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Kinematics - projectile with acceleration that depends on V

  1. Jan 26, 2015 #1
    1. The problem statement, all variables and given/known data
    A ship is travelling across the sea when its engines are cut. The ship slows down by a = -kV2. The ship gradually slows down from 5 knots/h to 12 knots/h over 20min. Determine the distance traveled in nautical miles. One knot is 1 nautical mile per hour.


    2. Relevant equations
    Kinematics

    3. The attempt at a solution

    Vi = .2 [nmiles / min]
    Vf = .1167 [nmiles / min]

    nmiles= nautical miles

    a = dV/dt = -kV2

    (1/V2)dV = -k dt

    integrate over Vi to Vf and 0 to 20min.

    therefore k = .1786 [1/nmiles]

    units seem to add up if a = -kv^2 the units will become nmiles/min^2

    confident that k is correct. Not sure though.

    Now for V.

    a= v dv/dx = -kv2

    -kv2 = v dv/dx

    -kdx = (1/v) dv

    integrating from Xi to Xf and Vi to Vf

    -k(Xf - Xi) = ln(Vf - Vi)

    Natural log of a negative number, impossible.
    Where am I going wrong?
     
    Last edited: Jan 26, 2015
  2. jcsd
  3. Jan 26, 2015 #2
    Wait. Did I take the integral wrong, does it become

    -k(Xf - Xi) = ln(Vf/Vi)

    ? Is that the correct answer?
     
  4. Jan 26, 2015 #3

    Nathanael

    User Avatar
    Homework Helper

    Did you mean to say the ship slows from 12 knots/h to 7 knots/h?
    If so, then I agree with your value of k

    Looks good to me :) I get the same
     
  5. Jan 26, 2015 #4
    Yes indeed I did mean to say that.

    Thank you Sir!
     
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