# Missile Landing Radius Problem (Orbital mechanics)

1. Aug 31, 2009

### springBreeze

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. Aug 31, 2009

### kyiydnlm

If you replace g by $$\frac{GM}{r^2}$$, things become much easier.