Projectile fired from earth. Conservation of energy

[auk411]

Homework Statement

A projectile is shot directly away from Earth's surface. Neglect the rotation of Earth.
What multiple of Earth's radius R_e gives the radial distance a projectile reaches if its (a) initial speed is .5 the escape speed from earth, (b) its initial kinetic energy is .5 of the kinetic energy required to escape Earth? (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?

Homework Equations

v =$$.5\sqrt{2GM/R}$$

The Attempt at a Solution

Total mechanical energy = 1/2 mv^2 - GMm/R =?

I know that I'm going to need to plug in the escape velocity for v^2. However, that is all I know.
(As for b and c, I don't even know what the relevant equations are. )

 

[Bhumble]

I'm not sure why it states mechanical energy in part of the problem but I'm assuming this is kinetic energy.
$$K = \frac{1}{2} mv^2$$
$$v = \sqrt{\frac{2GM}{R}}$$

a) $$v = 0.5\sqrt{\frac{2GM}{R}}$$ Solve for R
b) $$\frac{1}{2}mv^2 = \frac{1}{2}\frac{GMm}{R}$$ Solve for R
c) Start with your given formula for v (without the 0.5 that you added) and get to $$\frac{1}{2}mv^2$$ on the LHS

 

[auk411]

So does anyone know how to answer this cause the person before doesn't know how to answer it.

For one thing, part a is just wrong. That is not true equation.

... Unless I'm just totally confused.

 

[Bhumble]

You're correct that part a) is wrong. I should have had the 0.5 on the RHS of the equation. And the same for part b). I'll edit my original post to reflect this.