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Energy and force question

  1. Apr 23, 2015 #1
    1. The problem statement, all variables and given/known data

    An archer pulls a bowstring back 0.440 m, and the spring constant of the bow is 545 N/m. Find the amount of work the archer did to draw the bow.

    2. Relevant equations

    F = ma
    Ei = Ef

    3. The attempt at a solution

    It seems as though there are two ways of doing this: using Newton's laws or using conservation of energy.

    a) Newton's laws:

    W = Fd

    F = -kx = 240 N

    d = 0.440 m

    W = Fd = 106 J

    b) Conservation of energy

    Ei = Ef

    Wapply = Ufinal

    Wapply = (1/2)kx2 = 53 J



    As one can see, these are two different answers. In fact, the latter is exactly half of the former. What am I doing wrong? Is either answer even right?
     
  2. jcsd
  3. Apr 23, 2015 #2

    Doc Al

    User Avatar

    Staff: Mentor

    This is incorrect. This calculation assumes the force is constant, but it's not: it varies from 0 to the maximum value.

    This is correct.

    If you find the work correctly, by integrating ##\int F dx## since F is not constant, you'll get the same answer as you did in (b).

    (Since the force varies linearly, you can cheat a bit and use the average force times the distance. The average force is 1/2 of the max force, so that will also give you the correct answer.)
     
  4. Apr 23, 2015 #3
    Was just writing pretty much what Doc Al said. You'll also notice that in your first method, W = Fd = Fx = -kx^2 ...twice your second answer.
     
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