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Are change in momentum and impulse the same thing?

February 6, 2011

I’ve been playing around with the 20 minute pulse check thing we’ve been doing at school and I’ve been playing around with various prompts, but still one of my favorites is “what question do you still have?”

When I first did this, we collected all the questions, and tried to summarize a general question to tweet. The thing that was interesting is that we didn’t answer any of the questions. Part of me thinks this is a good thing—I want students to start to see that they simply forumlating a question can be a huge positive step, and thatthey have the ability to answer their own questions.

But at the same time, some questions have been so good, or some students seem to be so confused that I have decided to on occasion, begin to answer their questions. So I have the kids write questions on notecards, and then write a short reply on the notecard and return it the next day.

One of the most frequent comments I hear from students is “but I don’t have a question” and this can come with either two flavors: 1. I think I’ve got it so well I have no questions, or 2. I’m so confused I don’t even know what to ask. Both of these are problematic to me, and I’ve been trying to get students to see that you should always have a question in mind to guide your thinking. If you don’t have a question, it’s a sure sign you don’t understand as well as you think you do, and if you don’t even know what to ask, its a sign to me that you need to build your confidence that it’s ok to ask those most basic questions that students are so afraid to share.

This week, as we are studing momentum and impulse, I got a number of variations on the following question:

Since impulse is the same thing as change in momentum, why do we need the name impulse? Why not just use change in momentum?

This is a very interesting question, since it shows some of my students’ conceptions of mathematics are still a bit naive (which is to be expected).

I’ve avoided using the symbol \vec{J} for momentum just to avoid adding another symbol to their brains. They all know how to show the equivalence of impulse and momentum using N2:

\vec{a}=\frac{\vec{F}_{net}}{m}\\\frac{\vec{\Delta v}}{\Delta t}=\frac{\vec{F}_{net}}{m}\\m\vec{\Delta v} =\vec{F}_{net}\Delta t\\m\vec{v}_f-m\vec{v}_i=\vec{F}_{net}\Delta t\\\vec{\Delta{p}}=\vec{F}_{net}\Delta t\\change\;in\;momentum = impulse

We’ve discussed before the various meanings of an equal sign in physics, and this time we came back to this idea and talked about how change in momentum and impulse are two different human defined quantities that turn out through experiment to be measurably the same, and that we can see this equality through N2, but that it isn’t any more correct to say they “are the same” than it would be to go around talking about the 2\pi r of a circle instead of circumference.

I need to find a way to assess whether my students are developing a deeper understanding of what an equation is trying to say as relationship. This is the heart of the modeling curriculum, and certainly all sorts of proportional reasoning questions help students to see that equations are really just very short, precise summaries of relationships between various quantities. We also spend a lot of time exploring when a particular model is valid; still I think when students see an equation, they find it very hard to resist the urge to think “what do I plug into this thing?” without asking “what is this thing trying to say about how the world works.”

14 Comments leave one →
  1. February 6, 2011 11:34 am

    I like the focus on the equals sign here. When two things are “equal” sometimes it means that two different things are found to be the same in nature and that equivalence tells us something really interesting about the universe.

    I really like the “six ideas that shaped physics” way of doing this. Essentially every interaction is an opportunity to swap momentum and impulse is just an accounting trick to keep track of a continuous exchange of momentum.

    Of course we could start calling the units poms like I suggested here: http://andyrundquist.blogspot.com/2010/12/momentum-units.html

    • February 6, 2011 11:51 am

      Andy,
      This is a great idea. We already do talk about the meaning of the equal sign, but I don’t stress that equality of different things can be a clue into the nature of the universe. I’ve seen Six ideas before, and will give that another look. And I love the idea of poms of momentum—right around now, units start to get pretty crazy with Newtons, Joules and kg m/s all looking almost exactly the same. You could even tell students that a pom is shorthand for Parcel of Momentum.

      And it occurs to me that this way of introducing students to a conservation law, in terms of poms of momentum being transferred via impulses might be the perfect way to set them up to see energy conservation in terms of energy transfer via work, heat, and radiation.

      • February 6, 2011 12:14 pm

        love the parcel of momentum (wish I’d thought of that). It’s so interesting how students view units. I’ve said before how awed a student was when I said g was 9.8 meters per second (pause) per second. Of course I like 22 mph per second.

  2. February 7, 2011 12:04 am

    The post recalls these kid-friendly phrases:
    “the same”
    “the same but not the same”

    Impulse and momentum change are “the same but not the same.” Numerically equal, but they are entirely different quantities. IIRC, M&I stresses this point.

    For a=delta v/delta t, they are “the same.” For a=Fnet/m, they are “the same but not the same.”

    I hope this makes sense.

    • February 7, 2011 10:24 am

      I’m not sure that the “six ideas” author would quite agree with the “same but not the same”. He stresses that force is really just a collection of momentum swaps to ease the accounting. I do get what you’re saying, though.

      • February 7, 2011 7:02 pm

        Andy,
        I used the pom idea today to help develop a final synthesis of momentum for my kids, and they loved it. They call pronounce name like you pronounce ‘pwn’, which is their favorite word in the universe. This coupled with velocity mass bar charts (I’ll explain these in a later post) is the perfect way to lay the ground work for conservation of energy.

        • February 7, 2011 9:32 pm

          very cool, Frank. I look forward to hearing about the bar charts.

  3. February 7, 2011 9:34 pm

    whoops, got lost in the thread. Meant to say “very cool, John, can’t wait to hear about the bar charts”

  4. April 14, 2011 1:29 pm

    Hi guys, I’m an entry level physics student and searched the internet to elaboration on the concept, “are impulse and momentum one in the same?” This thread was inspiring for two reasons, (1) you all seem passionate in teaching which is a deep breath of fresh air. Secondly, the discussion about what makes the two concepts differ, and focusing on the sums being equal in value but only value,really helped. I encourage discussions just like you all submitted here to help students like me gain better understanding. Nice work.

    • April 18, 2011 7:20 pm

      Thanks so much for the very kind words. Good luck with your physics studying. While you’ve probably found you can find a lot of good insights by googling things (you’ll also find a lot of useless garbage this way), you might also try your hand at posting questions at physics stackexchange, which is a great community of physics enthusiasts who are usually quite willing to answer questions.

  5. issie permalink
    March 1, 2012 7:39 am

    i am still confused about the mathematical signs for direction.are they always the same for impulse and change in momentum?or are they the opposite signs as in N3?

    • March 1, 2012 7:50 am

      To answer this, you need to think carefully about the object you are talking about. Let’s start with a single object experiencing some external force F_{net,\;ext}.

      By Newton’s second law, we know:
      \vec{a}=\frac{\vec{F}_{net,\;ext}}{m}

      and knowing \vec{a}=\frac{\Delta \vec{v}}{\Delta t} we can substitute this expression for \vec{a} and cross-multiply:

      m \Delta \vec{v}=\vec{F}_{net,\;ext}\Delta t

      The term on the left is the change in momentum of that object. The term on the right is the impulse on that object, exerted by an external force. So the impulse experienced by an object and the changein momentum of the same object are always in the same direction.

      When you have two objects colliding with each other, the forces they exert on each other are always the same size and opposite in direction, by Newton’s 3rd law. (F_{a \; on \;b}=-F_{b \; on \; a}). Combining this with the previous result tells us that for two objects colliding with each other will experience impulses that are opposite each other.

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