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Plane landing with retarding force

  1. Jun 15, 2011 #1
    1. The problem statement, all variables and given/known data
    A light plane (mass = M) makes an emergency landing on a short runway. With its engine off, it lands at speed v0. A hook on the plane snags a cable attached to a sandbag (mass = m) and drags the sandbag along. The coefficient of friction between the sandbag and the runway is μ, and the plane's brakes give a retarding force of Fb. How far will the plane go before it stops?
    Data: v0 = 43.0 m/s; M = 839 kg; m = 97 kg; μ = 0.32; Fb = 1208 N.


    2. Relevant equations
    F=mdv/dt
    F=dp/dt=mdv/dt


    3. The attempt at a solution
    -Fb-μmg=(M+m)dv/dt
    Im pretty sure these are the correct forces, however, the solution to the diff eq does not yield a function (i.e. e^-(something) that goes to zero) that will give a stopping point. My thought is I am missing a part of the equation or have to incorporate momentum.
    Thanks.
     
  2. jcsd
  3. Jun 15, 2011 #2
    Notice that the left hand side if the equation you determined is constant:

    -Fb-μmg=(M+m)dv/dt

    You can now integrate this to find v(t) or use constant acceleration kinematic equations.
     
  4. Jun 16, 2011 #3
    That was just my guess at it, however, it cannot be correct because when you integrate for v, then again for x you get a parabolic function which does not go to zero, hence it does not stop, so it is wrong. Suggestions for fixing this?
     
  5. Jun 16, 2011 #4
    Plane making an emergency landing!

    1. The problem statement, all variables and given/known data
    A light plane (mass = M) makes an emergency landing on a short runway. With its engine off, it lands at speed v0. A hook on the plane snags a cable attached to a sandbag (mass = m) and drags the sandbag along. The coefficient of friction between the sandbag and the runway is μ, and the plane's brakes give a retarding force of Fb. How far will the plane go before it stops?
    Data: v0 = 43.0 m/s; M = 839 kg; m = 97 kg; μ = 0.32; Fb = 1208 N


    2. Relevant equations
    F=dp/dt


    3. The attempt at a solution
    not sure how to set this up I know the forces are:
    μmg=Ff
    (M+m)g=N
    Fb=retarding force
    and initial momentum = Mv0
    other than that I have no idea how to set up the differential equation to obtain V(t)
    suggestions?
     
  6. Jun 16, 2011 #5

    tiny-tim

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    Last edited by a moderator: Apr 26, 2017
  7. Jun 16, 2011 #6
    Re: Plane making an emergency landing!

    Ok, so then it would would be 1/2mv^2=Fb+μmgx?
    then I would solve for x(t) by using v^2=(dx/dt)^2
    is this correct?
     
  8. Jun 16, 2011 #7

    tiny-tim

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    yes, except you've missed out some brackets :redface:
    uhh? :confused: v is given

    get some sleep! :zzz:​
     
  9. Jun 16, 2011 #8
    OK, we have established it is constant acceleration and we need to find the distance before the plane stops. The solution to your equations already exist.

    Could we apply this constant acceleration kinematic equation:

    vf^2 = vo^2 + 2ad

    where,

    vf = final velocity = 0
    vo = initial velocity (given)
    a = accleration = -Fb - μmg / (M + m)
    d = distance

    solve for 'd' and note the sign of 'a' is negative.
     
  10. Jun 16, 2011 #9

    gneill

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    Re: Plane making an emergency landing!

    When the plane hooks the sandbag, it sounds like an inelastic collision to me. This occurs immediately before the dragging and braking... so, re-evaluate the remaining KE.
     
  11. Jun 16, 2011 #10
    Re: Plane making an emergency landing!

    Care to elaborate at all? Because the conservation approach is not working unless I am missing something?
     
  12. Jun 16, 2011 #11

    gneill

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    Re: Plane making an emergency landing!

    Plane has initial velocity. Grabs sandbag. That's an inelastic collision, so use conservation of momentum to work out the new velocity of the plane+sandbag. Plane+sandbag has kinetic energy. Go from there with conservation of energy and work-energy.
     
  13. Jun 16, 2011 #12
    Re: Plane making an emergency landing!

    Kraigandrews, if you solve it and get it right (eventually), could you post what answer it is? I'd like to know if I got it right :)

    @gneill: it's not really an inelastic collision, is it? Sounds like you're thinking of an inelastic collision as one where the objects stick to each other, which is true most of the time, but the real definition of an inelastic collision is where energy isn't conserved. And I'm pretty sure energy is conserved when the plane is just snagging a sandbag cable ... unlike a bullet slamming into a block of wood, or two cars crashing together, to quote other common conservation-of-momentum-but-not-energy problems / inelastic collision problems. Because, if energy is lost in the plane snagging the sandbag, where does the energy go?
     
    Last edited: Jun 16, 2011
  14. Jun 16, 2011 #13

    gneill

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    Re: Plane making an emergency landing!

    Plane grabs sandbag. Plane and sandbag move as one thereafter. That's an inelastic collision, no matter what color the plane is. Where the energy goes is for the crash inspectors to worry over. :smile:
     
  15. Jun 16, 2011 #14
    Re: Plane making an emergency landing!

    Ok, so I have tried this several ways and have not been able to get it using energy:

    .5M(v0^2)=(Fb+(μ(M+m)g))x solving for x
    also
    .5M(v0^2)=(Fb+(μ(m)g))x solving for x
    and some other variations of that.

    so I'm really not sure what to do from here.
     
  16. Jun 16, 2011 #15

    gneill

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    Re: Plane making an emergency landing!

    Question: How will you know when you've "got it"?
     
  17. Jun 16, 2011 #16
    Re: Plane making an emergency landing!

    It's homework set online, you enter the answer, tells you if its right or wrong.
     
  18. Jun 16, 2011 #17

    gneill

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    Re: Plane making an emergency landing!

    Did you give any thought to my suggestion that before the plane and sandbag start braking the velocity, that there is an inelastic collision to consider?
     
  19. Jun 16, 2011 #18
    Re: Plane making an emergency landing!

    Ok finally got, your approach was correct, I just was forgetting to neglect the mass of the plane when calculating the work done by friction. Thank you very much
     
  20. Jun 16, 2011 #19

    gneill

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    Re: Plane making an emergency landing!

    Happy to be of service :smile:
     
  21. Jun 16, 2011 #20

    berkeman

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    (Two threads merged. Please do not multiple-post here.)
     
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