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Shell fragments exploding - NSL for system of particles

  1. Oct 29, 2008 #1
    A shell is shot with initial velocity [tex]\vec{}v_{0}[/tex] is 20ms[tex]^{-1}[/tex] at an angle [tex]\vartheta[/tex] = 60 degrees. At the top of the trajectory, the shell explodes into two fragments of equal mass. Fragment1 has a speed of 0 immediately after the explosion and falls vertically. How far from the gun does fragment2 land assuming terrain is level and there is no air drag?

    Equations used:
    F = ma
    v[tex]^{2}[/tex] = v(of origin)[tex]^{2}[/tex] a + 2a(y-y[tex]_{o}[/tex])

    p = mv

    Initial middle
    p=20ms[tex]-{1}[/tex]m p = [tex]\frac{1}{2}[/tex]mv + [tex]\frac{1}{2}[/tex]m x 0
    => 20m => [tex]\frac{1}{2}[/tex]mv + [tex]\frac{1}{2}[/tex]mx0

    => 20m = [tex]\frac{1}{2}[/tex]mv
    => 20 = [tex]\frac{0.5mv}{m}[/tex] = [tex]\frac{1}{2}[/tex]v = 40ms[tex]-{1}[/tex]

    so the middle position is p = 40ms[tex]-{1}[/tex]

    breaking this down to x y components:
    for x:
    cos[tex]\vartheta[/tex] = [tex]\frac{x}{20}[/tex]
    cos [tex]\vartheta[/tex] x 20 = x
    [tex]\frac{1}{2}[/tex] x 20 = x
    x = 10

    for y:
    sin[tex]\vartheta[/tex] = [tex]\frac{y}{20}[/tex]
    sin[tex]\vartheta[/tex] x 20 = y
    [tex]\sqrt{\frac{3}{2}}[/tex] x 20 = y
    [tex]\frac{20 x [tex]\sqrt{3}[/tex]}{2}[/tex] = 10 x [tex]\sqrt{3}[/tex]
    y = 10 x [tex]\sqrt{3}[/tex]

    using v[tex]^{2}[/tex] = v[tex]_{o}[/tex][tex]^{2}[/tex] a + 2a(y-y[tex]_{o}[/tex])
    0 = (10 x [tex]\sqrt{3}[/tex])[tex]^{2}[/tex] + 2 x 9.8ms[tex]-{1}[/tex](y - 0)
    0 = 100 x 3 + 2 x 9.8 x y
    y = [tex]\frac{300}{19.6}[/tex] = 15.3

    fragment1 dropped from a height of 15.3 meters

    I am lost as to what equation to use to calculate the distance fragment 2 traveled from here. I know its simple but somehow I am not able to think of what to do next...... I need the distance in the x direction that fragement2 traveled.


  2. jcsd
  3. Oct 29, 2008 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    It looks like you calculated the momentum for the fragment based on the initial velocity. This is the wrong approach. What you need to do is:

    1.) Calculate the velocity of the shell at the top of its arc
    2.) Calculate the momentum of the shell at the top of its arc
    3.) Calculate the momentum of the second fragment based on conservation of momentum
    4.) Calculate fragment 2's velocity based on its momentum
    5.) Calculate how far it will go before it hits the ground
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