CollisionsDeriving 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 + 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

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.

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.