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Homework Help: Distance traveled by a block due to unbalanced force

  1. Oct 5, 2014 #1
    1. The problem statement, all variables and given/known data
    A 60 N block in horizontal plane is pushed horizontally by force P = t^3 + 10t. The frictional resistance f = 6 - t^2. What is the distance traveled by the block after 2 seconds.

    2. Relevant equations
    The net force F = (t^3 + 10t) - (6 - t^2) = t^3 + t^2 + 10t - 6.

    3. The attempt at a solution
    There are actually three questions in this problem, acceleration, velocity and distance, all after 2 seconds. I encountered no problem with the acceleration which is from F = Ma
    a = (g/W) (t^3 + t^2 + 10t - 6) and after 2 seconds, a = 2.398 m/s

    Also no problem with the velocity by taking dv / dt = a, it is straightforward to integrate dv = a dt from known acceleration equation above. The equation of velocity is v = (g/W) [ (t^4 / 4) + (t^3 / 3) + 5t^2 - 6t ].

    My question is the distance, with equation ds = v dt.

    I have difficulty deciding which one to take for the initial time. Since at t = 0, the frictional resistance f is greater than the horizontal force P. I initially took t from zero to 2, but found that motion will start to impend not until 0.552 590 6474 seconds. Integration to distance equation will yield 0.7 for limits from 0 to 2 and 0.8 if limits is from 0.552 590 6474 seconds to 2 seconds. So which one is the right answer, I need your opinion. Thanks.
  2. jcsd
  3. Oct 5, 2014 #2


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    Hello Romel, welcome to PF.
    Please note that 'relevant equations' means something different than what you present there: more like Fnet = Fext - Ffriction, etcetera. But you do mention a few under 3.

    I believe your ##F = ma## at t=2 (**). (As a physicist, I resent the author's wording 'A 60 N block' because that doesn't tell me the mass, but never mind).

    I would be surprised if your integration to get v is correct. After all, the thing does not move for a while, and you integrate from 0 to 2.

    You yourself only become suspicious when working on the displacement. Better late than never, I would say. But you should realize that if s is 0 for a while, then v = 0 for that while too!

    And I just love the 0.552 590 6474 ! You sure it's not 0.552 590 6473 ? :) What I mean is that you can't present so many digits (the 60, the 2, the coefficients in P and Ffriction are all single-digit). Best is here to give it a name like t1, work with that (if you're lucky it might even cancel at some later moment -- not in this exercise) and fill in a value only at the very last step.

    (**) But - just to be sure - can you show me how (28 - 2)(g/W) gives 2.4 ?

    And I also get quite different v and s ! Are you working in some strange system of units ?

    [edit]After a lot of work on my side, I see the mixup. Do yourself a favor and check what you are doing. If that's done and OK, check what you are posting !
    Last edited: Oct 5, 2014
  4. Oct 5, 2014 #3


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    There is another discussion lurking here: the exercise clearly states that 'The frictional resistance f = 6 - t^2'

    This is very unsatisfactory. It would be dead wrong if it would say 'The frictional force f = 6 - t^2'

    If I am lenient to the author, I expect he has written somewhere in the chapter something along the line that frictional resistance is the maximum friction force magnitude. But he (/she?) could have been less devious by writing ##|f_{max}|## instead of f.

    (It is dead wrong in another respect: there are no units)
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