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Homework Help: Missile Landing Radius Problem (Orbital mechanics)

  1. Aug 31, 2009 #1
    1. The problem statement, all variables and given/known data
    At time t=0, there is a missile at a height h directly above the perfectly round Earth moving tangential to the surface of Earth. What must be the maximum velocity of the missile at t=0 if it must land within radius r on Earth directly below its initial position by the time it strikes the ground?

    2. Relevant equations

    3. The attempt at a solution

    I have tried using some orbital mechanics equations by visualizing the trajectory of the missile as a parabola. At t=0, the object is at apoapsis and at final time (when it strikes the ground), it's at periapsis. I tried equations such as

    momentum = radius at apoapsis * velocity at apoapsis
    semi-latus rectum = momentum^2/gravitational constant

    in order to find the velocity at apoapsis but I have no idea what the radius at apoapsis should be. The apoapsis radius must be greater than h and less than h+R_earth but other than that, I have no clue on how to find it.
  2. jcsd
  3. Aug 31, 2009 #2
    If you replace g by [tex]\frac{GM}{r^2}[/tex], things become much easier.
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