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Flight of charged projectiles

  1. May 16, 2012 #1
    I am an undergraduate physics student and found myself thinking about a scenario where two positively charged projectiles are fired at an angle to the horizontal at a velocity with a certain separation. So they will be subject to a gravitational force and perpendicular to this, an electrostatic force.
    The distance dependent acceleration is something I have not covered before and am unsure as to how to calculate where the projectiles will land.

    Lets give the projectiles a mass of 1kg and a charge of +1mC. Say they are fired at 45degrees to the horizontal with a velocity of 40m/s. They are fired in the same direction at the same time but separated by 1m. If we say this happens on a co-ordinate system, we can say they are fired from +/- 0.5 x, and parallel to the y axis.

    I have worked out that the time of flight is 5.77s and that the y co-ordinates of the impact points are +/- 160.2m.

    How can I set up the calculus to solve for x?
     
  2. jcsd
  3. May 16, 2012 #2

    haruspex

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    The gravitational force is vertical. By symmetry, the electrostatic force is always horizontal, in the x direction. If you're ignoring air resistance, there's no interaction between these. So you can treat the particles as repelling, and moving, purely in the x direction for the known flight time.
    Again by symmetry, the repulsion on each ~ 1/x^2:
    x'' ~ x^-2
    Can you solve that?
     
  4. May 16, 2012 #3
    Yes but neither the acceleration or the separation are constant so I am a bit confused.
     
  5. May 16, 2012 #4

    haruspex

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    Quite so. The acceleration (x direction) is as I indicated, is inversely proportional to the square of the separation (x direction). Can you solve that differential equation?
     
  6. May 17, 2012 #5
    Sorry, yes you did show that.
    Ok so I solve and get x=-c.ln(x)+x0.
    Again I am unsure how to solve this.
     
  7. May 17, 2012 #6

    haruspex

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