vf12 + vf22 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+ √v12 - vf12
v12 = {vf1+ √v12 - vf12
v12 - vf12 = vf22
√v12 - vf12 = vf2
Take square root
and substitute
To group like terms, square both sides} x {vf1+ √ v12 - vf12 }
Now FOIL
v12 = vf12 + 2vf1 √ v12 - vf12 + (v12 –vf12)
0 = 2vf1 √(v12 - vf12 )
v12 = vf12 + 2vf1 √v12 - vf12 + (v12 –vf12)
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.
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What the CoM is doing?
It is moving rightward at
constant speed.
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What if the masses aren’t
equal?
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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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