# Why do we need both momentum and energy?

Since seeing Brian’s post about asking students this question, I’ve been thinking about how my students might answer. In one section of Honors Physics, we’re just wrapping up a second look at momentum and energy transfer (mainly using problems that involve a transition between using both in the same analysis). My Honors kids have written answers on scraps of paper to questions from my “Internet friends” before, so they were ready for another round (they also enjoy signing their responses with their classmates names and the reading of the answers that I do when they’ve all been collected—the voices in the class are pretty unique, so they usually know who has written which response anyway).

Earlier in this unit, one of my students was surprised when he needed to use momentum to solve a problem (a ballistic pendulum). He said, “I thought once we hit energy we were done. I thought we could use energy for everything, now.” Today we thought a bit about how, though we developed all 3 of our fundamental principles (Newton’s Laws, momentum, energy) in the first semester, we spent more than half of the time on only one of them (Newton’s Laws). It makes sense that we’d want to come back to the other two in the second semester and do some fine-tuning of how we talk and thinking about them.

Anyway, here’s what the students wrote in response to the question, “Why do we need both momentum and energy?”

Can make 2 equations that help solve for 2 variables. 2 ways to view/deal w/ situation.

Momentum and energy don’t exist. They are made up tools to model things.

Because I want to… But also because it allows us to look at a situation from multiple views and find more unknown variables.

Treating situations as examples of “Momentum” and “Energy” allows us to more easily understand them… the two don’t actually exist.

Because we care about the transfer of momentum and energy, and when one transfers the other must too, so we can find the change of the other if we know one.

Momentum and energy are completely different even though they are modeled by using the same components they give us different information. BUT to answer your question, we don’t need either. You just make us do physics.

Without momentum and energy, roller coaster would be awful. I mean I don’t need it I could just be a couch potato.

Because through momentum, we can find the impulse thus relating to the average net force.

Because momentum is a Socialist pig and Energy is the savior of society (capitalism).

We need both momentum and energy because energy includes other variables besides mass and velocity. When we need to include other elements (spring force, for example) we use energy, but it’s much simpler to use momentum when only velocity and mass are involved.

We had some more good conversation after reading these responses. I especially liked that more than one student noted that we’ve really just made up momentum and energy. We talked some about being able to cleverly choose snapshots (states) so that there is no change in thermal energy between them and the convenience of having conservation of momentum to pick up the slack. We talked a little bit about infinite sets of numbers (for example: the set of odd numbers is infinite; the set of even numbers is infinite; the two sets have no numbers in common). We thought a very little bit about problems where, while both Conservation of Momentum and Conservation of Energy would each be satisfied by an infinite number of solutions, the two infinite sets would only have one solution in common (the one that actually happens).

When we find a spare bit of time in the next few weeks, I’m hoping to get them thinking about the Newton’s Cradle problem.

## 6 thoughts on “Why do we need both momentum and energy?”

1. Dan d

When my kids ask me why we need both momentum and KE, I ask which would hurt more, being hit by a speeding bullet or a car going 2 mph (about 1m/s). They always say the bullet. A 0.02kg bullet going 300m/s has a momentum of 7 kg-m/s, a 1300kg car going 1m/s has a momentum of 1300, so the car should do far more damage. They’re always perplexed until I suggest we look at KE. The car’s is 650, the bullet’s is 1225. The moral of the story is that momentum is a better model for some things, KE for others.

1. I definitely agree that energy can be thought about in terms of possible pain… 😉 It’s tough to talk too much about it without getting into pressure, though.

Another good “which would hurt more” for momentum is thinking about objects that bounce off (or not) other objects (like heads, etc).

In any case, the change in the quantity is probably more useful to think about than the quantity itself, right?

2. Dan D

I always loved having this conversation with my students. I really feel that it can help tie all kinematics, momentum and energy together and give the kids a better sense that we don’t make LAWS, we just sort of do our best to approximate what’s happening. I also like to bring up the fact that to really understand an interaction, you MUST look at it from multiple points of view (like anything in life). Like the example above, momentum gives a skewed picture when considering such vastly different scenarios as a slow moving heavy thing and a fast moving tiny thing.

I’ve done it by posing a problem like I described above then breaking up into a socratic seminar or philosophical chairs (or similar). Some times a discussion breaks out immediately and we just go with it until the point when some student demands we test it. If they don’t go that way, I generally prod them to give me ‘simple, organic proof’ (“Forget the physics, give me observational proof – real life proof that what you’re saying is correct”). We then hash out the appropriate method for testing and have at it.

Some of the more advanced students will appreciate a conversation about the fact that KE is just a velocity integral of momentum, Force is just a time derivative of momentum, and of course work is a position integral of force. These days I’m teaching Physics First, so most of my kids don’t benefit from the detailed math, but still benefit from the discussion. We just replace the words ‘time derivative’ and ‘velocity integral’ with simpler terms like ‘over the course of time’ or ‘taking velocity more into consideration’.

1. Do they come up with the relationship about how KE = (p^2)(2m)? That rarely comes up in my classes, but when it does, they think it’s a conspiracy or something.