Vertical Projectile: Solved, Energy, Max Height

[cscott]

[SOLVED] Vertical projectile

Homework Statement

Is it true that a projectile launched vertically with initial velocity $v_0$ has total energy $$E = \frac{1}{2}m{v_0}^2 - G\frac{Mm}{R}$$ where R is the radius of the earth?

And to find the maximum height I set the total energy equal to $$-G\frac{Mm}{x+R}$$ where x is height, and solve for x?

 

[Gokul43201]

Yes, and yes...but do you understand why?

 

[cscott]

I think I do. I was just concerned because when I work it out for x I don't get a meaningful answer so I guess I messed up my algebra.

But when we approximate PE with mgx we get no PE term for total energy. This is because in that case our reference point is the surface of the Earth and not infinity, correct?

 

[cscott]

Is it correct that max height = $$x = \frac{-R^2v^2}{Rv^2-2GM}$$?

 

[cscott]

I get the correct expression for escape velocity from the above so I think it's OK.

 

[Gokul43201]

Is it correct that max height = $$x = \frac{-R^2v^2}{Rv^2-2GM}$$?

Yes.