# Escape speed and distance

1. Apr 26, 2010

### popo902

1. The problem statement, all variables and given/known data
A projectile is shot directly away from Earth's surface. Neglect the rotation of the Earth. What multiple of Earth's radius RE gives the radial distance (from the Earth's center) the projectile reaches if (a) its initial speed is 0.225 of the escape speed from Earth and (b) its initial kinetic energy is 0.225 of the kinetic energy required to escape Earth? (Give your answers as unitless numbers.)

2. Relevant equations

3. The attempt at a solution
ok so i have no idea what to do now
i found the escape speed of earth
but what now?
If i had to guess i'd multiply.225 to my KE and the Escape V

so i'd end up with this energy equation
-GMm/R + 1/2m(.225)v^2 = 0
then cancel small m
so

-GM/R + 1/2(.225)V^2 = 0
then solve for...R?
is that right? or am i not on track?

2. Apr 26, 2010

### zachzach

So $$E_i = \frac{-GMm}{R} + \frac{1}{2}mv_{0}^2$$

$$v_{esc} = [\frac{2GM}{R}]^{1/2} \rightarrow v_0 = (0.225)[\frac{2GM}{R}]^{1/2}$$

$$E_i = \frac{-GMm}{R} + \frac{1}{2}m[\frac{2GM}{R}(0.051)]$$

When at its max height, V = 0 so

$$E_f = \frac{-GMm}{r}$$

Set equal and use algebra to solve for r.

Last edited: Apr 26, 2010