Solving for Planet Islander's Radius from a Projectile Launch

[raisatantuico]

Homework Statement

a projectile of mass m is launched vertically from the surface of Planet Islander at a speed that is one-third the escape speed from the surface. If the projectile reaches a maximum height that is a distance h from the surface of the planet, what is the radius of planet islander?

Homework Equations

Escape Speed = square root of G*mass/radius^2

The Attempt at a Solution

Is this the correct equation:

1/3 (G*mass/r + h) = G*mass/r

so the radius is equal to 9h? or is it 3h? or h/9?

I am confused as what to what equations to equate? and where the height comes in?

Please help!

 

[Filip Larsen]

Your formula for the escape speed and none of the solutions you suggest are correct. You need that, and a consideration of the potential and kinetic energies involved at two interesting points of the projectiles flight in order to solve this problem.

 

[raisatantuico]

is energy conserved in this problem? so we can equate energy at the peak and energy just before it is launched?

 

[Filip Larsen]

Yes, mechanical energy is conserved if you make the assumption that there are no atmosphere.