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auk411
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Homework Statement
A projectile is shot directly away from Earth's surface. Neglect rotation of Earth. What multiple of Earth's radius R_e gives the radial distance a projectile reaches if its initial speed is 1/2 of the escape speed from Earth.
Homework Equations
Ki + Ui = Kf + Uf,
where K is the kinetic energy and u is the gravitational potential energy.
v0 =[tex]\sqrt{Gm/2R}[/tex] ... this is the initial velocity when its initial speed is 1/2 of the escape speed.
The Attempt at a Solution
We know that energy is conserved. Then what I do is:
[tex]\frac{1}{2}[/tex]mv02 - [tex]\frac{GMm}{R_e}[/tex] = 0
I plug in v0 =[tex]\sqrt{Gm/2R}[/tex] for v0 in the previous equation. I do not get the right answer. What is wrong with this equation?
Apparently, this is the right equation:
[tex]\frac{1}{2}[/tex]mv02 - [tex]\frac{GMm}{R_e}[/tex] = [tex]\frac{GMm}{r}[/tex], where r is just some radial distance from earth.
Now, I do not understand where [tex]\frac{GMm}{r}[/tex] comes from.
Also, why doesn't the [tex]\frac{1}{2}[/tex]mv02 - [tex]\frac{GMm}{R_e} =\frac{-GMm}{2r}[/tex], since that is what my textbook says is what the total mechanic energy is for satellites. Isn't this thing going to be become a satellite? So why couldn't I use the aforementioned equation?
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