Putting a simple momentum equation into words

  1. Check the accuracy of my wording for these first two equations, then, if you could, try to explain the last equation for me.

    Ʃ F = dp/dt
    "The total net force acting on a particle is equal to the rate of change of momentum over time."

    ∫ Ʃ F dt = ∫ dp/dt dt
    "The sum of the total net force acting on a particle over a time interval is equal to the sum of the rate of change of momentum over a time interval."

    My physics book breaks up differentials like this very regularly so as to make it clear for integration. But my question is, is this step a meaningful equation, or is this simply mathematical notation?
    Ʃ F dt = dp

    Thanks!
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    hi tolove! :smile:
    yes, except …

    "total" and "net" mean the same, so you needn't use both

    personally, i'd say that "rate" means "over time", so i'd leave that out also
    "sum" is definitely wrong

    it's an integral (and there's no simple-english alternative to that word) :wink:

    and you need to use the word "same" … "over the same time interval"
    it's mathematically meaningful (it's a statement about mathematical objects called "differentials"),

    but it has no physical meaning, except as a limit or an approximation: it would never apply to an actual time interval (unless everything is constant) :smile:
     
  4. To correct the first one,
    Ʃ F = dp/dt
    "The net force acting on a particle is equal to the change of momentum over time."

    and for ∫ Ʃ F dt = ∫ dp/dt dt,

    "The integral of the total force acting on a particle over a certain time interval is equal to the integral of the rate of change of momentum over the same time interval."

    And if I wanted to remove the word "integral," I could put the Reimann sum notation into words.

    These don't make good bedtime stories. Thank you for clearing up that third equation for me! That has been driving me nuts.
     
  5. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    hi tolove! :smile:
    oops! :redface:

    i should have been clearer … i meant leave out "over time" … you do need the words "rate of"!
    the integral of a rate of change is just the original function, so you could shorten that to:

    "The integral of the total force acting on a particle over a certain time interval is equal to the change of momentum over the same time interval."

    (and of course you can shorten "The integral of the total force" to "The impulse" :wink:)
    how? :confused:

    (without taking several paragraphs and making it really confusing :redface:)
     
  6. Yep! By making it long and confusing.
     
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