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Work-Energy: Determining x- and y-components of wind force

  1. Feb 28, 2014 #1
    1. The problem statement, all variables and given/known data

    2m34fq0.png

    2. Relevant equations
    Constant acceleration equations: Δx = v,xi *t + 0.5*(a,x)*t^2
    Δy = v,yi *t + 0.5*(a,y)*t^2
    K = 0.5*m*v^2
    U = mgh
    W,nonconservative forces = ∫ ∑ F dx
    Conservation of Energy: K1 + U1 = K2 + U2
    ∑F = ma

    3. The attempt at a solution (Note that in my notation, I denoted a subscript with a comma preceding it)
    F,w = wind force
    F,wx and F,wy are constant
    ∑F,x = -F,wx = m*a,x
    ∑F,x = -F,wx - mg = m*a,y

    Δx = v,xi *t + 0.5*(a,x)*t^2 --> x,f = d1*cos(θ)
    v,xi = v,a *cos(θ,1) = 27.8 m/s = 83.6*v,xi

    Solve for a,x --> a,x = -0.199 m/s^2
    -F,wx = m*a,x (Solve for F,wx)

    Δy = v,yi *t + 0.5*(a,y)*t^2 (Solve for y,f)

    Now I am stuck! Any help would be greatly appreciated…thank you in advance! :)
     
    Last edited: Feb 28, 2014
  2. jcsd
  3. Mar 1, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    You know everything apart from ay, where is the problem?

    To write subscripts, you can use [noparse]ay[/noparse] -> ay. Alternatively, you can use LaTeX.
     
  4. Mar 2, 2014 #3
    Is this work correct?? Thank you in advance!

    x=97.9cos(32)=83.02 m
    y=97.9sin(32)=51.88 m
    Also resolve Va into its x and y components:
    Vx=28cos(7)=27.79 m/s
    Vy=28sin(7)=3.412 m/s

    Motion in x direction
    Let Ax be acceleration in x direction
    x=(1/2)(Ax)(t)^2 + (Vx)(t)
    83.02=(1/2)Ax(3.02)^2 + (27.79)(3.02)
    -.906=(1/2)Ax(3.02)^2
    -.20=Ax
    F=ma
    Wind force in x direction = 80(-.20) = -16 Newtons

    Motion in y direction
    Let Ay be acceleration in y direction (positive Ay is acceleration down)
    y=(1/2)(Ay)(t)^2 + (Vy)(t)
    51.88=(1/2)Ay(3.02)^2 + 3.412(3.02)
    41.58=(1/2)Ay(3.02)^2
    9.12=Ay
    Acceleration due to gravity is 9.8
    9.8-9.12 = .68 (acceleration up)
    Wind force in y direction = 80(.68) = 54 Newtons (blowing up)

    (Energy the skier has when he lands)
    Vfx will be his landing velocity in the x direction.
    Vfy will be his landing velocity in the y direction.

    Vfx = (Ax)(t) + Vx = (-.2)(3.02) + 27.79 = 27.2 m/s
    Vfy = (Ay)(t) + Vy = (9.12)(3.02) + 3.412 = 31.0 m/s

    K = kinetic energy when landing = (1/2)mv^2 = (1/2)(80)(27.2^2+31.0^2)
    K = 68,000 J

    N = Normal force when landing due to skiers mass = mgcos(32)
    f = friction force = (2.5)(80)(9.8)cos(32) = 1660 Newtons

    Force due to horizontal component of wind = 16cos(32) = 14 Newtons
    Force due to vertical component of wind = 54sin(32) = 29 Newtons

    Total force on skier = 1660+14+29 = 1700 Newtons

    (Work-energy theorem)
    Fz = K + (mgz)sin(32) (The last term is because the skier will continue to acquire energy from the force of gravity as he descends down the slope)
    1700*(d2) = 68000 + (80)*(9.8)*(d2)*sin(32)
    1700*(d2) = 68000 + 415*(d2)
    1285*(d2) = 68000
    d2 = 53 meters
     
  5. Mar 2, 2014 #4

    mfb

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    I think you are supposed to ignore the wind between B and C (otherwise you would get a different normal force, too).

    The numbers don't fit, and you cannot add horizontal and vertical forces like that.

    There are a lot of units missing. I did not check the numbers (that's something a computer can do), but the approach is good.
     
  6. Mar 2, 2014 #5
    The numbers are good, I just checked them! Looking at the prompt again, it says to ignore the b component of the wind force (the component perpendicular to the ramp). So would the sum of the nonconservative forces be (Fw)a + f (the a component of the wind force and the friction force)? (Essentially, the sum of the forces in the a direction--the direction parallel to the ramp)?
     
  7. Mar 2, 2014 #6
    Also, is this a correct conversion from x-y to a-b coordinates, given the two relative coordinate systems?

    169npg2.jpg
     
  8. Mar 2, 2014 #7

    mfb

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    Staff: Mentor

    Ah, right.
    All your forces are conservative here.
    Wind just adds another force to consider, noting magical.

    The conversion looks fine.
     
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