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oddjobmj
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Homework Statement
A projectile of mass m is fired from the surface of the Earth at an angle α from the vertical, where α = 0.46 radians. The initial speed v0 is equal to √{GM/Rearth}. Calculate the maximum height that the projectile will reach. Neglect air resistance and the Earth's rotation. Express the result as the ratio to the radius of the Earth. [Hint: Apply the conservation laws to the orbit of the projectile.]
Homework Equations
E=[itex]\frac{1}{2}[/itex]m[itex]\dot{r}[/itex]2+Ueff(r)
Ueff(r)=[itex]\frac{L^2}{2mr^2}[/itex]+U(r)
The Attempt at a Solution
Intuitively speaking the mass will go into orbit and because we're not necessarily near the surface of the Earth the potential is not mgh. Also, the orbit must be bound or the answer would be infinity. If the mass is in orbit I could compare the effective potential of two points in the orbit but I don't see how to make a useful comparison with the information I am provided. What would be the variable of comparison?
Should I be comparing the initial kinetic energy to that of a mass in orbit? I can't seem to figure out how to cancel out v02 if that is the case:
[itex]\frac{1}{2}[/itex]mv02=[itex]\frac{1}{2}[/itex]m[itex]\dot{r}[/itex]2+Ueff(r)
Any suggestions? Thank you for your help.