Work done by friction on a skier and resulting distance the skier travels

  1. 1. The problem statement, all variables and given/known data
    A skier starts from rest on a 20m high, 20° slope. μk=0.210
    Find the horizontal distance traveled by the skier.
    From this, for the equations below we know that:
    yf = 0
    vi = 0
    vf = 0

    2. Relevant equations
    Wnet = Wnc + Wg = ΔKE
    Wnet = -fkd
    Wnc = ΔKE + ΔPE
    Wnc = ΔKE + mg(yf - yi)
    KE = 1/2mv2
    PE = mgy
    fk = μkmg

    3. The attempt at a solution
    So I went with the work of a non-conservative force
    Wnc = (KEf - KEi) + (PEf - PEi)
    Wnc = (1/2mvf2 - 1/2mvi2) + (mgyf - mgyi)
    From given, I eliminated all 0 quantities, leaving me with
    Wnc = -mgyi
    Then plugged in Wnet = -fkd = -μkmgd so,
    kmgd = -mgyi
    eliminated like terms (m, g):
    kd = -yi
    and solved for d
    d = yik
    and plugged in the knowns
    d = 20m/0.210
    d = 95.2m
    I realize this is the distance traveled from the top of the hill to the end of motion, but all they want is the horizontal distance, so now I have to solve for the distance traveled from the top of the hill to the bottom of the hill. The only thing I think that changes between the above work and the distance from the top to bottom is the final velocity which will be nonzero.
    So my question is, how do I find the distance traveled from the top of the hill to the bottom?
    Or am I going about this wrong? Is there a more direct way to find just the horizontal distance traveled?

    Oh I should add, the answer given by the book is 40.3m
     
  2. jcsd
  3. Doc Al

    Staff: Mentor

    Is this the full statement of the problem?

    I'm guessing that after the skier gets to the bottom of the slope, she skis over a horizontal stretch of ground? Perhaps it's that horizontal distance that they want.
     
  4. AH! I figured it out...

    I had to find the length of the bottom of the triangle formed by the horizontal and the slope of the hill. sin20° = 20/x (where x is the hypotenuse, or the length of the slope of the hill)
    from that I got x ≈ 58.5m
    Then,
    cos20° = x/58.5m (where x is the length of the bottom of the triangle)
    x ≈ 54.9m
    then subtract that from the total length traveled by the skier, 95.2m - 54.9m = 40.3m which is the answer given by the book.
     
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