“It still works!”
“Can we do this every day?”
Then we went back to the front of the room (desks and whiteboard) to talk about what they had just found; it’s conservation of momentum, where momentum (p) is defined as mass times velocity, or p=mv. Turns out that momentum is conserved in situations like those we explored, which we usually write mathematically as
This means that if one object loses momentum, the other must gain the same amount, and vice versa (I use loses and gains for now even though that’s not fully correct; really, if one changes positively the other changes negatively, but a negative change can be ‘more’ momentum if momentum was negative in the first place. We will revisit this idea later).
The next 20 minutes in class were spent deriving conservation of momentum from Newton’s laws. I gave them instructions just as I wrote I would yesterday;
- Draw a free body diagram (FBD) for each car during the collision
- Set up Newton’s 2nd law for each car
- Play around.
An interesting question came up when they started to draw free body diagrams; should we draw it for the situation we just did or other situations? My answer: Yes. This helped them see that each situation has the same FBD, hence the general model (and I will point this out again on Monday when we reflect on the whole process).
Some students had trouble figuring out which direction each force should be; they like to say that during the inelastic collision the moving cart experienced a force in its direction of motion when it collides with the other cart (presumably because they assume that velcro sticking together means the cart at rest pulls on the moving one), when in fact the moving cart experiences a backward force. I lead them to the correct direction by covering the FBD of one car and asking about what happens to the other. Oh. the cart slows down! So the force must be backward! That is usually all it takes.
Another common problem is students including friction (which is indeed a bit of a problem…external forces!). However, I just ask how big friction is in comparison to the force of the collision. Small? Yep. Ridiculously small, actually. So let’s ignore it for now.
To get to the point where students realize the forces are the same, I brought out my (awesome) hacked together ‘force carts,’ shown at right. How does the force of the red on the blue compare to the force of the blue on the red? The same. What if we change the mass? The same. What about direction? Opposite!
In future years they will have seen the force carts already; I just had this idea recently so they were seeing them for the first time. I did hear the question “Didn’t we learn that already?” Yes, yes you did.
This got most of them to the point below;
Some noticed they didn’t have the negative and asked (because conservation of momentum, with the negative, was on the board), I simply pointed out that the accelerations were opposite in direction and thus a negative for one was necessary to equate them.
What now? Well, what is acceleration? That question tended to be enough to get them to the point of
Many realized right away that the time was the same for each (in fact, most were not labeling them separately for each cart and, without actually thinking about it, assumed they were the same…at which point I said, “Are you sure those times are the same?” They are used to me questioning correct answers at this point, meaning they don’t assume when I question that it means they are wrong. So most of them thought for a bit and then concluded they are in fact the same). For the others I referred the force cart graph and simply pointed out the width of the peaks.
In hindsight, the only think I think I would change about today (and, really, most of this experience) is that I did a fair amount of guiding for this last part, geting them to derive conservation of momentum. I think it would have been good if I didn’t tell them where to start and let them play around a bit more. Next year I plan to do more goal-less problems, which will make this sort of exercise easier, I believe. I did have one group start looking into energy for the derivation, but they quickly hit a dead end. It would have been fun to see students start and finish more or less all on their own.
Well, that’s it for this activity. What an experience! At this point I am even more excited for the summer Modeling training as I have some experience to fall back on, but I really want to gain expertise for implementing modeling more in my regular physics classes.