Momentum is King!
About a week ago, I saw a great tweet by Andy Rundquist describing Newton’s laws in terms of momentum.
Andy’s three laws of momentum were:
- Particles swap momentum.
- Interactions determine swap rate.
- If you don’t swap, you don’t change.
There’s something incredibly pleasing about seeing Newton’s laws in this form. If I want to get all zen about it, I would say the emphasis on swapping and interactions remind us how how connected everything is, while Newton’s laws as they are regularly phrased tend to focus on an isolated particle.
As a reminder, in my class, we state Newton’s 3 laws as:
- An object’s velocity will remain constant unless it is acted upon by an net force.
- The acceleration of an object is proportional to the net force and inversely proportional to the inertia of the object.
- If object A exerts a force on object B, then object B will exert a force on object A. These two forces will be of the same time, equal in size, and opposite in direction.
As an exercise, I asked my students to figure out which of Newton’s Laws corresponded to each of Andy’s laws, and the kids dispatched this task in short order. We also talked about why swap was such a good word in these laws, rather than something else like “give each other” or ” trade.” Swap implies a 1 for 1 exchange. If particle gains X POMs of momentum from particle B, particle B must lose X POMs of momentum.
Then we sort of got on a tangent, and began to talk about the a person on the earth.
So what is the change in momentum of this person. Some students quickly jump to the conclusion that it’s zero, since the speed doesn’t change. The picture helps them to quickly see this mistake.
How fast is the person moving anyway? Here’s a great chance for us to do some simple estimation. Let’s let the earth be 25000 miles in circumference, and suppose that there are about 2000 m in every mi. So, we have that the person must travel about m in 24 hours (approximately 100,000 seconds), which gives a speed of about 625 m/s (compare this to 723 m/s for the exact answer).
So what’s the change in momentum between these two times? Let’s assume a mass of 50kg, and that motion to the left is negative.
So then I ask, what would happen if a person in this room suddenly received 60,000 POM of momentum? “They would die”, is the rather obvious response from my students. Pretty quickly, my student sees that the reason we don’t get hurt, even though our momentum changes by this much is that the transfer takes place over 12 hours. So
So the force required to cause this change in momentum turns out to average 1.2 N to the right. Soon my students compare this to the gravitational force, 500 N, and realize it’s tiny. And then we think how this force comes to be, and my students recognize it must be that the normal force is just slightly smaller than the gravitational force, so over any small enough time interval, your velocity is approximately constant, but over large enough time intervals, the change is significant. And we get all this before we’ve done anything with centripetal motion. Momentum is king!
Finally, as a special treat, Andy put together this great screencast that explains how conservation of energy really is just a result of momentum conservation (There’s a tiny bit of calculus here).
So I wonder, could I retool the introductory course to start with momentum, and use that as the central principle from which we explain a whole range of phenomena? Possibly, but I think there would be significant hurdles to overcome, particularly in the abstract nature of these ideas. But this seems to be idea for an second year AP course.