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02 April 2013

Avoiding the negative sign in impulse = change in momentum

Classic question: I have two identical balls.  One is a "happy ball" which rebounds upon impact with the floor, reaching almost the same height from which it was dropped; one is a "sad ball" which essentially stops upon hitting the floor.  It's a reasonable assumption that upon hitting the floor, both remain in contact with the floor for about the same amount of time.

Which ball applies a larger force to the floor?

The correct approach, of course, is to use the impulse-momentum formula.  The time of collision is the same for each ball, so whichever ball experiences a larger change in momentum applies a larger force.

So which ball changes its momentum by more?  It's often useful to make up values here... assume 1 Ns of momentum for both balls right before they collide.

The classic approach is to define up as the positive direction, and to define the change in momentum as pfinal - pinitial.  The sad ball has a momentum change equal to (0 - (-1 Ns)) = 1 Ns; the happy ball has a momentum change equal to (1 Ns - (-1 Ns) = 2 Ns.  The happy ball changes its momentum by more, thus applies a larger force to the ground.

I don't believe in negative signs where they can be avoided.  So how have my freshmen answered this same question without the negative sign?

They recognize that the sad ball goes from 1 Ns to 0 Ns, so changes its momentum by 1 Ns.  That's easy.

The happy ball also starts the collision by coming to rest, losing 1 Ns of momentum.  But then, the happy ball has to rebound, too.  It has to not only lose its original 1 Ns to stop, but it also has to gain 1 Ns in order to get back up to speed.  So its total momentum change is 2 Ns: 1 Ns to stop, 1 Ns to rebound.

Explained this way (and then practiced a few times), the students are entirely comfortable with this conceptual approach.  The follow up question is to make sure they understand why the happy ball doesn't have zero momentum change -- after all, the happy ball had the same amount of momentum before and after collision!  But they quickly recognize that "zero momentum change" means the object never collided with anything.  Rebounding takes more momentum change than just stopping.

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