How Long Does a Momentum “Exchange” Take?

1. HowLong Does a Momentum “Exchange” Take? In every case we’ve ever observed where two objects, A and B, interact, their changes in linear momentum were equal in magnitude but opposite in direction:PB = –PA These individual changes in momentum PA, and PB, are called impulses. And each exchange of momentum involves a pair of impulses—equal in magnitude but opposite in direction. Question: The interaction that produced each pair of impulses lasted just a short while. Then the masses either separated or became one mass. But what about that “short while?” Was it the same amount of timefor each mass? In other words, can object A stop touching object B before B stops touching A? OSU PH 211, Before Class 13

2. Conclusion: In an exchange of impulses, whatever A does to B, B does just the oppositeto A—and in the same amount of time. But is time the only factor that determines the amount of P each object gets? Can a 1-second interaction give, say, 10 kg·m/s of impulse to object A (and of course, –10kg·m/s to B), but then another 1-second interaction could give 20 kg·m/s? Or 100? Of course. Clearly, the amount of P depends both on the duration and on the intensity of the interaction. The duration, t, is a scalar quantity (time) that is the same for both A and B. The “intensity,” therefore, must be a vector: equal in magnitude but acting oppositely in direction on each object, in order to produce the respective vector directions of the P for each object. This vector “intensity” of interaction is what we call a force, F. Thus: P = Ft OSU PH 211, Before Class 13

3. Now let’s get very specific with names: We name a force FAB when it is the action by object A on object B, over some time interval, t, which causes a change in B’s momentum, PB: FABt = PB And FBA is the action by object B on object A, over the time interval, t, which causes a change in A’s momentum, PA: FBAt = PA But we have already observed that (because momentum is conserved in this universe), PB = –PA In other words: FABt = –FBAt But matching impulses happen in the same t, so: FAB = –FBA Conclusion: Anytime there is a force, there are actually two forces—a matched pair:FAB = –FBA This is Newton’s Third Law. OSU PH 211, Before Class 13

4. Newton’s Third Law in words: “For every action, there is an equal but opposite reaction.” It describes the very nature of a force: A force must have two parties participating—with a force acting equally but oppositely on each party. Again, notice the units of force…. OSU PH 211, Before Class 13