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Kinetics Problem: Non-constant force (calculus)

  1. Dec 10, 2014 #1
    1. The problem statement, all variables and given/known data
    A rock with mass m slides with initial velocity v0 on a horizontal surface. A retarding force F that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock (F = -kv1/2) . a) Find expression for the velocity of the rock as a function of time. b) Find expression for the position of the rock as a function of time. c) In terms of m,k, and V0 , at what time will the rock come to rest? d) In terms of m ,k and V0, what is the distance of the rock from its starting point when it comes to rest?

    2. Relevant equation
    f=ma A= dv/dt

    3. The attempt at a solution

    I solved parts a and b and I'm fairly certain my answers are correct.

    F=−kv^(1/2)

    a=dv/dt= (-kv^(1/2))/m

    dv/(v^1/2)=(-kdt)/(m)

    Integrate both sides and solve for the constant C

    V= (k2 t2 )/ (4m2 ) - (ktv01/2)/m + v0

    Then I integrated again to find position as a function of time.

    The only way I can think to solve parts c and d is to solve V(t)=0 but this would require the quadratic equation and be very messy. Then, to solve part d, I would have to plug in answer to c into x(t) which is even messier. Is there another way? Am I missing something?

    Any help would be great. Thank you.
     
    Last edited: Dec 10, 2014
  2. jcsd
  3. Dec 10, 2014 #2

    haruspex

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    Not at all messy, it turns out.
    Slightly. After integrating dv/(v^1/2)=(-kdt)/(m), and determining the constant, substitute v = 0 without converting it to the quadratic form.
     
  4. Dec 10, 2014 #3
    I made a calculation error and didn't realize until now that solving v(t) with a quadratic actually comes out neatly.
     
  5. Dec 10, 2014 #4
    Thanks for your help
     
  6. Dec 10, 2014 #5

    haruspex

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    My pleasure.
     
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